On Complex Dimensions and Heat Content of Self-Similar Fractals

Abstract

Complex fractal dimensions, defined as poles of appropriate fractal zeta functions, describe the geometric oscillations in fractal sets. In this work, we show that the same possible complex dimensions in the geometric setting also govern the asymptotics of the heat content on self-similar fractals. We consider the Dirichlet problem for the heat equation on bounded open regions whose boundaries are self-similar fractals. The class of self-similar domains we consider allows for non-disjoint overlap of the self-similar copies, provided some control over the separation. The possible complex dimensions, determined strictly by the similitudes that define the self-similar domain, control the scaling exponents of the asymptotic expansion for the heat content. We illustrate our method in the case of generalized von Koch snowflakes and, in particular, extend known results for these fractals with arithmetic scaling ratios to the generic (in the topological sense), non-arithmetic setting.

1. Introduction

In this work, we study the heat content of regions with self-similar, fractal boundaries. Given a region $\Omega \subset {\mathbb{R}}^N$, here an open bounded set, its boundary $\partial \Omega$ is said to be self-similar if it is the invariant set of an iterated function system consisting only of (necessarily contractive) similitudes of Euclidean space. On such regions, we consider the Dirichlet problem for the heat equation in the sense of Perron–Wiener–Brelot with boundary conditions chosen as a model for heat flow into the region. The heat content is the integral of this solution over the region as a function of the time parameter and we will concern ourselves with the asymptotic expansion of this quantity.

The main results of this work are to provide explicit formulae for the heat content of regions with self-similar fractal boundary in terms of the poles of associated scaling zeta functions and the residues of a heat zeta function, up to an order determined by estimates of the remainder in an initial decomposition. These formulae take the form of a sum over the possible complex dimensions of the boundary with coefficients determined by an associated heat zeta function. We prove these results in the context of solutions to scaling functional equations, for which we state and prove sufficient conditions for requisite asymptotic estimates needed to explicitly compute inverse Mellin transforms in terms of limits of sums of residues of the integrand at poles of a meromorphic function.

1.1. Background

The von Koch snowflake (the leftmost shape in Figure 1) is a self-similar fractal based on von Koch’s planar curve with nowhere defined tangents [1,2]. The heat content of this fractal was analyzed using the renewal theorem in the work of Fleckinger, Levitin, and Vassiliev [3,4]; see also [5]. Following these developments, van den Berg and collaborators analyzed the heat content for generalized von Koch fractals [6,7,8,9]; see also the geometric analysis of these fractals by Paquette and Keleti [10]. Feller established the renewal theorem in the field of probability [11], and it has found many applications in the study of fractals, such as the work of Strichartz on self-similar measures [12,13,14] and the work of Kigami and the second author on the Weyl problem for Laplacians on self-similar sets [15]. The heat content of self-similar fractals has also been analyzed recently in the context of Brownian motion [16] (also by means of Feller’s renewal theorem) and in connection with fractal curvature measures [17].

Tube functions, connected to the theory of complex dimensions as established by the second author and collaborators (see, e.g., [18,19]), were analyzed for the von Koch snowflake by the second author and Pearse [20]. Recently, the first author extended this analysis to self-similar fractals, including these generalized von Koch fractals, and determined their possible complex dimensions using scaling zeta functions and scaling functional equations [21]. Scaling zeta functions appear in an earlier form in the work of the second author with Pearse and Winter [20,22,23]; see also [18,19,24]. The framework of scaling functional equations is essentially a multiplicative analogue of Feller’s renewal theorem. Furthermore, the notion of a (scaling) functional equation was employed by Deniz, Koçak, Özdemir, and Üreyen in [25] to provide a new proof of tube formulae for self-similar fractal sprays (originally established in [22,23]).

In this work, we extend the study of scaling functional equations and, in particular, establish the precise analysis of admissible remainder terms (Section 3.2) under which the methods in [18,19] can be used to deduce explicit formulae for solutions to these scaling functional equations. We apply our results to the study of the short-time asymptotics of the heat content of regions whose (fractal) boundaries are the invariant set of a self-similar system. In particular, we provide explicit formulae for asymptotics of the heat content of general self-similar fractals, provided there are some separation conditions on the underlying self-similar system, in terms of estimates of the remainder in the self-similar decomposition of the heat content. In the application of our results to generalized von Koch fractals, we recover the known leading order results in the lattice case [6,7] and, furthermore, our formulae extend to the nonlattice case. See Section 2.2.2 for the definition and further discussion of this dichotomy. Moreover, our results demonstrate that the change in behavior between these cases can be precisely explained by the structure of the possible complex dimensions (viz. the poles of the associated scaling zeta function) between these two cases.

1.2. Main Results

Let $\Omega \subset {\mathbb{R}}^N$ be a bounded open set and suppose that its boundary $\partial \Omega$ is the invariant set of a self-similar system $\Phi$, that is, a finite set of nontrivial contractive similitudes on ${\mathbb{R}}^N$. Suppose, further, that $(\partial \Omega ,\Omega )$ is an osculant fractal drum ([21], Definition 5.2), which is to say that for some self-similar system $\Phi$, $\Omega$ is given by a partition of the form

$$\Omega =\left(\underset{\phi \in \Phi }{\uplus }\phi [\Omega ]\right)\uplus {\Omega }_R,$$

where ⊎ denotes a disjoint union, ${\Omega }_R:=\Omega \setminus \left({\cup }_{\phi \in \Phi }\phi [\Omega ]\right)$, and with the additional property that if $y\in \phi [\Omega ]$, then $d(y,\partial \Omega )=d(y,\phi [\partial \Omega \left]\right)$. In other words, $\Omega$ is given by a union of similar copies, up to a residual set ${\Omega }_R$, with the property that the closest point in the boundary to the image of a point in $\Omega$ lies in the corresponding image of the boundary under the same mapping. (Note that as the invariant set of the self-similar system $\Phi$, by definition, $\partial \Omega$ is exactly the union of these images.) See also Definition 3 in Section 2.2.3.

We consider the following Dirichlet problem for the heat equation on $\Omega$, considered in the sense of Perron–Wiener–Brelot (PWB) in order to analyze an arbitrary open subset of ${\mathbb{R}}^N$. Namely,

$$\left\{\begin{array}{cccc}\hfill {\partial }_{t}u-C\Delta u& =0\hfill & \hfill & \mathrm{in}\Omega \times [0,\infty ),\hfill \\ \hfill u& =0\hfill & \hfill & \mathrm{on}\Omega \times \left\{0\right\},\hfill \\ \hfill u& =1\hfill & \hfill & \mathrm{on}\partial \Omega \times (0,\infty ),\hfill \end{array}\right.$$

(1)

where $C>0$. By $E_{\Omega }$, we denote the total heat content of $\Omega$, that is, $E_{\Omega }\left(t\right):={\int }_{\Omega }{u}_{\Omega }(x,t)dx,$ where $u_{\Omega }$ is the PWB solution to Problem 1 on $\Omega$. Under the hypothesis that $(\partial \Omega ,\Omega )$ is an osculant fractal drum, we assume that the heat content on $\Omega$ is approximated by the sum of the heat content of the analogous Dirichlet problems on each of the images of $\Omega$ under the mappings of the associated self-similar system, up to some decomposition remainder. Explicitly, we define

$$R_{\Omega }\left(t\right):=E_{\Omega }\left(t\right)-\sum _{\phi \in \Phi }{E}_{\phi [\Omega ]}\left(t\right),$$

and we suppose that $|R_{\Omega }\left(t\right)|=O\left(t^{(N-{\sigma }_0)/2}\right)$ as $t\to 0^{+}$, where ${\sigma }_0\in \mathbb{R}$ is the smallest such parameter for which such an estimate holds. See also Definition 8. We note that this hypothesis holds for the classic von Koch snowflake and, more generally, for lattice generalized von Koch snowflakes (with parameters chosen so that the boundary is topologically simple), as well as for many nonlattice generalized snowflakes. See Section 4.3 and especially Theorems 10 and 11 therein for the application of our results to such examples.

We state and prove two admissibility conditions for this decomposition remainder $R_{\Omega }\left(t\right)$, namely, Theorems 1 and 2 in Section 3.2, depending on $\Phi$ and ${\sigma }_0$ under which the following explicit formulae for $E_{\Omega }$ may be deduced. In the pointwise setting, the $k^{\mathrm{th}}$ antiderivative of the heat content, $E_{\Omega }^{\left[k\right]}\left(t\right)$, for $k\ge 2$, is shown to satisfy

$$E_{\Omega }^{\left[k\right]}\left(t\right)=\sum _{\omega \in {\mathcal{D}}_{\Phi }\left({\mathbb{H}}_{{\sigma }_0}\right)}Res\left({\displaystyle \frac{t^{(N-s)/2+k}}{{((N-s)/2+1)}_k}}{\widehat{\zeta }}_{\Omega }(s/2;\delta );\omega \right)+{\mathcal{R}}^k\left(t\right),$$

for all $t\in (0,\delta )$ for fixed $\delta >0$ and where the remainder term satisfies the estimate ${\mathcal{R}}^k\left(t\right)=O\left(t^{(N-{\sigma }_0)/2-\epsilon +k}\right)$ as $t\to 0^{+}$, for any given $\epsilon >0$. Here, ${\mathcal{D}}_{\Phi }\left({\mathbb{H}}_{{\sigma }_0}\right)$ denotes the set of poles of an associated scaling zeta function (given explicitly by ${\zeta }_{\Phi }\left(s\right)={(1-{\sum }_{\phi \in \Phi }{\lambda }_{\phi }^s)}^{-1}$, where ${\lambda }_{\phi }$ is the scaling ratio of $\phi$) in the open right half-plane ${\mathbb{H}}_{{\sigma }_0}:=\left\{s\in \mathbb{C}|\Re \left(s\right)>{\sigma }_0\right\}$. These are also the possible complex dimensions of the relative fractal drum $(\partial \Omega ,\Omega )$ [21]. The function ${\widehat{\zeta }}_{\Omega }(s/2;\delta )$ is a heat zeta function, given by Definition 11 in Section 4.1, and for which we prove an explicit formula in Theorem 7 (also in Section 4.1).

Furthermore, this identity is valid for any $k\in \mathbb{Z}$ when interpreted as a distributional identity, and, in particular, the heat content itself is given by

$$E_{\Omega }\left(t\right)=\sum _{\omega \in {\mathcal{D}}_{\Phi }\left({\mathbb{H}}_{{\sigma }_0}\right)}Res\left(t^{(N-s)/2}{\widehat{\zeta }}_{\Omega }(s/2;\delta );\omega \right)+{\mathcal{R}}^{\left[0\right]}\left(t\right)$$

as $t\to 0^{+}$. In the case that the poles $\omega$ of ${\zeta }_{\Phi }$ in the half-plane ${\mathbb{H}}_{{\sigma }_0}$ are simple, this expansion simplifies to an expression of the form

$$E_{\Omega }\left(t\right)=\sum _{\omega \in {\mathcal{D}}_{\Phi }\left({\mathbb{H}}_{{\sigma }_0}\right)}{a}_{\omega }{t}^{(N-\omega )/2}+{\mathcal{R}}^{\left[0\right]}\left(t\right),$$

as $t\to 0^{+}$, where ${\mathcal{D}}_{\Phi }\left({\mathbb{H}}_{{\sigma }_0}\right)$ denotes the set of possible complex dimensions belonging to the open right half-plane ${\mathbb{H}}_{{\sigma }_0}$ (i.e., with the real part strictly larger than ${\sigma }_0$) and, for each $\omega \in {\mathcal{D}}_{\Phi }\left({\mathbb{H}}_{{\sigma }_0}\right)$, the constant $a_{\omega }$ is determined by the residue of the heat zeta function at $\omega /2$. The pointwise result is the content of Theorem 8 in Section 4.2 and the distributional result (without specialization to the case of $k=0$) is the content of Theorem 9, also in Section 4.2. The formal details of these results, including the definition of these sums (defined as an appropriate symmetric limit) and the remainder terms, are discussed in Section 3.3 (in the general setting) and in Section 4.2 (within the context of the heat content).

These results are obtained through the study of solutions to scaling functional equations, which heat contents are shown to satisfy by virtue of their scaling properties (viz. Proposition 2 in Section 2.1.2 and Corollary 1 in Section 3.1.3) when there exists a decomposition induced by a self-similar system for the heat content with suitably estimated remainder (cf. Definition 8 in Section 3.1.3). In fact, they are stated and proved in the context of general solutions to scaling functional equations (viz. Theorems 3 and 4 in Section 3.3) with admissible remainders such that the Mellin transform of the remainder term and the scaling zeta function ${\zeta }_{\Phi }$ associated to the scaling functional equation are jointly languid (see Definition 9 in Section 3.2). The first admissibility criterion occurs when ${\sigma }_0$, from the remainder estimate, is strictly smaller than a lower bound for the poles of ${\zeta }_{\Phi }$, here called the lower similarity dimension of $\Phi$ (Definition 5 in Section 3.2.2). The second criterion is when the scaling ratios of $\Phi$ lie in the lattice case (Definition 1 in Section 2.2.2).

Lastly, we illustrate our results in Section 4.3 in the case of generalized $(n,r)$-von Koch fractals, such as the examples in Figure 1 in Section 1, the definition of which is recalled in Section 4.3.1. The admissibility criteria are shown to be satisfied when either $n\ge 5$ or in the lattice case (see Definition 1 in Section 2.2.2). In particular, the ordinary von Koch snowflake (the leftmost shape in Figure 1) falls in the lattice case. Furthermore, the poles of ${\zeta }_{{\Phi }_{n,r}}$ (and, thus, the complex dimensions of the generalized von Koch fractal) are simple, the heat content is given by a distributional expansion of the form

$$E_{\Omega }\left(t\right)=\sum _{\omega \in {\mathcal{D}}_{{\Phi }_{n,r}}\left({\mathbb{H}}_0\right)}{r}_{\omega }{t}^{(2-\omega )/2}+\mathcal{R}\left(t\right),$$

where $r_{\omega }$ is a constant determined by the residue of the heat zeta function; see Theorem 11 in Section 4.3. The pointwise versions of these results are given in Theorem 10, also in Section 4.3. The estimates for the decomposition remainder were established in [6] and our results here extend the explicit formulae in [6] to the nonlattice case (see Definition 1 in Section 2.2.2) when $n\ge 5$, extending the results of [9]. Moreover, we establish a precise connection between the spectral (here, regarding the heat content) complex dimensions and the underlying geometric complex dimensions.

1.3. Organization of the Work

This work is organized as follows. Section 2 contains the relevant background information for stating and proving the main results, including an overview of Perron–Wiener–Brelot solutions to the heat equation (the details of which are relegated to Appendix A and the references therein) and the geometry of self-similar fractals and their complex dimensions. An additional appendix regarding Mellin transforms and some relevant properties for the proofs of our results is contained in Appendix B while Appendix C contains the relevant information regarding growth estimates known as languidity. Section 3 contains the discussion of scaling functional equations as well as the theorems and proofs of our main results regarding their solutions in the general setting. The applications of these results to the heat content of regions with self-similar fractal boundaries are contained in Section 4, including the illustration thereof in the case of generalized von Koch fractal domains.

2. Preliminaries

2.1. The Heat Equation on Bounded Open Sets

Let $Y\subset {\mathbb{R}}^{N+1}$ be an arbitrary bounded open set. To consider the heat equation on an arbitrary open set, care must be taken to interpret the prescribed boundary conditions. In this work, we study a solution in the sense of Perron–Wiener–Brelot (PWB) per Definition A1, the details of which we include in Appendix A. We provide a brief discussion of PWB solutions and our choice of boundary conditions in Section 2.1.1. In Section 2.1.2, we recall the definition of heat content and discuss some of its relevant properties (stated and proved in the context of PWB solutions) necessary for deducing our results on heat content asymptotics in Section 4.

2.1.1. An Overview of Perron–Wiener–Brelot Solutions

In brief, a Perron–Wiener–Brelot (PWB) solution adapts Perron’s famous method for solving the Laplace equation [26] to the context of the heat equation. It is named, additionally, for the contributions of Wiener [27,28] and Brelot [29,30]. For a more thorough background discussion concerning the contributors to heat potential theory, see the introduction of Appendix A.

A PWB solution to the heat equation exists when the infimum of a class of functions called hypertemperatures (defined by an appropriate variational formulation of the heat equation instead expressed as an inequality, per Definition A1), which are bounded from below by the boundary conditions for appropriate boundary limits (see Definition A2), coincides with the supremum of a class of similarly defined hypotemperatures which are bounded from above by the boundary conditions for appropriate boundary limits. This coinciding function is called a PWB solution. The boundary conditions to a Dirichlet problem are called resolutive when a PWB solution exists (see Definition A1), which occurs if the boundary conditions are continuous or, more generally, when they are integrable with respect to a family of parabolic measures associated to points on the boundary (see Definition A4).

The advantage of the PWB approach, over a weak/variational formulation, is that it yields a solution which is pointwise-defined everywhere and continuously differentiable in the interior. Convergence to prescribed boundary values is a pointwise limit, though only on a subset of the boundary called the regular essential boundary (see Appendix A.2).

We restrict our attention to a cylindrical open set, the Cartesian product of a fixed spatial region $\Omega \subset {\mathbb{R}}^N$ (without loss of generality a connected open set) and $(0,\infty )$. The Dirichlet problem for the heat equation is then given by

$$\left\{\begin{array}{cccc}\hfill {\partial }_{t}u-C\Delta u& =0\hfill & \hfill & \mathrm{in}\Omega \times (0,\infty ),\hfill \\ \hfill u& =f\hfill & \hfill & \mathrm{on}\Omega \times \left\{0\right\},\hfill \\ \hfill u& =g\hfill & \hfill & \mathrm{on}\partial \Omega \times (0,\infty ).\hfill \end{array}\right.$$

(2)

Here, C is a positive constant called the diffusivity constant. In order for a solution u to exist in the PWB sense, the prescribed boundary values specified by the functions $f:\Omega \to \mathbb{R}$ and $g:\partial \Omega \to \mathbb{R}$ must be resolutive (per Definition A3). Letting $F:\partial (\Omega \times (0,\infty \left)\right)\to \mathbb{R}$ represent the shared boundary conditions, that is, with ${F|}_{t=0}\equiv f$ and ${F|}_{x\in \partial \Omega }\equiv g$, an equivalent characterization is that F is integrable with respect to each element of a family of parabolic measures relative to $\Omega \times (0,\infty )$ (per Definition A4).

In what follows, we will specialize to the following boundary conditions. Initially, we suppose that the PWB solution to the heat equation inside of $\Omega$ is zero. On the boundary of $\Omega$ for all time, we assume that the boundary $\partial \Omega$ is held at a constant temperature, which we suppose to be equal to one, as given by Problem 1 in Section 1.2. These boundary conditions model the flow of heat into $\Omega$ from its boundary, supposing that the edges are held at a constant temperature. These boundary conditions define a resolutive boundary function for any bounded open set, so that a PWB solution exists, and furthermore, they are scale invariant.

To see the former, let $Y=\Omega \times (0,\Omega )$ and define the function $F(x,t):={\mathbb{1}}_{\partial \Omega }\left(x\right)$, which is a measurable function for each of the parabolic measures ${\omega }_{Y,p}$ (Definition A4), $p\in Y$, since each measure is a Borel measure and $\partial \Omega$ is closed (whence its indicator function is measurable). Further, F is integrable since the measure of the (essential) boundary (see Appendix A.2) of Y is equal to one, i.e., ${\omega }_{Y,p}\left({\partial }_{e}Y\right)=1$, since they are probability measures. Thus, F is parabolically integrable and Problem 1 in Section 1.2 admits a PWB solution. We also have that a solution to Problem 1 is unique due to the results of Widder [31] since the solution is nonnegative (which follows from the minimum principle for the heat equation).

The scale invariance of these boundary conditions will be essential to establishing the relationship between Problem 1 and the analogous problem on the image $\phi [\Omega ]$ under a similitude $\phi$ of Euclidean space. By (parabolically) scale invariant, we mean that $F(x,t)=F(\phi \left(x\right),{\lambda }^{2}t)$ for any similitude $\phi$ of ${\mathbb{R}}^N$ whose scaling ratio is $\lambda >0$. That F is scale invariant follows immediately from it being piecewise constant (and the fact that ${\lambda }^2·0=0$). In general, the results of this work may be generalized to parabolically integrable, scale-invariant choices of boundary conditions.

2.1.2. Heat Content and Its Properties

Let $u_{\Omega }$ be the PWB solution to the Dirichlet Problem 2 in Section 2.1 on an open bounded set $\Omega \subset {\mathbb{R}}^N$ with resolutive boundary conditions (per Definition A3). The heat content $E_{\Omega }$ inside any measurable set $F\subseteq \Omega$ is defined as

$$E_{\Omega }(t;F):={\int }_F{u}_{\Omega }(x,t)dx.$$

When $F=\Omega$, we write $E_{\Omega }\left(t\right)=E_{\Omega }(t;\Omega )$ and call it the total heat content.

In the case of Problem 1 in Section 1.2, we note that the heat content $E_{\Omega }(t;F)$ is uniformly bounded for all $t\in [0,\infty )$ and any measurable set F. This follows as a corollary of the maximum principle for the heat equation, the boundedness of the region $\Omega$, and elementary properties of the Lebesgue measure. Since the boundary conditions of Problem 1 are bounded by one, we have the following explicit estimate.

Proposition 1

(Heat Content Boundedness). Let $u_{\Omega }$ be the PWB solution to Problem 1 in Section 1.2. Then, for any $t\in [0,\infty )$ and any measurable set $F\subseteq \Omega$, we have $|E_{\Omega }(t;F)|\le {\left|\Omega \right|}_N$.

Next, we deduce a scaling property for the heat content. This will require imposing (parabolically) scale-invariant boundary conditions, the simplest of which are constant boundary conditions.

Proposition 2

(Heat Content Scaling Property). Let $u_{\Omega }$ and $u_{\phi [\Omega ]}$ be PWB solutions to Problem 2 in Section 2.1.1 on a bounded open set Ω and $\phi [\Omega ]$ in ${\mathbb{R}}^N$, respectively, where φ is a similitude of ${\mathbb{R}}^N$ with scaling ratio $\lambda >0$. In particular, suppose that f and g are chosen so that the boundary conditions are resolutive (in the sense of Definition A3) and so that $u_{\Omega }$ is unique, e.g., if f and g are both nonnegative (in which case $u_{\Omega }\ge 0$, which is sufficient by [31]).

Suppose that the boundary functions f and g have the following scale invariance properties with respect to φ: for any $x\in \Omega$, f is scale invariant (i.e., $f\left(\phi \right(x),0)=f(x,0)$) and g is parabolic scale invariant (i.e., $g(\phi \left(x\right),{\lambda }^{2}t)=g(x,t)$). Then,

$$E_{\phi [\Omega ]}\left(t\right)={\lambda }^N{E}_{\Omega }(t/{\lambda }^2).$$

Proof. 

First, we have that for any $x\in \Omega$ and for all $t>0$,

$$u_{\phi \left[X\right]}(\phi \left(x\right),{\lambda }^{2}t)=u_{\Omega }(x,t).$$

(3)

To see this, we note that the function $v(x,t):=u_{\phi [\Omega ]}(\phi \left(x\right),{\lambda }^{2}t)$ satisfies the heat equation for any $x\in \Omega$ and for all $t>0$, since $({\partial }_t-\Delta )v={\lambda }^2({\partial }_t-\Delta )u_{\phi [\Omega ]}=0$ on the points $\left(\phi \right(x),t)\in \phi [\Omega ]\times (0,\infty )$.

Furthermore, v satisfies the same boundary conditions as the function $u_{\Omega }$. Let us use the notation in the discussion of Problem A3: let $u_{\Omega }=u_{Y,F}$ and $u_{\phi [\Omega ]}=u_{\phi \left[Y\right],F}$, where $Y=\Omega \times (0,\infty )$ and where F represents respective boundary conditions f and g when appropriately restricted. Let $p_n=(x_n,t_n)\to q=(y,s)$ be a sequence of points in Y approaching the boundary point q, and in the case of a semi-singular boundary point, we assume that $t\to s^{+}$. Let $p^{\prime }=(\phi \left(x_n\right),{\lambda }^{2}t)$ and $q^{\prime }=(\phi \left(y\right),{\lambda }^{2}t)$ denote the corresponding points under parabolic transformation.

By assumption, at any regular boundary point q (respectively, $q^{\prime }$), we have that

$$\begin{array}{ccc}\hfill u_{Y,F}\left(p_n\right)& \to F\left(q\right)\hfill & \hfill \mathrm{as}{p}_n\to q\in {\partial }_{e}Y,\\ \hfill u_{\phi \left[Y\right],F}\left(p_n^{\prime }\right)& \to F\left(q^{\prime }\right)\hfill & \hfill \mathrm{as}{p}_n\to q\in {\partial }_{e}Y.\end{array}$$

In other words, $u_{Y,F}$ (respectively, $u_{\phi \left[Y\right],F}$) converges on the regular essential boundary in the sense of Definition A2. We note that if q is a regular boundary point, then so too is $q^{\prime }$ for its corresponding boundary. This can be seen from the barrier criterion (see for instance ([32], Theorem 8.46)). Namely, if there is a barrier w defined near the point q, then the corresponding function $w(\phi \left(x\right),{\lambda }^{2}t)$ will be a barrier at $q^{\prime }$.

By definition, $v\left(p\right)=u_{\phi \left[Y\right],F}\left(p^{\prime }\right)$, so $v\left(p\right)\to F\left(q^{\prime }\right)$. Under the parabolic scale-invariance assumption, $F\left(q^{\prime }\right)=F\left(q\right)$. Thus, we conclude by uniqueness that $v(x,t)=u_{\Omega }(x,t)$, which is exactly (3). Using properties of the Lebesgue integral and (3), it follows that

$$\begin{array}{cc}\hfill E_{\phi [\Omega ]}\left(t\right)& ={\int }_{\Omega }{u}_{\phi [\Omega ]}(\phi \left(x\right),t)d\phi \left(x\right)\hfill \\ \hfill & ={\lambda }^N{\int }_{\Omega }{u}_{\phi [\Omega ]}(\phi \left(x\right),{\lambda }^2(t/{\lambda }^2))dx\hfill \\ \hfill & ={\lambda }^N{\int }_{\Omega }{u}_{\Omega }(x,t/{\lambda }^2)dx\hfill \\ \hfill & ={\lambda }^N{E}_{\Omega }(t/{\lambda }^2),\hfill \end{array}$$

as required. □

Note that an important part of this scaling property is that the heat content in a region is related to a rescaled problem’s heat content (cf. $E_{\lambda \Omega }\left(t\right)$), not a restricted heat content (cf. $E_{\Omega }(t;\lambda \Omega )$).

2.2. Geometry of Self-Similar Fractals

2.2.1. Self-Similar Iterated Function Systems

Let $(X,d)$ be a complete metric space. An iterated function system on $(X,d)$ is a finite set $\Phi$ of contraction mappings on X. Explicitly, this means that for each map $\phi \in \Phi$, there exists a constant $r_{\phi }\in (0,1)$ such that for all $x,y\in X$, $d(\phi \left(x\right),\phi \left(y\right))\le r_{\phi }d(x,y)$. Hutchinson showed that there is a unique nonempty set $K=K_{\Phi }$ which is compact and satisfying

$$K=\bigcup _{\phi \in \Phi }\phi \left[K\right],$$

which is called the attractor or invariant set of $\Phi$; it is compact and obtained as the closure of the set of fixed points of finite compositions of mappings in the iterated function system [33]. For example, many classical fractals may be obtained through this framework using the space of nonempty, compact subsets of Euclidean space ${\mathbb{R}}^N$ equipped with the Hausdorff metric. This includes Cantor sets, fractal snowflakes, Sierpinski carpets, Menger sponges, and more [34,35,36]. In what follows, we shall be focused on sets like these in Euclidean space, rather than elements of more general complete metric spaces.

All of these specific examples share an additional property, self-similarity. The mappings of an iterated function system are required to be contractions, but for self-similar (iterated function) systems, we shall also require that the mappings are similitudes. A self-similar system $\Phi$ on a complete metric space $(X,d)$ is a finite set of (nontrivial) contractive similitudes, $\Phi :={\left\{{\phi }_k:X\to X\right\}}_{k=1}^m$. Explicitly, for each of the mappings ${\phi }_k$, $k=1,\dots ,m$, to be both a contraction and a similitude, we must have that for every $x,y\in X$, there is some $r_k\in (0,1)$, called the scaling ratio of ${\phi }_k$, so that $d({\phi }_k\left(x\right),{\phi }_k\left(y\right))=r_{k}d(x,y)$. Note that we have ruled out the presence of a constant mapping (by requiring $r_k>0$) and have enforced contractivity (by imposing that $r_k<1$). Given a nonempty compact set $X\subset {\mathbb{R}}^N$, X is said to be a self-similar set if there exists a self-similar system $\Phi$ such that X is the invariant set or attractor of $\Phi$.

2.2.2. Lattice/Nonlattice Dichotomy

In the study of self-similar fractals with multiple scaling ratios, there is a lattice/nonlattice dichotomy in behavior depending on whether or not the distinct scaling ratios are arithmetically related or not. This dichotomy has also been called the arithmetic/non-arithmetic dichotomy, and has been discussed in the work of Lalley in [37,38,39], in the work of Strichartz on self-similar measures [12,13,14], in the work of the second author and collaborators on fractal harps, fractal drums, and the theory of complex dimensions (such as in [15,18,19,22,23,24]), and for the generalized von Koch snowflakes, by van den Berg and collaborators in their work on heat content [6,7,8,9], among others. See [40] and the references therein for more information about this dichotomy in fractal geometry.

Definition 1

(Lattice/Nonlattice Dichotomy). A set of (distinct) scaling ratios ${\left\{{\lambda }_k\right\}}_{k=1}^K$ is said to be in the lattice case if the group

$$G=\prod _{k=1}^K{\lambda }_k^{\mathbb{Z}}$$

is a discrete subgroup of the positive real line with respect to multiplication. In this case, there exists a generator ${\lambda }_0$ such that for every $k=1,\dots ,K$, there exists some positive integer $m_k$ such that ${\lambda }_k={\lambda }_0^{m_k}$. In this case, the (distinct) scaling ratios are then said to be arithmetically related.

If G is not a discrete subgroup of the positive real line (in which case it is a dense subgroup), then the (distinct) scaling ratios are said to be non-arithmetically related. This is called the nonlattice case.

Note that if $\Phi$ is a self-similar system, the set ${\left\{{\lambda }_{\phi }\right\}}_{\phi \in \Phi }$ is necessarily the set of distinct scaling ratios of the similitudes in $\Phi$, rather than viewed as a multiset. It has cardinality less than or equal to $|\Phi |$. By a slight abuse of language, to say that the scaling ratios of the mappings in $\Phi$ are arithmetically related is to say that the set of scaling ratios, i.e., the distinct scaling ratios, are arithmetically related.

An important feature of this dichotomy is that the complex dimensions of self-similar fractals behave very differently in the lattice and nonlattice case. Indeed, very briefly, in the lattice case, they are periodically distributed (with the same oscillatory period) along finitely many vertical lines, whereas in the nonlattice case, they are quasiperiodically distributed (with countably many pseudoperiods); see Figure 2 for depictions of these two cases. The structure of complex dimensions has important implications for the asymptotics of the quantities on regions with self-similar fractal boundary. For more information about the structure results in the different cases, see ([18], Theorem 2.16) for the one dimensional setting and ([18], Theorem 3.6) for a generalized result which may be applied to self-similar sets in higher dimensions.

2.2.3. Relative Fractal Drums

A relative fractal drum, a notion originally introduced in [19], is a geometric object that considers a fractal of interest, here, the boundary $\partial \Omega$, relative to some set which is close to the fractal in an appropriate sense. For this work, the relative set is simply the interior region $\Omega$ on which we study Problem 2 and the closeness condition is automatic from the definition of the boundary of a set.

More formally, let $X,\Omega \subset {\mathbb{R}}^N$. Suppose that $\Omega$ is open, has finite Lebesgue measure, and has the property that there exists $\delta >0$ such that

$$X\subseteq {\Omega }_{\delta }:=\left\{x\in {\mathbb{R}}^N|\exists y\in \Omega ,d(x,y)<\delta \right\},$$

a $\delta$-neighborhood of $\Omega$. Then, the pair $(X,\Omega )$ is called a relative fractal drum ([19], Definition 4.1.2). This notion of a relative fractal drum generalizes both the standard notion of a fractal string [18,41] and of an ordinary fractal drum (or drum with fractal boundary) in ${\mathbb{R}}^N$ [18,19], as well as of any bounded subset of ${\mathbb{R}}^N$. Furthermore, it is very useful for the calculation of fractal zeta functions and the associated complex dimensions of many bounded subsets (and relative fractal drums) of ${\mathbb{R}}^N$; see [19].

Here, the relative fractal drums of this work are of the form $(\partial \Omega ,\Omega )$, where $\Omega \subset {\mathbb{R}}^N$ is a bounded open set (hence having finite Lebesgue measure). Note that any $\delta >0$ is sufficient for ${\Omega }_{\delta }$ to contain $\partial \Omega$. Often, one may assume that $\Omega$ is connected since we may independently study each of the relative fractal drums $(\partial U,U)$ for each connected component U of $\Omega$. (In other words, $\Omega$ may be chosen to be a bounded domain of ${\mathbb{R}}^N$.) Note that, in general, $\Omega$ has at most countably many connected components.

Since our fractal sets (i.e., the boundaries of the given region) will be defined via iterated function systems, we will impose conditions which control the way in which this relative set interacts with the iterated function system. The first one is a separation condition called the open set condition, originally introduced by Moran [42].

Definition 2

(Open Set Condition [42]). An iterated function system $\Phi ={\left\{{\phi }_i\right\}}_{i=1}^m$ on ${\mathbb{R}}^N$ satisfies the open set condition if there exists a nonempty, open set $U\subset {\mathbb{R}}^N$ (called a feasible open set for Φ) such that

1. 

$U\supset {\displaystyle \bigcup _{i=1}^m}{\phi }_i\left[U\right]$;

2. 

For each $i,j\in \left\{1,\dots ,m\right\}$ with $i\ne j$, ${\phi }_i\left[U\right]\cap {\phi }_j\left[U\right]=\varnothing$.

We need slightly more information about such a feasible open set when studying a relative fractal drum $(X,\Omega )$. Namely, we need to know that the images of this set remain closest to the corresponding images of the attractor under the mapping. This condition, introduced as the osculating set condition ([21], Definition 5.2), ensures that points in the image $\phi [\Omega ]$, for $\phi \in \Phi$, remain closest to the corresponding image $\phi \left[X\right]$ under the same contraction $\phi$, as opposed to another distinct image.

Definition 3

(Osculating Sets and Osculant Fractal Drums [21]). Let $\Phi ={\left\{{\phi }_i\right\}}_{i=1}^m$ be an iterated function system on ${\mathbb{R}}^N$, and let X be its attractor. A nonempty, open set $\Omega \subset {\mathbb{R}}^N$ is said to be an osculating set for Φ if the following hold:

1. 

Φ satisfies the open set condition (Definition 2) with respect to Ω;

2. 

For each $i=1,\dots ,m$, if $y\in {\phi }_i[\Omega ]$, then $d(y,X)=d(y,{\phi }_i\left[X\right])$.

A relative fractal drum $(X,\Omega )$ is called an osculant fractal drum if there exists an iterated function system Φ for which X is its attractor and Ω is an osculating set thereof.

An example of an osculating set for a generalized von Koch fractal (the definition of which is recalled and discussed in Section 4.3.1) is depicted in Figure 3. The interior is divided according to threefold symmetry. Each of the three generalized von Koch curves which form the boundary—say $X_k$, for $k=1,2,3$—is viewed relative to the interior components—say ${\Omega }_k$, for $k=1,2,3$—with a set of measure zero having been removed. One such region ${\Omega }_3$ is shaded in the bottom left, and the images under the mappings of the self-similar system of the corresponding region ${\Omega }_1$ in the top third are depicted in the top third of Figure 3. Each relative fractal drum $(X_k,{\Omega }_k)$ is osculant.

2.2.4. Tube Zeta Functions and Complex Dimensions

Fractal zeta functions arise out of the use of the Mellin transform to analyze the oscillatory behavior characteristic of fractals in the space of scales. The versions of fractal zeta functions developed for the higher-dimensional theory, distance and tube zeta functions, extend the theory of complex (fractal) dimensions to this setting [19], building on the development of fractal zeta functions in [18,41]. In this work, we will focus on relative tube zeta functions, considering a boundary relative to its corresponding interior region.

To start, the tube function of the relative fractal drum $(X,\Omega )$ (also called a relative tube function) is the Lebesgue measure of a tubular neighborhood of X intersected with $\Omega$ as a function of the distance parameter defining the neighborhood, that is, the function $V_{X,\Omega }\left(t\right):={\left|X_t\cap \Omega \right|}_N$ for all $t>0$. Here, we denote by ${\left|·\right|}_N$ the Lebesgue measure in ${\mathbb{R}}^N$ and we need only impose that $\Omega$ is measurable, since a tubular neighborhood (as a union of open balls) is necessarily an open set, even if X itself is arbitrary. For the fractal drums we consider, $\Omega$ is open and, thus, measurable.

A truncated Mellin transform, also called a restricted Mellin transform, is an ordinary Mellin transform but with restriction of the domain to an interval subset of $(0,\infty )$; see ([21], Definition 4.1) or Appendix B. Letting $(X,\Omega )$ be a relative fractal drum and $\delta >0$, the relative tube zeta function ${\tilde{\zeta }}_{X,\Omega }$ of X relative to $\Omega$ is given by

$${\tilde{\zeta }}_{X,\Omega }(s;\delta ):={\mathcal{M}}^{\delta }\left[t^{-N}{V}_{X,\Omega }\right]\left(s\right),$$

for all $s\in \mathbb{C}$ for which the analytic continuation of the integral transform is well defined, where ${\mathcal{M}}^{\delta }$ is the truncated Mellin transform with respect to the interval $(0,\delta )$. Note that this is equivalent to ([19], Definition 2.2.8). This tube zeta function will typically possess a meromorphic continuation in the complex plane, and its poles are of great importance to the geometry of the fractal.

The singularities (here, the poles) of this fractal zeta function are called complex dimensions [18,19]. (Strictly speaking, other types of singularities are of interest and should be considered as complex dimensions; for the self-similar sets we consider here, however, only poles will occur, and thus we may restrict our definition, for simplicity.) As we show in Section 4, they will play the role of exponents in explicit formulae for heat content. Additionally, we note that the singularities (viz. the poles) of the tube zeta function are independent of the cutoff parameter $\delta$, that they are geometric invariants of a given fractal set, and further, that the singularities of different variants of fractal zeta functions, most notably, the tube and distance zeta functions, are the same provided that the relative Minkowski dimension of the relative fractal drum is strictly less than the ambient dimension N, as will be the case in our applications in this work [19].

To formally define complex dimensions, we must specify the window W, a subset of the complex plane to which the function permits a meromorphic extension. Let $(X,\Omega )$ be a relative fractal drum and let ${\tilde{\zeta }}_{X,\Omega }$ be the relative tube zeta function of X. If $W\subset \mathbb{C}$, then the complex dimensions of X relative to $\Omega$ contained in the window W, denoted by ${\mathcal{D}}_X\left(W\right)$, are the poles of ${\tilde{\zeta }}_{X,\Omega }$ contained within W.

3. Analysis of Scaling Functional Equations

3.1. Scaling Functional Equations

First, we define scaling operators which act on $C^0\left({\mathbb{R}}^{+}\right)$, where ${\mathbb{R}}^{+}:=(0,\infty )$. A pure scaling operator $M_{\lambda }$, where $\lambda \in {\mathbb{R}}^{+}$, shall act by precomposition of the function with scaling, namely,

$$M_{\lambda }\left[f\right]\left(x\right):=f(x/\lambda ).$$

(4)

This convention of division by the scaling factor shall be convenient for our applications. A general scaling operator L is any finite linear combination of such pure scaling operators with real coefficients, i.e., $L={\sum }_{k=1}^K{a}_k{M}_{{\lambda }_k}$, where ${\lambda }_k\in {\mathbb{R}}^{+}$, and $a_k\in \mathbb{R}$ for each $k=1,\dots ,K$. Later, we will add the constraint that the multiplicities $a_k$ be positive and integral.

Given functions $f,R\in C^0\left({\mathbb{R}}^{+}\right)$ and a scaling operator L, a scaling functional equation for f with remainder R is an identity of the form $f=L\left[f\right]+R$. If $R\equiv 0$, the scaling functional equation is said to be exact, and if not, then it is said to be an approximate scaling functional equation. Given such a scaling functional equation, it can be solved directly by means of truncated Mellin transforms [21] or it can be converted into an additive functional equation through changes of variables and then solved by means of the renewal theorem of Feller [11].

3.1.1. Scaling Zeta Functions and Similarity Dimensions

Given a scaling operator L, we define a scaling zeta function associated to L as in [21]. Namely, given a scaling operator $L={\sum }_{k=1}^K{a}_k{M}_{{\lambda }_k}$, the scaling zeta function ${\zeta }_L$ associated to L is the analytic continuation in $\mathbb{C}$ of the function defined by

$${\zeta }_L\left(s\right):={\displaystyle \frac{1}{1-{\sum }_{k=1}^K{a}_k{\lambda }_k^s}}$$

(5)

for all $s\in \mathbb{C}\setminus {\mathcal{D}}_L$, where ${\mathcal{D}}_L$ the (discrete) set of singularities of ${\zeta }_L$. This function plays a key role in describing solutions to scaling functional equations, owing to its relation to the Mellin transform of such functions and the role of its singularities. For example, the complex dimensions of an osculant, self-similar fractal drum (with appropriate estimates) is a subset of the poles of such an associated scaling zeta function (viz. ([21], Theorem 5.5)).

Given a self-similar system $\Phi$, we can define a scaling operator $L_{\Phi }$ associated to $\Phi$ (and, thus, also a scaling zeta function associated to $\Phi$). Letting ${\lambda }_{\phi }$ denote the scaling ratio of a similitude $\phi \in \Phi$, we define the scaling operator $L_{\Phi }$ of the self-similar system $\Phi$ by

$$L_{\Phi }:=\sum _{\phi \in \Phi }{M}_{{\lambda }_{\phi }}.$$

(6)

The scaling zeta function associated to $\Phi$ is simply the scaling zeta function associated to this operator $L_{\Phi }$, i.e., ${\zeta }_{L_{\Phi }}$, which we will denote by ${\zeta }_{\Phi }$. We note that the multiplicities of $L_{\Phi }$ are positive and integral, and that the scaling ratios ${\lambda }_{\phi }$ lie in $(0,1)$ since the mappings in $\Phi$ are nontrivial and contractive.

For such scaling zeta functions, we can give a precise bound for the locations of their poles. For this purpose, we define the upper and lower similarity dimensions of a self-similar system $\Phi$, which will define a vertical strip in the complex plane containing any such singularity.

Firstly, an upper similarity dimension is an extension of the definition of the similarity dimension of a set, defined by means of a Moran scaling equation.

Definition 4

((Upper) Similarity Dimension). Let Φ be a self-similar system and let ${\left\{{\lambda }_{\phi }\right\}}_{\phi \in \Phi }$ denote the set of scaling ratios of the similitudes $\phi \in \Phi$. Then there is a unique real solution D to Moran’s equation,

$$1=\sum _{\phi \in \Phi }{\lambda }_{\phi }^D.$$

(7)

This number $D={dim}_S(\Phi )$ is called the (upper) similarity dimension of Φ.

Under the imposition of the open set condition (Definition 2 in Section 2.2), it follows from Moran’s theorem [42] that the similarity dimension for a self-similar system of mappings in ${\mathbb{R}}^N$ coincides with the Minkowski dimension (and with the Hausdorff dimension) of the invariant set of $\Phi$. This invariant set is called the self-similar set associated with $\Phi$. Consequently, it follows that $D={dim}_S(\Phi )$ satisfies the bounds $0\le D\le N$. Further, supposing that $|\Phi |\ge 2$, it follows that $D>0$.

We will also write ${dim}_S(\Phi )={\overline{dim}}_S(\Phi )$ when we wish to explicitly differentiate the upper similarity dimension from its lower counterpart (to be defined presently). Under the imposition of the open set condition (Definition 2), the (upper) similarity dimension of a self-similar system is exactly the similarity dimension of its invariant set by Moran’s theorem [42]. It is called an upper similarity dimension since it equals the largest real part of the poles of ${\zeta }_{\Phi }$, with all its other poles having equal or smaller real parts, which follows by ([18], Theorem 3.6).

Next, we define the lower similarity dimension of a self-similar system. The (upper) similarity dimension is a pole of ${\zeta }_{\Phi }$ with largest real part (per ([18], Theorem 3.6)), so the lower similarity dimension will be the infimum of these poles.

Definition 5

(Lower Similarity Dimension). Let Φ be a self-similar system and let ${\zeta }_{\Phi }$ be its associated scaling zeta function (i.e., ${\zeta }_{\Phi }\left(s\right)={(1-{\sum }_{\phi \in \Phi }{\lambda }_{\phi }^s)}^{-1}$, where ${\lambda }_{\phi }$ is the scaling ratio of φ). Then the lower similarity dimension of Φ, denoted by ${\underline{dim}}_S(\Phi )$, is defined by

$${\underline{dim}}_S(\Phi ):=inf\left\{\Re \left(\omega \right)|\omega \in {\mathcal{D}}_{\Phi }\left(\mathbb{C}\right)\right\},$$

(8)

where ${\mathcal{D}}_{\Phi }\left(\mathbb{C}\right)$ denotes the set of poles of ${\zeta }_{\Phi }$ in $\mathbb{C}$ (given by the complex solutions of the equation ${\zeta }_{\Phi }{\left(\omega \right)}^{-1}=0$).

If ${\mathcal{D}}_{\Phi }\left(\mathbb{C}\right)$ were to contain points with arbitrarily large negative real parts, then we would have that ${\underline{dim}}_S(\Phi )=-\infty$. However, we have the following lower bound for ${\underline{dim}}_S(\Phi )$, obtained in [18]. Given a self-similar system $\Phi$, let ${\left\{r_k\right\}}_{k=1}^M$ denote the distinct scaling ratios of the mappings in $\Phi$ with corresponding multiplicities $m_k$. Suppose that the scaling ratios are ordered by size, i.e., $r_1>r_2>\dots >r_M$. Then there is a unique real solution $D_{\ell }$ to the equation

$${\displaystyle \frac{1}{m_M}}{\left(r_M^{-1}\right)}^{D_{\ell }}+\sum _{k=1}^{M-1}{\displaystyle \frac{m_k}{m_M}}{\left({\displaystyle \frac{r_k}{r_M}}\right)}^{D_{\ell }}=1,$$

(9)

and by ([18], Theorem 3.6), we have that $D_{\ell }\le {\underline{dim}}_S(\Phi )$. We summarize these properties of the similarity dimensions as Proposition 3, which is an immediate corollary of ([18], Theorem 3.6), noting that, in light of (5), ${\zeta }_{\Phi }$ is the reciprocal of a Dirichlet polynomial.

Proposition 3

(Similarity Dimension Bounds). Let Φ be a self-similar system and let ${\zeta }_{\Phi }$ be its associated scaling zeta function. Let $D_{\ell }$ be the unique real solution to (9) corresponding to Φ. Then, for any $\omega \in {\mathcal{D}}_{\Phi }\left(\mathbb{C}\right)$, the set of poles of ${\zeta }_{\Phi }$ in $\mathbb{C}$, we have that

$$-\infty <D_{\ell }\le {\underline{dim}}_S(\Phi )\le \Re \left(\omega \right)\le {dim}_S(\Phi ).$$

(10)

If $|\Phi |\ge 2$, then ${dim}_S(\Phi )\ge D_{\ell }>0$. Furthermore, under the imposition of the open set condition (Definition 2 in Section 2.2), we have that ${dim}_S(\Phi )\le N$.

Note that the upper bound in (10) is achieved, since $D={dim}_S(\Phi )$ is a pole of ${\zeta }_{\Phi }$, but that the lower bound $D_{\ell }\le {\underline{dim}}_S(\Phi )$ need not be achieved in general. A sufficient condition for this lower similarity dimension to be the infimum of the real parts of the poles of ${\zeta }_{\Phi }$ is the full rank nonlattice case (called the generic nonlattice case in [18]), i.e., when the multiplicative group $G={\prod }_{k=1}^M{r}_k^{\mathbb{Z}}$ has rank M. Note that this occurs for almost every (with respect to the Lebesgue measure) tuple of distinct scaling ratios $(r_1,\dots ,r_M)\in {(0,1)}^M$.

3.1.2. Establishing Scaling Functional Equations

A scaling functional equation can be established for a function $f_A\left(t\right)=f(t;A)$, where $t\in {\mathbb{R}}^{+}$ and A is a geometric object such as a relative fractal drum or a bounded subset of Euclidean space, through two elements: a decomposition induced by a self-similar system $\Phi$ and a scaling law satisfied by f. To the former, a decomposition of f induced by $\Phi$ is an identity of the form

$$f(t;A)=\sum _{\phi \in \Phi }f(t;\phi \left[A\right])+R\left(t\right),$$

where R is some error quantity called a decomposition remainder. A scaling law for a family of functions ${\left\{f_A\right\}}_{A\in \mathcal{A}}$ is an identity of the form $f(t;\phi \left[A\right])=f(t/{\lambda }_{\phi }^{\alpha };A)$, where $\alpha \in \mathbb{R}$ is fixed (for tube functions $\alpha =1$ and for heat contents, per Corollary 1 in Section 3.1.3, $\alpha =2$) and ${\lambda }_{\phi }$ is the scaling ratio of $\phi$. Note that we assume for each $A\in \mathcal{A}$, so too is $\phi \left[A\right]\in \mathcal{A}$. Letting $L_{\Phi }^{\alpha }={\sum }_{\phi \in \Phi }{M}_{{\lambda }_{\phi }^{\alpha }}$, it follows that $f_A$ satisfies the scaling functional equation $f_A=L_{\Phi }^{\alpha }\left[f_A\right]+R$.

More precisely, let $\Phi$ be a self-similar system on ${\mathbb{R}}^N$ (or, more generally, on a complete metric space) and let $\mathcal{A}$ be a collection of objects with the following closure property: If $A\in \mathcal{A}$, then $\phi \left[A\right]$ is well defined and $\phi \left[A\right]\in \mathcal{A}$ for each $\phi \in \Phi$. For brevity, we will say that $\mathcal{A}$ is $\Phi$-closed. Next, let ${\left\{f_A\right\}}_{A\in \mathcal{A}}$ be a family of functions $f_A:I\to G$ for some set I and an additive semigroup $(G,+)$. We will identify f with the family and write $f(t;A)=f_A\left(t\right)$, for all $t\in I$ and $A\in \mathcal{A}$.

Let $X\subset {\mathbb{R}}^N$ be a self-similar set, let $\Phi$ be a self-similar system having X as its invariant set (or attractor), and let $f_A\left(t\right)=f(t;A)$ be a function defined on $\mathbb{R}$ and certain subsets $A\subseteq {\mathbb{R}}^N$. Here, we will only require that if $f_A$ is defined on A, then $f_{\phi \left[A\right]}$ is defined for any image $\phi \left[A\right]$ of A under the maps $\phi \in \Phi$.

Definition 6

(Induced Decomposion). Let Φ be a self-similar system on ${\mathbb{R}}^N$, let $\mathcal{A}$ be a Φ-closed collection of objects, and let $\mathcal{F}={\left\{f_A\right\}}_{A\in \mathcal{A}}$ be a family of functions $f_A:I\to G$ for some set I and an additive semigroup $(G,+)$. Let also $R:I\to G$ be a function.

A self-similar system Φ is said to induce a decomposition of f on I with remainder R if for all $t\in I$ and $f_A\in \mathcal{F}$,

$$f(t;A)=\sum _{\phi \in \Phi }f(t;\phi \left[A\right])+R\left(t\right).$$

(11)

In this work, I is always a subset of $\mathbb{R}$ and the codomain is the field $\mathbb{R}$ with its standard addition operation. Further, A is some geometric structure, either a bounded subset of ${\mathbb{R}}^N$ or, more generally, a relative fractal drum $(X,\Omega )$ in ${\mathbb{R}}^N$. If $A\subset {\mathbb{R}}^N$, $\phi \left[A\right]$ is its pointwise image. For a relative fractal drum, we define $\phi \left[\right(X,\Omega \left)\right]:=\left(\phi \right[X],\phi [\Omega \left]\right)$, noting that this also defines a relative fractal drum.

An induced decomposition will be paired with a scaling law satisfied by the function $f(t;A)$. A linear scaling (invariance) law would be of the form $f(\lambda t;\lambda A)=f(t;A)$, for each $\lambda \in {\mathbb{R}}^{+}$, where ${\mathbb{R}}^{+}:=(0,\infty )$. However, depending on the context, the real and set parameters may scale differently. In the case of heat contents, the function will obey a quadratic scaling law: $f({\lambda }^{2}t;\lambda A)=f(t;A)$, where $\alpha =2$. We provide a general notion for defining the scaling relation which allows one to specify the parameter $\alpha$ dictating this relationship.

Definition 7

($\alpha$-Scaling Law). Let $\mathcal{F}={\left\{f(·;A)\right\}}_{A\in \mathcal{A}}$ be a family of functions $f_A:{\mathbb{R}}^{+}\to \mathbb{R}$ where $\mathcal{A}$ is a $\left\{\phi \right\}$-closed collection of objects for any similitude φ of ${\mathbb{R}}^N$. Given $\alpha \in \mathbb{R}$ (and in this work, $\alpha >0$), we say that $\mathcal{F}$ has an α-scaling law if for any similitude φ with scaling ratio $\lambda >0$ and for any $f_A\in \mathcal{F}$, $f(t;\phi \left[A\right])=f(t/{\lambda }^{\alpha };A)$, for all $t\in I$.

We conclude this section with the main idea: A scaling law and an induced decomposition for a family of functions together lead to the scaling functional equation of a single function.

Proposition 4

(Scaling Functional Equation). Let $\mathcal{F}={\left\{f_A\right\}}_{A\in \mathcal{A}}$ be a family of functions on I satisfying an α-scaling law. Suppose that Φ is a self-similar system which induces a decomposition of $\mathcal{F}$ on I with remainder term R. Define the scaling operator $L_{\Phi }^{\alpha }$ by $L_{\Phi }^{\alpha }:={\sum }_{\phi \in \Phi }{M}_{{\lambda }_{\phi }^{\alpha }}$. Then, each function $f_A\in \mathcal{F}$ in the family satisfies the scaling function equation $f_A=L_{\Phi }^{\alpha }\left[f_A\right]+R$ on I with error term R, which is to say that for all $t\in I$,

$$f_A\left(t\right)=L_{\Phi }^{\alpha }\left[f_A\right]\left(t\right)+R\left(t\right).$$

We say that Φ induces a scaling functional equation for $f_A$ with operator $L_{\Phi }^{\alpha }$.

Proposition 4 is, in essence, an untangling of definitions meant to lead to a single key concept: A self-similar system induces a scaling functional equation for functions with scaling laws. It is, however, a consequence of the definitions that any function in this family satisfies an induced scaling functional equation. The proof is merely to apply the $\alpha$-scaling law to each term of an induced decomposition and then use the definition of the scaling operator $L_{\Phi }^{\alpha }$ to rewrite the decomposition in terms of a single function.

3.1.3. Scaling Functional Equations for Heat Content

Proposition 2 in Section 2.1 implies that the total heat content $E_{\Omega }\left(t\right)=E(t;\Omega )$, viewed as a family of functions, satisfies a 2-scaling law in the sense of Definition 7, provided a fixed set of boundary conditions for which Proposition 2 holds. We state this in our specific case, i.e., for Problem 1 in Section 1.2.

Corollary 1

(2-Scaling Law of Heat Content). Given any bounded open set $A\subset {\mathbb{R}}^N$ and any similitude φ on ${\mathbb{R}}^N$, let $E_A\left(t\right)$ be the total heat content for the PWB solution $u_{\Omega }$ of Problem 1 in Section 1.1 for each of the sets $\Omega =A$ and $\Omega ={\phi }^{\circ k}\left[A\right]$, $k\in \mathbb{N}$. Then, the family of normalized functions $t^{-N/2}{E}_A\left(t\right)$ (with respect to the Φ-closed set $\mathcal{A}=\left\{{\phi }^{\circ k}\left[A\right]|k\in {\mathbb{N}}_0\right\}$) satisfies a 2-scaling law in the sense of Definition 7, viz. for every $A\in \mathcal{A}$ and $t>0$,

$${(t/{\lambda }^2)}^{-N/2}{E}_A(t/{\lambda }^2)=t^{-N/2}{E}_{\phi \left[A\right]}\left(t\right).$$

Proof. 

The constant, nonnegative boundary conditions of Problem 1 ensure that $u_{\Omega }$ is unique and they are clearly scale invariant and parabolically scale invariant in their respective variables. Because $\phi$ is a similitude, every set ${\phi }^{\circ k}\left[A\right]$ is bounded—using the bound ${\lambda }^{k}C$, where C is a bound such that A is contained in a ball of radius C—and open, which follows because $\phi$ is injective and its inverse is also a similitude and, hence, continuous.

Applying Proposition 2 to an arbitrary set $A\in \mathcal{A}$, we have that

$$\begin{array}{c}\hfill t^{-N/2}{E}_{\phi \left[A\right]}\left(t\right)=t^{-N/2}{\lambda }^N{E}_A(t/{\lambda }^2)={(t/{\lambda }^2)}^{-N/2}{E}_A(t/{\lambda }^2),\end{array}$$

as desired. □

Suppose now that our region $\Omega$ has a self-similar boundary $\partial \Omega$ corresponding to a self-similar system $\Phi$ and that $\Omega$ is an osculating set (see Definition 3 in Section 2.2) for $\Phi$. In order to use the scaling property to obtain a scaling functional equation, we will need an induced decomposition in the sense of Definition 6 in Section 3.1. For later use, we define the remainder which occurs as the difference of the heat content and the sum of scaled copies. Given a suitable estimate of this quantity, we will be able to obtain explicit formulae for the heat content.

Definition 8

(Decomposition Remainder). Let $u_{\Omega }$ be the PWB solution to Problem 1 in Section 1.2 on the bounded open set $\Omega \subset {\mathbb{R}}^N$. Suppose that $\partial \Omega$ is the attractor of a self-similar system Φ and that the relative fractal drum $(\partial \Omega ,\Omega )$ is osculant.

We define the decomposition remainder of $E_{\Omega }$ to be the quantity

$$R_{\Omega }\left(t\right):=E_{\Omega }\left(t\right)-\sum _{\phi \in \Phi }{E}_{\phi [\Omega ]}\left(t\right).$$

(12)

The normalized remainder is the quantity $R\left(t\right):=t^{-N/2}{R}_{\Omega }\left(t\right)$, for all $t>0$.

Note that in the case of Problem 1, we can deduce that $R_{\Omega }$ is bounded for all $t\ge 0$ as a corollary of Proposition 1 in Section 2.1. This will provide an upper bound for the abscissa of absolute convergence of its truncated Mellin transform, ${\mathcal{M}}^{\delta }\left[t^{-N}{R}_{\Omega }\left(t\right)\right]$, as by Lemma A1, we may deduce that this transform, as a function of the complex variable s, is holomorphic in the right half plane ${\mathbb{H}}_{N/2}$. This helps establish some technical preliminaries for later proofs, allowing us to concern ourselves solely with asymptotic estimates of the form $R_{\Omega }\left(t\right)=O\left(t^{(N-{\sigma }_0)/2}\right)$ as $t\to 0^{+}$, for some ${\sigma }_0\in \mathbb{R}$. When ${\sigma }_0$ is the smallest such parameter, it corresponds to ${\sigma }_0=2{\sigma }_{ac}$, where ${\sigma }_{ac}$ is the abscissa of absolute convergence of the normalized remainder. (Division by two is for normalization owing to the quadratic nature of the heat content scaling law.)

The most precise results occur when there is no error term $R_{\Omega }$, viz. when $R_{\Omega }\equiv 0$. In this case, the results of this work apply where estimates for the remainder terms may be chosen of the form $O\left(t^{\tau }\right)$ as $t\to 0^{+}$, for any $\tau >0$. This occurs when the set X partitions into disjoint, self-similar copies (up to sets of measure zero). The novelty of this method, though, lies in its ability to handle cases where this is not true, but with explicit estimates for the degree to which the decomposition is not exact, such as in the case of generalized von Koch fractals. The key part of using this scaling function approach is obtaining good estimates for the remainder term (viz. for small values of ${\sigma }_0$), as the expansion will be explicit up to an order determined by these estimates. For the reader wishing to establish such functional equations for a particular class of self-similar sets, when studying heat content, see [3,7].

3.2. Admissible Remainders

As part of our analysis of scaling functional equations, we will be required to show that certain functions are languid, a type of growth condition detailed in Appendix C. Languidity was introduced in [41] and then refined in [18,19] in the context of proving explicit tube formulae and other geometric, dynamical, and spectral formulae for generalized fractal harps (or strings) or relative fractal drums, respectively. In so doing, we will need to constrain the types of remainders which can appear in order for these growth conditions to be established. In this section, we introduce terminology for such admissible remainders relative to a given self-similar system and we state and prove two types of sufficient conditions for admissibility in Section 3.2.2 and Section 3.2.3, respectively.

3.2.1. Admissibility and Joint Languidity

The admissibility of a remainder is essentially tied to a need for joint languidity. Two functions, the Mellin transform of the remainder ${\zeta }_R$ and the scaling zeta function ${\zeta }_{\Phi }$, must share the same screen and the same horizontal contours on which they are both uniformly bounded as part of the languidity hypotheses (i.e., hypotheses L1 and L2 in Appendix C.1). Here, a screen $S:\mathbb{R}\to \mathbb{R}$ is a bounded, Lipschitz continuous function which (by a slight abuse of notation) is identified with the set

$$\mathcal{S}=\left\{S\left(\tau \right)+i\tau \in \mathbb{C}|\tau \in \mathbb{R}\right\}\subset \mathbb{C},$$

just as in [18,19,41].

Definition 9

(Joint Languidity). Let ${\zeta }_f$ and ${\zeta }_g$ be functions which are both holomorphic in a half-plane ${\mathbb{H}}_{{\sigma }_1}$ and both admitting a necessarily unique analytic continuation to a half-plane ${\mathbb{H}}_{{\sigma }_0}$, where ${\sigma }_0<{\sigma }_1$, except perhaps for a discrete set ${\mathcal{D}}_{f,g}\left({\mathbb{H}}_{{\sigma }_0}\right)$ at which either ${\zeta }_f$ or ${\zeta }_f$ may be singular.

We say that ${\zeta }_f$ and ${\zeta }_g$ are jointly languid with exponent $\kappa \in \mathbb{R}$ if there exists a screen S (contained in ${\mathbb{H}}_{{\sigma }_0}\setminus {\mathcal{D}}_{f,g}\left({\mathbb{H}}_{{\sigma }_0}\right)$) such that the following hold.

• There is a sequence of admissible heights (i.e., a doubly infinite sequence ${\left\{{\tau }_n\right\}}_{n\in \mathbb{Z}}$ which satisfies (A12) in Appendix C) for which ${\zeta }_f$ and ${\zeta }_g$ both satisfy languidity hypothesis L1 with exponent κ and with respect to S using this sequence.
• Both ${\zeta }_f$ and ${\zeta }_g$ satisfy languidity hypothesis L2 with exponent κ and with respect to S.

In this work, the functions we consider will have a necessarily unique meromorphic continuations to ${\mathbb{H}}_{{\sigma }_0}$ and ${\mathcal{D}}_{f,g}\left({\mathbb{H}}_{{\sigma }_0}\right)$ is the union of the sets of poles of ${\zeta }_f$ and ${\zeta }_g$ in ${\mathbb{H}}_{{\sigma }_0}$. In the event that ${\zeta }_g$ is holomorphic in ${\mathbb{H}}_{{\sigma }_0}$, then ${\mathcal{D}}_{f,g}\left({\mathbb{H}}_{{\sigma }_0}\right)={\mathcal{D}}_f\left({\mathbb{H}}_{{\sigma }_0}\right)$ consists only of poles of ${\zeta }_f$.

Given a scaling functional equation of the form $f=L\left[f\right]+R$, the remainder term R will be called admissible when its Mellin transform ${\zeta }_R$ and the scaling zeta function ${\zeta }_L$ associated to L are jointly languid in the following precise sense.

Definition 10

(Admissible Remainders and Screens). Let R be a function such that its truncated Mellin transform ${\zeta }_R(s;\delta )={\mathcal{M}}^{\delta }\left[R\right]\left(s\right)$ is holomorphic in the half-plane ${\mathbb{H}}_{{\sigma }_0}$. Let L be a scaling operator with associated scaling zeta function ${\zeta }_L$.

We say that R is an admissible remainder for L if there exists a screen S such that ${\zeta }_L$ and ${\zeta }_R$ are jointly languid with exponent $\kappa =0$ with respect to S in the sense of Definition 9. Any such screen S is called an admissible screen for R and L (or their respective zeta functions). Given a self-similar system Φ, we say that R is an admissible remainder for Φ if it is admissible for the associated operator $L_{\Phi }$ (and similarly for admissible screens).

We note that ${\zeta }_R(s;\delta )={\mathcal{M}}^{\delta }\left[R\right]\left(s\right)$ is holomorphic in the half-plane ${\mathbb{H}}_{{\sigma }_0}$ if, for instance, R is continuous on $(0,\delta ]$ and $R\left(t\right)=O\left(t^{-{\sigma }_0}\right)$ as $t\to 0^{+}$ (per Lemma A1).

3.2.2. Lower Dimension Criterion

The easiest way to obtain a nontrivial admissible screen occurs when ${\zeta }_R$ is holomorphic in a half-plane to the left of the vertical strip containing all of the poles of ${\zeta }_L$. This ensures that a screen may be chosen to the left of any pole of ${\zeta }_L$, but also in a region where ${\zeta }_R$ is holomorphic. Supposing that $L=L_{\Phi }$ is the scaling operator associated to a self-similar system, we have that this strip is bounded explicitly by the similarity dimensions of $\Phi$: if $\omega \in {\mathcal{D}}_{\Phi }$, then ${\underline{dim}}_S(\Phi )\le \Re \left(\omega \right)\le {dim}_S(\Phi )$. So, the first criterion is to ensure an estimate for R that guarantees that the abscissa of absolute convergence of ${\zeta }_R$ is strictly smaller than the explicit bound $D_{\ell }$ for the lower similarity dimension, which is used to estimate ${\zeta }_{\Phi }$.

Theorem 1

(Lower Dimension Criterion for Admissibility). Let Φ be a self-similar system and let $D_{\ell }\le {\underline{dim}}_S(\Phi )$ be the lower similarity dimension bound as in Proposition 3 in Section 3.1.1. For any $R\in C^0\left({\mathbb{R}}^{+}\right)$, if there exists ${\sigma }_0<D_{\ell }$ such that as $t\to 0^{+}$, $R\left(t\right)=O\left(t^{-{\sigma }_0}\right)$, then R is an admissible remainder for Φ and any screen of the form $S_{\epsilon }\left(\tau \right)\equiv {\sigma }_0+\epsilon$, with $0<\epsilon <D_{\ell }-{\sigma }_0$, is admissible.

Proof. 

Let $D_{\ell }\le {\underline{dim}}_S(\Phi )$ be the lower similarity dimension bound for $\Phi$ and let $S_{\epsilon }\left(\tau \right)\equiv {\sigma }_0+\epsilon$ be a constant screen in ${\mathbb{H}}_{{\sigma }_0}$, where $\epsilon >0$ is such that ${\sigma }_0+\epsilon <D_{\ell }$. We will show that ${\zeta }_R$ and ${\zeta }_{\Phi }$ are jointly languid with exponent $\kappa =0$ on $S_{\epsilon }$.

To start, we have that ${\zeta }_{\Phi }$ is strongly languid with exponent $\kappa =0$ by Proposition A2. This guarantees that for any ${\sigma }_{-}$, there exists a sequence of admissible heights ${\left\{T_n\right\}}_{n\in \mathbb{Z}}$ (i.e., with $T_{-n}\to -\infty$ and $T_n\to \infty$ as $n\to \infty$ and with $T_n>0>T_{-n}$ for each $n\ge 1$) and some screen $S_{-}$ with $sup{S}_{-}<{\sigma }_{-}$ such that ${\zeta }_{\Phi }$ is uniformly bounded on horizontal contours of the form $[S_{-}\left(T_n\right)+i{T}_n,{\sigma }_{+}+i{T}_n]$, where ${\sigma }_{+}>max({\sigma }_{-},{dim}_S(\Phi ))$. In particular, ${\zeta }_{\Phi }$ will be uniformly bounded on the subsets $I_n:=[{\sigma }_0+\epsilon +i{T}_n,{\sigma }_{+}+i{T}_n]$. This establishes languidity hypothesis L1 for ${\zeta }_{\Phi }$.

Next, we note that the estimate on R implies that the function ${\zeta }_R$ is holomorphic in the open right half-plane ${\mathbb{H}}_{{\sigma }_0}$ by Lemma A1. By Corollary A1, we also have that ${\zeta }_R$ is bounded on the screen $S_{\epsilon }\left(\tau \right)\equiv {\sigma }_0+\epsilon >{\sigma }_0$ as well as on any vertical strip of the form ${\mathbb{H}}_{{\sigma }_0+\epsilon }^{{\sigma }_{+}}$. This establishes both languidity hypotheses L1 and L2 with exponent $\kappa =0$ and on the screen $S_{\epsilon }$ with respect to the intervals $I_n\subset {\mathbb{H}}_{{\sigma }_0+\epsilon }^{{\sigma }_{+}}$, with shared sequence of admissible heights.

It remains to show that ${\zeta }_{\Phi }$ is bounded on the screen $S_{\epsilon }$. To this end, we enforce that ${\sigma }_0+\epsilon <D_{\ell }$, in which case for any s on the screen, $\Re \left(s\right)={\sigma }_0+\epsilon <D_{\ell }$. Let ${\left\{r_k\right\}}_{k=1}^m$ denote the set of unique scaling ratios of $\Phi$, arranged in decreasing order $r_1\ge \dots \ge r_M$, and let $m_k$ denote the multiplicity of $r_k$. Define the function

$$p\left(t\right)={\displaystyle \frac{1}{m_M}}{\left(r_M^{-1}\right)}^t+\sum _{k=1}^{M-1}{\displaystyle \frac{m_k}{m_M}}{\left({\displaystyle \frac{r_k}{r_M}}\right)}^t,$$

(13)

which is readily seen to be strictly increasing with range $(0,\infty )$. Note that $p\left(D_{\ell }\right)=1$ by definition. We will obtain a bound for ${\zeta }_{\Phi }\left(s\right)$ when $\Re \left(s\right)={\sigma }_0+\epsilon <D_{\ell }$ using this function.

To that end, let $f\left(s\right)={\zeta }_{\Phi }{\left(s\right)}^{-1}=1-{\sum }_{k=1}^M{m}_k{r}_k^s$ be the denominator and let $\sigma =\Re \left(s\right)$. Then

$$\begin{array}{cc}\hfill |r_M^{-s}/m_{M}f\left(s\right)+1|& =\left|r_M^{-s}/m_M-\sum _{k=1}^{M-1}{m}_k/m_M{(r_k/r_M)}^s\right|\hfill \\ \hfill & \le p\left(\sigma \right)<p\left(D_{\ell }\right)=1.\hfill \end{array}$$

By the reverse triangle inequality, we have that

$$\begin{array}{c}\hfill 1>p\left(\sigma \right)\ge \left||r_M^{-s}/m_{M}f\left(s\right)|-1\right|=\left|r_M^{-\sigma }/m_M\left|f\left(s\right)\right|-1\right|,\end{array}$$

which may be rewritten as $\left|\right|f\left(s\right)|-m_M{r}^{\sigma }|\le m_M{r}^{\sigma }p\left(\sigma \right)<m_M{r}_M^{\sigma }$. Once again using the reverse triangle inequality, we find that the lower bound for $\left|f\right|$ is furnished by

$$\begin{array}{c}\hfill \left|f\left(s\right)\right|\ge m_M{r}_M^{\sigma }-\left|\right|f\left(s\right)|-m_M{r}^{\sigma }|\ge m_M{r}_M^{\sigma }-m_M{r}^{\sigma }p\left(\sigma \right)>0.\end{array}$$

It follows that when $\Re \left(s\right)={\sigma }_0+\epsilon$ is fixed, $|{\zeta }_{\Phi }\left(s\right)|\le C_{\Phi }$, with $C_{\Phi }={(m_M{r}_M^{\sigma }-m_M{r}^{{\sigma }_0+\epsilon }p({\sigma }_0+\epsilon ))}^{-1}$. This establishes hypothesis L2 for ${\zeta }_{\Phi }$, and, thus, ${\zeta }_{\Phi }$ and ${\zeta }_R$ are jointly languid on the screen $S_{\epsilon }$ (with exponent $\kappa =0$). □

In order to accommodate $\alpha$-scaling laws with $\alpha >0$ (in the sense of Definition 7 in Section 3.1), we will consider scaling operators of the form $L_{\Phi }^{\alpha }:={\sum }_{\phi \in \Phi }{M}_{{\lambda }_{\phi }^{\alpha }}$, where $\Phi$ is a self-similar system. In this case, the zeta function associated to $L_{\Phi }^{\alpha }$ is exactly the function ${\zeta }_{\Phi }\left(\alpha s\right)$, where ${\zeta }_{\Phi }\left(s\right)$ is the scaling zeta function associated to $\Phi$ itself.

This change in variables amounts to a simple rescaling of the bound with respect to the lower similarity dimension of $\Phi$ a remainder must satisfy in order to be admissible by Theorem 1. Namely, when $R\left(t\right)=O\left(t^{-{\sigma }_R}\right)$ as $t\to 0^{+}$ for some ${\sigma }_R<D_{\ell }/\alpha \le {\underline{dim}}_S(\Phi )/\alpha$, we have that ${\zeta }_{\Phi }\left(\alpha s\right)$ has poles in the strip ${\underline{dim}}_S(\Phi )/\alpha \le \Re \left(s\right)\le {dim}_S(\Phi )/\alpha$ and ${\sigma }_R$ lies strictly to the left of this bound. Equivalently, if we state the estimate for R in the form $R\left(t\right)=O\left(t^{-{\sigma }_0/\alpha }\right)$ (as $t\to 0^{+}$), where ${\sigma }_0/\alpha ={\sigma }_R$, then ${\sigma }_0$ must satisfy the bound ${\sigma }_0<D_{\ell }$ directly. The estimates for ${\zeta }_{\Phi }$ apply to its rescaled analogue and, thus, the proof of Theorem 1 yields the following corollary.

Corollary 2

(Rescaled Lower Dimension Criterion). Let Φ be a self-similar system and let $D_{\ell }\le {\underline{dim}}_S(\Phi )$ be the lower similarity dimension bound as in Proposition 3 in Section 3.1.1. Given $\alpha >0$, define the scaling operator $L_{\Phi }^{\alpha }:={\sum }_{\phi \in \Phi }{M}_{{\lambda }_{\phi }^{\alpha }}$.

For any $R\in C^0\left({\mathbb{R}}^{+}\right)$, if there exists ${\sigma }_R<D_{\ell }/\alpha$ such that as $t\to 0^{+}$, $R\left(t\right)=O\left(t^{-{\sigma }_R}\right)$, then R is an admissible remainder for $L_{\Phi }^{\alpha }$ and any screen of the form $S_{\epsilon }\left(\tau \right)\equiv {\sigma }_R+\epsilon$, with $0<\epsilon <D_{\ell }/\alpha -{\sigma }_R$, is admissible.

Proof. 

By Lemma A1, we have that ${\zeta }_R\left(s\right)$ is holomorphic in ${\mathbb{H}}_{{\sigma }_R}$ and, thus, ${\zeta }_R(z/\alpha )$ is holomorphic when $\Re \left(z\right)>\alpha {\sigma }_R$. Further, ${\zeta }_R\left(s\right)$ is bounded on any vertical strip of the form ${\mathbb{H}}_a^b$, ${\sigma }_R<a\le b<\infty$, so ${\zeta }_R(z/\alpha )$ is bounded on the corresponding strips ${\mathbb{H}}_{a^{\prime }}^{b^{\prime }}$, $\alpha {\sigma }_R<a^{\prime }\le b^{\prime }<\infty$.

Since $\alpha {\sigma }_R<D_{\ell }$, by repeating the proof of Theorem 1 (with ${\sigma }_0=\alpha {\sigma }_R$), we obtain that ${\zeta }_{\Phi }\left(z\right)$ and ${\zeta }_R(z/\alpha )$ are jointly languid on screens of the form $S_{\epsilon }\left(\tau \right)\equiv \alpha {\sigma }_R+\epsilon$ when $0<\epsilon <D_{\ell }-\alpha {\sigma }_R$. Taking $z=\alpha s$, we see that the functions ${\zeta }_{\Phi }\left(\alpha s\right)$ and ${\zeta }_R\left(s\right)$ are jointly languid on screens of the form $S_{{\epsilon }^{\prime }}^{\prime }\left(\tau \right)\equiv {\sigma }_R+{\epsilon }^{\prime }$ where ${\epsilon }^{\prime }=\epsilon /\alpha \in (0,D_{\ell }/\alpha -{\sigma }_R)$. □

3.2.3. Lattice Criterion

The next criterion is related to when we have explicit knowledge of the locations of the singularities of ${\zeta }_{\Phi }$. In particular, in the lattice case (see Definition 1 in Section 2.2.2), we can explicitly show that all of the poles lie on one of finitely many vertical lines (and are distributed periodically along these lines with a shared period for each line); see, for instance, ([18], Theorem 3.6). Thus, we can easily choose screens within this region which will never encounter singularities of ${\zeta }_{\Phi }$, with distance to any pole bounded by the distance of the real part of the screen to the real part of the closest of the finitely many exceptional points.

Theorem 2

(Lattice Criterion for Admissiblility). Let Φ be a self-similar system and suppose that its distinct scaling ratios ${\left\{{\lambda }_{\phi }\right\}}_{\phi \in \Phi }$ are arithmetically related (see Definition 1 in Section 2.2.2). Let R be a continuous function on ${\mathbb{R}}^{+}$ with the estimate that $R\left(t\right)=O\left(t^{-{\sigma }_0}\right)$ as $t\to 0^{+}$, for some ${\sigma }_0\in \mathbb{R}$. Then, for all but finitely many $\sigma >{\sigma }_0$, $S_{\sigma }\left(\tau \right)\equiv \sigma$ is an admissible screen. Consequently, there are admissible screens of the form $S_{{\sigma }_0+\epsilon }$ for any $\epsilon >0$ sufficiently small.

Proof. 

By definition, in the lattice case, there exist some ${\lambda }_0\in {\mathbb{R}}^{+}$ such that for each $\phi \in \Phi$, there is a positive integer $k_{\phi }$ such that ${\lambda }_{\phi }={\lambda }_0^{k_{\phi }}$. Under this assumption, we may explicitly write the denominator of ${\zeta }_{\Phi }$ as the Dirichlet polynomial

$$P\left(s\right)=1-\sum _{\phi \in \Phi }{\lambda }_{\phi }^s=1-\sum _{\phi \in \Phi }{\lambda }_0^{k_{\phi }s}.$$

Under the change of variables $s={log}_{{\lambda }_0}z$ (so that ${\lambda }_0^s=z$), we have that

$$P\left({log}_{{\lambda }_0}z\right)=1-\sum _{\phi \in \Phi }{z}^{k_{\phi }}.$$

This is precisely a polynomial in the variable z (since $|\Phi |=n<\infty$), so by the fundamental theorem of algebra it has finitely many roots (exactly $K=max\left({\left\{k_{\phi }\right\}}_{\phi \in \Phi }\right)$) in $\mathbb{C}$, counting multiplicity.

Denote these roots by $Z={\left\{z_j\right\}}_{j=1}^K$. Any solution of $P\left(\omega \right)=0$ must then be of the form ${\lambda }_0^{\omega }=z_j$, for some $z_j\in Z$. We note that $0\notin Z$ since the polynomial equals one when $z=0$; so, there exists a branch of the logarithm for which $log{z}_j$ is well defined for each $z_j\in Z$ and, without loss of generality, we may find a single branch defined for each $z_j\in Z$ and for ${\lambda }_0$ since Z is finite. The logarithm is multivalued, however, with ${\lambda }_0^{\omega }=e^{\omega log{\lambda }_0}=e^{\omega log{\lambda }_0+2\pi im}$ for any $m\in \mathbb{Z}$. So, we will obtain as solutions to $P\left(\omega \right)=0$ exactly the points ${\omega }_{j,m}$ of the form ${\omega }_{j,m}=log\left(z_j\right)/log\left({\lambda }_0\right)+2\pi im/log\left({\lambda }_0\right)$, where $m\in \mathbb{Z}$ is arbitrary.

Observe that there are finitely many real parts, ${\sigma }_j=\Re (log\left(z_j\right)/log\left({\lambda }_0\right))$, $j=1,\dots ,K$, at which these poles occur and that the imaginary parts are all distributed with the same period, $2\pi /log{\lambda }_0$. We will show, starting with hypothesis L1, that for any screen of the form $S_{\sigma }\left(\tau \right)\equiv \sigma$, where $\sigma \in \mathbb{R}$ and $\sigma \ne {\sigma }_j$ for each $j=1,\dots ,K$, ${\zeta }_{\Phi }$ is languid with exponent $\kappa =0$ with respect to $S_{\sigma }$.

The next two steps of the proof are the same as in the proof of Theorem 1. In short, by the strong languidity of ${\zeta }_{\Phi }$, for any ${\sigma }_{-}$, there exists a sequence of admissible heights ${\left\{T_n\right\}}_{n\in \mathbb{Z}}$ on which ${\zeta }_{\Phi }$ is bounded on horizontal intervals of the form $[{\sigma }_{-}+i{T}_n,{\sigma }_{+}+i{T}_n]$, where ${\sigma }_{-}$ is arbitrarily small and ${\sigma }_{+}>max({dim}_S(\Phi ),{\sigma }_{-})$. Choosing ${\sigma }_{-}=\sigma$ establishes hypothesis L1 with respect to the screen $S_{\sigma }$ for ${\zeta }_{\Phi }$. Secondly, we have that ${\zeta }_R$ satisfies hypotheses L1 and L2 for the sequence of heights as above on any screen of the form $S_{\sigma }$, $\sigma >{\sigma }_0$, because ${\zeta }_R$ is a Mellin transform and is holomorphic in ${\mathbb{H}}_{{\sigma }_0}$.

It only remains to show that ${\zeta }_{\Phi }$ satisfies hypothesis L2 on $S_{\sigma }$ when $\sigma$ is not one of the exceptional values ${\sigma }_j$, $j=1,\dots ,K$. We will show that $f\left(\tau \right)$ is bounded from below by a strictly positive constant on all of $\mathbb{R}$, and, thus, ${\zeta }_{\Phi }$ will be bounded on $S_{\sigma }$. As we have shown, $P(\sigma +i\tau )\ne 0$ whenever $\sigma \notin {\left\{{\sigma }_j\right\}}_{j=1}^K$, with $\sigma ,\tau \in \mathbb{R}$. Let $f\left(\tau \right)=\left|P\right(\sigma +i\tau \left)\right|$, for all $\tau \in \mathbb{R}$. Given any compact interval $[a,b]$, we have that $f\left(t\right)$ must be nonzero and bounded from below by a strictly positive constant on $[a,b]$. This follows by continuity and the intermediate value theorem, noting that if $f\left(\tau \right)=0$ with $\tau \in \mathbb{R}$, then $P(\sigma +i\tau )=0$, which is a contradiction since, by hypothesis, $\sigma \notin {\left\{{\sigma }_j\right\}}_{j=1}^K$.

We will use the (multiplicative) periodicity of f to show that $f\left(t\right)=\left|P\right(\sigma +i\tau \left)\right|$ cannot become arbitrarily small as $\left|\tau \right|\to \infty$. Recall that

$$P(\sigma +i\tau )=1-\sum _{j=1}^K{\lambda }_0^{k_j({\sigma }_0+i\tau )}=1-\sum _{j=1}^K{\lambda }_0^{k_j\sigma }·e^{i(k_{j}log{\lambda }_0)\tau }.$$

Under the transformation $\tau \mapsto (2\pi m/log{\lambda }_0)\tau$, for any $m\in \mathbb{Z}$, we have that each exponential is invariant since $k_j$ is an integer. Thus, $P(\sigma +i\tau )=P(\sigma +i(2\pi m/log{\lambda }_0)\tau )$. Choosing $m=-1$ shows us that the function $f\left(\tau \right):=\left|P\right(\sigma +i\tau \left)\right|$ is multiplicatively periodic with period $p=-2\pi /log{\lambda }_0=2\pi /log{\lambda }_0^{-1}$ or a rational multiple thereof, depending on the greatest common divisor of the integers $k_j$, $j=1,\dots ,K$. However, if they share a greatest common divisor GCD, it is possible to redefine ${\lambda }_0$ by ${\lambda }_0^{\prime }={\lambda }_0^{\mathrm{GCD}}$ and obtain a new set of integers without this property. (In other words, without loss of generality, this assumption may be incorporated into the definition of ${\lambda }_0$.) Note that $log\left({\lambda }_0^{-1}\right)>0$.

By the previous argument, $f\left(\tau \right)=\left|P\right(\sigma +i\tau \left)\right|$ is bounded from below on any compact interval. So, f is nonzero on the interval $[-1,1]$. Now, pick one full multiplicative period in $(0,\infty )$, say, $[p^{m_0},p^{m_0+1}]$ (if $p>1$) or $[p^{m_0+1},p^{m_0}]$ (if $p<1$). The function f must be bounded from below strictly away from zero as this is a compact interval. By periodicity, this same bound applies to any interval of the form $[p^m,p^{m+1}]$ or $[p^{m+1},p^m]$, respectively, for any $m\in \mathbb{Z}$. Since the function $p\mapsto p^t$ is surjective onto $(0,\infty )$, this implies that f is bounded uniformly from below by a strictly positive constant, the same as the first bound, when $\tau >0$. For $\tau <0$, we can use the starting interval $[-p^{m_0+1},-p^{m_0}]$ or $[-p^{m_0},-p^{m_0+1}]$, depending on whether $p>1$ or $p<1$, and the same argument to deduce that it is also bounded from below by a strictly positive constant on $(-\infty ,0)$. Taking the minimum of these three bounds shows that there exists a positive constant C such that $f\left(\tau \right)=\left|P\right(\sigma +i\tau \left)\right|>C$.

It follows that ${\zeta }_{\Phi }$ is bounded from above on $S_{{\sigma }_0}$, establishing hypothesis L2. Thus, we have established the joint languidity of ${\zeta }_{\Phi }$ and ${\zeta }_R$ on any screen of the form $S_{\sigma }$, where $\sigma >{\sigma }_0$ and where $\sigma \notin {\left\{{\sigma }_j\right\}}_{j=1}^K$. When choosing a screen $S_{{\sigma }_0+\epsilon }$, taking $\epsilon$ with $0<\epsilon <{min}_{j=1,...,K}\left(\right|{\sigma }_j-{\sigma }_0\left|\right)$ is sufficient. □

3.3. Explicit Formulae from Scaling Functional Equations

In the spirit of Proposition 4 in Section 3.1, we will obtain explicit formulae for functions which satisfy a scaling functional equation. Given a scaling operator L, a scaling functional equation is a relation of the form $f=L\left[f\right]+R$, where R is a remainder term. We obtain formulae which are valid up to an asymptotic order determined by estimates of an admissible remainder term (in the sense of Definition 10 in Section 3.2). We begin with the statement of our results, followed by the discussion and estimates needed for their proof.

We focus on scaling functional equations which are induced by a self-similar system on functions which satisfy certain scaling laws (see Proposition 4), as these are the main type of scaling functional equations that we will need for our application to heat content in Section 4. Explicitly, let $\Phi$ be a self-similar system, let $\alpha >0$, and define the scaling operator $L_{\Phi }^{\alpha }:={\sum }_{\phi \in \Phi }{M}_{{\lambda }_{\phi }^{\alpha }}$. The scaling functional equations we study herein will be stated in terms of such scaling operators. Note, though, that any general scaling operator with scaling ratios in $(0,1)$, each having positive, integral multiplicity can be written in this form.

There are two types of explicit formulae, namely, pointwise and distributional (just as in [18,19,41]). The former have the advantage of being simpler in their statements, but the disadvantage that they may only be valid for antiderivatives of the function satisfying the scaling functional equation and not the function itself. The distributional explicit formulae we obtain do not have this restriction, but they will require additional terminology to state and interpret in a weak formulation.

3.3.1. Pointwise Explicit Formulae

The pointwise explicit formulae we establish in this work for solutions to a scaling functional equation, say, $f\in C^0\left(\mathbb{R}\right)$, will be valid for its antiderivatives. To that end, we introduce the following notation and convention. Firstly, define $f^{\left[0\right]}:=f$. For any integer $k>0$, we define $f^{\left[k\right]}$ recursively by integrating the previous antiderivative and imposing the convention that $f^{\left[k\right]}\left(0\right)=0$. Namely, for $k>0$,

$$f^{\left[k\right]}\left(t\right):={\int }_0^t{f}^{[k-1]}\left(\tau \right)d\tau .$$

We will also denote ${\zeta }_f(s;\delta )={\mathcal{M}}^{\delta }\left[f\right]\left(s\right)$. In our applications, typically we have some function $F\left(t\right)$ of interest which must be normalized in order to satisfy a scaling law in the sense of Definition 7 in Section 3.1, so we will have that $f\left(t\right)=t^{-\beta }F\left(t\right)$ for a given real parameter $\beta$. For tube functions (cf. [21]), $\beta$ is the dimension N and for the heat content in Section 4, it will be $N/2$. When we state the explicit formula, we will do so in terms of the parameter $\beta$, noting that typically, $\beta =N$. Let $F^{\left[k\right]}$ be defined in the same manner as $f^{\left[k\right]}$, $k\ge 0$.

As an additional preliminary to stating the result, we define the Pochhammer symbol ${\left(z\right)}_w:=\Gamma (z+w)/\Gamma \left(w\right)$ for $z,w\in \mathbb{C}$. Note that when $w=k$ is a positive integer, this simplifies to ${\left(z\right)}_k=z(z+1)\cdots (z+k-1)$ or, when $w=0$, to ${\left(z\right)}_0=1$.

Lastly, we must give meaning to the summations which will appear in what follows. Let $\mathcal{D}\subset \mathbb{C}$ be a discrete (and, hence, at most countably infinite) subset with the property that for any $m\in \mathbb{N}$,

$${\mathcal{D}}_m:=\left\{\omega \in \mathcal{D}\left|\right|\Im \left(\omega \right)|\le m\right\}<\infty .$$

Then, we define

$$\sum _{\omega \in \mathcal{D}}{a}_{\omega }:=\underset{m\to \infty }{lim}\sum _{\omega \in {\mathcal{D}}_m}{a}_{\omega },$$

when the limit exists. Note that this summation is a symmetric limit of finite partial sums. That is to say, it is a sum over the elements of $\mathcal{D}$ with increasing large absolute values of imaginary parts, but with the upper and lower bounds taken at the same rate. As such, the convergence of these sums is comparatively more delicate; it may be that the limit, when taken independently, does not exist.

Theorem 3

(Pointwise Explicit Formula). Let Φ be a self-similar system, $\alpha >0$, $\beta \in \mathbb{R}$, and $f\left(t\right)=t^{-\beta /\alpha }F\left(t\right)\in C^0\left({\mathbb{R}}^{+}\right)$. Suppose that Φ induces the scaling functional equation $f=L_{\Phi }^{\alpha }\left[f\right]+R$ on $[0,\delta ]$, with admissible remainder term R (in the sense of Definition 10 in Section 3.2) and with corresponding screen S. Let ${\sigma }_R$ denote the abscissa of absolute convergence of ${\zeta }_R$, $D={dim}_S(\Phi )$ the (upper) similarity dimension of Φ, and suppose $(\beta /\alpha )+1>max(D/\alpha ,{\sigma }_R)$. Lastly, suppose that S is contained in the half-plane ${\mathbb{H}}_{{\sigma }_R}:=\left\{s\in \mathbb{C}|\Re \left(s\right)>{\sigma }_R\right\}$ and write $W_S$ for the window to the right of S.

Then, for every integer $k\ge 2$ and every $t\in (0,\delta )$, we have that

$$F^{\left[k\right]}\left(t\right)=\sum _{\omega \in {\mathcal{D}}_{\Phi }\left(\alpha W_S\right)}Res\left({\displaystyle \frac{t^{(\beta -s)/\alpha +k}}{{((\beta -s)/\alpha +1)}_k}}{\zeta }_f(s/\alpha ;\delta );\omega \right)+{\mathcal{R}}^k\left(t\right),$$

where ${\zeta }_f$ is given by (22) in Section 3.3.3. Moreover, the error term satisfies

$${\mathcal{R}}^k\left(t\right)=O\left(t^{(\beta /\alpha )-sup\left(S\right)+k}\right)$$

as $t\to 0^{+}$.

Note, in particular, that the order of the remainder directly controls the error of the approximation. As we are considering small values of t, the larger the exponent, the better the remainder estimate. The restriction of $k\ge 2$ is related to the exponent $\kappa =0$ of the languid growth of ${\zeta }_{\Phi }$, and, consequently, of ${\zeta }_f$.

3.3.2. Distributional Explicit Formulae

Put simply, a distribution is a continuous, linear functional on a prescribed space of functions called test functions. A standard distribution is a functional on the space $C_c^{\infty }$ of smooth, compactly supported functions on a given domain. The type of distributions we consider here, called tempered distributions, will be a subspace of these distributions.

We define as the space of test functions the class of Schwartz functions, or functions of rapid decrease. For a finite domain $(0,\delta )$, just as in ([18], Chapter 5) and ([19], Chapter 5), the space of test functions on $(0,\delta )$ is defined by

$$\mathcal{S}(0,\delta ):=\left\{\psi \in C^{\infty }(0,\delta )|\begin{array}{c}\forall m\in \mathbb{Z},\forall q\in \mathbb{N},\mathrm{as}t\to 0^{+},\hfill \\ t^m{\psi }^{\left(q\right)}\left(t\right)\to 0\mathrm{and}{(t-\delta )}^m{\psi }^{\left(q\right)}\left(t\right)\to 0\hfill \end{array}\right\}.$$

(14)

Note that $C_c^{\infty }(0,\delta )\subseteq \mathcal{S}(0,\delta )$, with a continuous embedding, whence by duality, the space ${\mathcal{S}}^{\prime }(0,\delta )$ of continuous, linear functionals on $\mathcal{S}(0,\delta )$ is a subset of the usual space of distributions, the dual of $C_c^{\infty }(0,\delta )$, and is called the space of tempered distributions on $(0,\delta )$. Typically, for a distribution F and a test function $\psi$, we write $F\left(\psi \right)=F,\psi$, denoting by $\langle ·,·\rangle$ the natural pairing of $\mathcal{S}(0,\delta )$ with its dual space.

In this distributional setting, the restriction of $k\ge 2$ may be relaxed in Theorem 3. However, equality must be interpreted in the sense of distributions: We say that $F=G$ in the sense of distributions if for any test function $\psi \in \mathcal{S}(0,\delta )$, $\langle F,\psi \rangle =\langle G,\psi \rangle$. In addition to the standard definitions for the differentiation and integration of distributions, we note the following relevant identity regarding residues at a point:

$$\langle Res(t^{\beta }G\left(s\right);\omega ),\psi \rangle =Res(\mathcal{M}\left[\psi \right](\beta +1)G\left(s\right);\omega ).$$

(15)

See ([19], Equation (5.2.5)), as well as the preceding discussion regarding the extension of $\psi$ to all of ${\mathbb{R}}^{+}$.

Secondly, we provide the definition for a distribution $\mathcal{R}$ to satisfy the error estimate $O\left(t^{\alpha }\right)$ as $t\to 0^{+}$ (defined as in ([19], Theorem 5.2.11) and similarly in [18,41]). Given a test function $\psi \in \mathcal{S}(0,\delta )$ and $a>0$, define the new test function ${\psi }_a\left(t\right):=a^{-1}\psi (t/a)$. We say that $\mathcal{R}\left(t\right)=O\left(t^{\alpha }\right)$ as $t\to 0^{+}$ if for all $a>0$ and for all $\psi \in \mathcal{S}(0,\delta )$,

$$\langle \mathcal{R},{\psi }_a\left(t\right)\rangle =O\left(a^{\alpha }\right),$$

(16)

as $t\to 0^{+}$, in the usual sense.

Theorem 4

(Distributional Explicit Formula). Let Φ be a self-similar system, $\alpha >0$, $\beta \in \mathbb{R}$, and $f\left(t\right)=t^{-\beta /\alpha }F\left(t\right)\in C^0\left({\mathbb{R}}^{+}\right)$. Suppose that Φ induces the scaling functional equation $f=L_{\Phi }^{\alpha }\left[f\right]+R$ on $[0,\delta ]$ with admissible remainder term R (in the sense of Definition 10) with corresponding screen S. Let ${\sigma }_R$ denote the abscissa of absolute convergence of ${\zeta }_R$ and $D={dim}_S(\Phi )$ the (upper) similarity dimension of Φ, and suppose $(\beta /\alpha )+1>max(D/\alpha ,{\sigma }_R)$. Lastly, suppose that S is contained in the half-plane ${\mathbb{H}}_{{\sigma }_R}$ and write $W_S$ for the window to the right of S.

Then, for any $k\in \mathbb{Z}$, we have that, in the sense of distributions, $F^{\left[k\right]}$ satisfies

$$\begin{array}{cc}\hfill F^{\left[k\right]}\left(t\right)& =\sum _{\omega \in {\mathcal{D}}_{\Phi }\left(\alpha W_S\right)}Res\left({\displaystyle \frac{t^{(\beta -s)/\alpha +k}}{{((\beta -s)/\alpha +1)}_k}}{\zeta }_f(s/\alpha ;\delta );\omega \right)+{\mathcal{R}}^{\left[k\right]}\left(t\right),\hfill \end{array}$$

(17)

as $t\to 0^{+}$ for a remainder distribution term ${\mathcal{R}}^{\left[k\right]}$. See (18) for the explicit identity of action on test functions. Here, ${\zeta }_f$ is as in Corollary 4 in Section 3.3.3. Moreover, the distributional remainder term satisfies the estimate

$${\mathcal{R}}^{\left[k\right]}\left(t\right)=O\left(t^{(\beta /\alpha )-sup\left(S\right)+k}\right)$$

as $t\to 0^{+}$, in the sense of (16).

Most precisely, (17) means that for any $\psi \in \mathcal{S}(0,\delta )$,

$$\begin{array}{cc}\hfill \langle F^{\left[k\right]},\psi \rangle & =\sum _{\omega \in {\mathcal{D}}_{\Phi }\left(\alpha W_S\right)}Res\left({\displaystyle \frac{\mathcal{M}\left[\psi \right]\left((\beta -s)/\alpha +k+1\right)}{{((\beta -s)/\alpha +1)}_k}}{\zeta }_f(s/\alpha );\omega \right)+\langle {\mathcal{R}}^{\left[k\right]},\psi \rangle .\hfill \end{array}$$

(18)

While this formulation is less direct compared to the pointwise expansion, it does have the advantage of requiring less regularity to leverage the expansion. Namely, it is valid when $k=0$, yielding a formula (in the sense of distributions) for the function f itself.

Corollary 3

(Distributional Explicit Formula, $k=0$). Let Φ be a self-similar system, $\alpha >0$, $\beta \in \mathbb{R}$, and $f\left(t\right)=t^{-\beta /\alpha }F\left(t\right)\in C^0\left({\mathbb{R}}^{+}\right)$. Suppose that Φ induces the scaling functional equation $f=L_{\Phi }^{\alpha }\left[f\right]+R$ on $[0,\delta ]$ with admissible remainder term R (in the sense of Definition 10) with corresponding screen S. Let ${\sigma }_R$ denote the abscissa of absolute convergence of ${\zeta }_R$ and $D={dim}_S(\Phi )$ the (upper) similarity dimension of Φ, and suppose $(\beta /\alpha )+1>max(D/\alpha ,{\sigma }_R)$. Lastly, suppose that S is contained in the half-plane ${\mathbb{H}}_{{\sigma }_R}$ and write $W_S$ for the window to the right of S.

Then, in the sense of distributions, we have that for $t\in [0,\delta )$,

$$F\left(t\right)=\sum _{\omega \in {\mathcal{D}}_{\Phi }\left(\alpha W_S\right)}Res\left(t^{(\beta -s)/\alpha }{\zeta }_f(s/\alpha ;\delta );\omega \right)+\mathcal{R}\left(t\right).$$

(19)

The distributional remainder term satisfies the estimate $\mathcal{R}\left(t\right)=O\left(t^{(\beta /\alpha )-sup\left(S\right)}\right)$ as $t\to 0^{+}$, in the sense of (16).

If we further assume that the poles of ${\zeta }_{\Phi }$ in $W_S$ are simple, then (19) simplifies to

$$f\left(t\right)=\sum _{\omega \in {\mathcal{D}}_{\Phi }\left(\alpha W_S\right)}Res({\zeta }_f(s/\alpha ;\delta );\omega )t^{(\beta -\omega )/\alpha }+\mathcal{R}\left(t\right).$$

In this formulation, and in particular, in the case when the self-similar system induces simple poles of its zeta function, the formula is a simple expansion with constant coefficients and powers determined by the poles $\omega$. The computation of the residues, however, is not a trivial matter and beyond the current scope of this work. Corollary 3 is a direct application of Theorem 4.

3.3.3. Formulae for the Zeta Functions

We will discuss the solution of scaling functional equations by means of truncated Mellin transforms (see Appendix B). In what follows, let $\delta >0$ be fixed and let ${\mathcal{M}}^{\delta }$ be the truncated Mellin transform on $(0,\delta )$. Given a function f which is locally (Lebesgue) integrable on $(0,\delta )$, define ${\zeta }_f(s;\delta ):={\mathcal{M}}^{\delta }\left[f\right]\left(s\right)$ for all $s\in \mathbb{C}$ for which the integral is convergent. We say that ${\zeta }_f(s;\delta )$ is a tamed Dirichlet-type integral (viz. ([19], Definition A.1.2)) if $d\mu \left(t\right)=f\left(t\right)dt$ is a local complex measure (i.e., its restriction to any compact subset is a complex measure) on $[0,\delta ]$ and if there exists $\sigma \in \mathbb{R}$ such that the defining integral is convergent (and, hence, holomorphic) on ${\mathbb{H}}_{\sigma }$. See Appendix B for more information. For example, if f is integrable, bounded away from 0, and $f\left(t\right)=O\left(t^{-\sigma }\right)$ as $t\to 0^{+}$, then ${\zeta }_f$ is holomorphic on ${\mathbb{H}}_{\sigma }$ (viz. Lemma A1).

Given a scaling operator $L:={\sum }_{k=1}^m{a}_k{M}_{{\lambda }_k}$, define

$${\xi }_{L,f}(s;\delta ):=\sum _{k=1}^m{a}_k{\lambda }_k^s{\mathcal{M}}_{\delta }^{\delta /{\lambda }_k}\left[f\right]\left(s\right).$$

(20)

If $\Phi$ is a self-similar system, then define ${\xi }_{\Phi ,f}:={\xi }_{L_{\Phi },f}$, where $L_{\Phi }$ is the scaling operator associated to $\Phi$. Lastly, we recall that ${\zeta }_L$ is the scaling zeta function of L, with singular set ${\mathcal{D}}_L$ (see Section 3.1).

The following is essentially ([21], Theorem 4.8), although we take the opportunity to state the theorem with slightly more precise and slightly more general hypotheses.

Theorem 5

(Zeta Functions of Solutions to General Scaling Functional Equations). Let $L:={\sum }_{k=1}^m{a}_k{M}_{{\lambda }_k}$ be a scaling operator and $\Lambda =max{\left\{{\lambda }_k^{-1}\right\}}_{k=1}^m$. Fixing $\delta >0$, let f and R be integrable functions on $(0,\Lambda \delta )$ and $(0,\delta )$, respectively, and suppose that ${\zeta }_f(s;\Lambda \delta ):={\mathcal{M}}^{\Lambda \delta }\left[f\right]\left(s\right)$ and ${\zeta }_R(s;\delta ):={\mathcal{M}}^{\delta }\left[R\right]\left(s\right)$ are tamed Dirichlet-type integrals (viz. ([19], Definition A.1.2)), with respective abscissae of absolute convergence ${\sigma }_f$ and ${\sigma }_R$, respectively.

Suppose that $L:={\sum }_{k=1}^m{a}_k{M}_{{\lambda }_k}$ is a scaling operator for which f satisfies the scaling functional equation $f=L\left[f\right]+R$ for all $t\in (0,\delta )$. Then, for all $s\in {\mathbb{H}}_{{\sigma }_R}\setminus {\mathcal{D}}_L\left(\mathbb{C}\right)$,

$${\zeta }_f(s;\delta )={\zeta }_L\left(s\right)({\xi }_{L,f}(s;\delta )+{\zeta }_R(s;\delta )),$$

(21)

where ${\xi }_{L,f}$, defined as in (20), is an entire function. Further, ${\zeta }_f$ is holomorphic in the half-plane ${\mathbb{H}}_{max(D,{\sigma }_R)}$, where D is the unique real pole of ${\zeta }_L$, and admits a meromorphic continuation to ${\mathbb{H}}_{{\sigma }_R}$ with any poles contained in ${\mathcal{D}}_L\left({\mathbb{H}}_{{\sigma }_R}\right)$. Moreover, these poles are independent of the choice of δ, i.e., ${\zeta }_f(s;\delta )$ and ${\zeta }_f(s;{\delta }^{\prime })$ have the same poles for any ${\delta }^{\prime }\in (0,\delta )$.

Proof. 

As tamed Dirichlet-type integrals, there exist ${\sigma }_f,{\sigma }_R\in [-\infty ,\infty )$ for which ${\zeta }_f(s;\Lambda \delta )$ and ${\zeta }_R(s;\delta )$ are holomorphic in ${\mathbb{H}}_{{\sigma }_f}$ and ${\mathbb{H}}_{{\sigma }_R}$, respectively, with the convention that ${\mathbb{H}}_{-\infty }=\mathbb{C}$; see ([19], Appendix A). Consequently, they are both holomorphic in ${\mathbb{H}}_{{\sigma }_M}$ where ${\sigma }_M:=max({\sigma }_f,{\sigma }_R)$. Additionally, we have that by Lemma A2, for each $M_{{\lambda }_k}$,

$${\mathcal{M}}^{\delta }\left[M_{{\lambda }_k}\left[f\right]\right]\left(s\right)={\lambda }_k^s{\mathcal{M}}^{\delta /{\lambda }_k}\left[f\right]\left(s\right).$$

Note that each function ${\mathcal{M}}^{\delta /{\lambda }_k}\left[f\right]$, $k=1,\dots ,m$, is holomorphic in ${\mathbb{H}}_{{\sigma }_f}$ since ${\zeta }_f(s;\Lambda \delta )$ is assumed to be a tamed Dirichlet-type integral for the largest interval of the form $(0,\delta /{\lambda }_k)$, $k=1,\dots ,m$. (It follows from Lemma A1 that the difference between ${\zeta }_f(s;{\delta }_1)$ and ${\zeta }_f(s;{\delta }_2)$ is an entire function for any ${\delta }_1,{\delta }_2\in (0,\Lambda \delta )$, so integration over a subset yields a function with the same holomorphicity properties.)

With these holomorphicity properties, the proof of ([21], Theorem 4.8) is applicable without further modification, noting that $E\left(s\right)={\xi }_{L,f}(s;\delta )$ and ${\sigma }_0={\sigma }_R$. The independence of the poles of ${\zeta }_f$ on the value of $\delta$ may be seen as a corollary of Lemma A1. □

We now specialize to operators of the form $L_{\Phi }^{\alpha }:={\sum }_{\phi \in \Phi }{M}_{{\lambda }_{\phi }^{\alpha }}$, where $\Phi$ is a self-similar system and $\alpha >0$. We require that $\alpha >0$, so that ${\lambda }_{\phi }^{\alpha }\in (0,1)$ for each $\phi \in \Phi$. When $\alpha =1$, this is exactly the scaling operator $L_{\Phi }$ associated to $\Phi$. For other values of $\alpha$, the effect on the associated zeta function is simply a rescaling of the input relative to ${\zeta }_{\Phi }$, namely, ${\zeta }_{L_{\Phi }^{\alpha }}\left(s\right)={\zeta }_{\Phi }\left(\alpha s\right)$. (This is an immediate consequence of the definition of these functions and elementary properties of exponents.) Note that it follows that if $D={dim}_S(\Phi )$ is the similarity dimension of ${\zeta }_{\Phi }$, an upper bound for the real parts of its poles by Proposition 3 in Section 3.1.1, then ${\zeta }_{L_{\Phi }^{\alpha }}$ is holomorphic in the half plane ${\mathbb{H}}_{D/\alpha }$. Further, the sets of poles are directly related: Any pole $\omega \in {\mathcal{D}}_{\Phi }$ exactly corresponds to the pole $\omega /\alpha \in {\mathcal{D}}_{L_{\Phi }^{\alpha }}$. These observations together yield the following corollary of ([21], Theorem 4.8).

Corollary 4

(Zeta Functions of Solutions to Self-Similar Scaling Functional Equations). Let $f,R\in C^0\left({\mathbb{R}}^{+}\right)$ and suppose that there exist ${\sigma }_R,{\sigma }_f\in \mathbb{R}$ such that $R\left(t\right)=O\left(t^{-{\sigma }_R}\right)$ and $f\left(t\right)=O\left(t^{-{\sigma }_f}\right)$ as $t\to 0^{+}$. Let Φ be a self-similar system, let $\alpha >0$, and let $L_{\Phi }^{\alpha }:={\sum }_{\phi \in \Phi }{M}_{{\lambda }_{\phi }^{\alpha }}$. If f satisfies the scaling functional equation $f=L_{\Phi }^{\alpha }\left[f\right]+R$ for all $t\in (0,\delta )$, then

$${\zeta }_f(s;\delta )={\zeta }_{\Phi }\left(\alpha s\right)({\xi }_{L_{\Phi }^{\alpha },f}(s;\delta )+{\zeta }_R(s;\delta ))$$

(22)

and ${\zeta }_f(s;\delta )$ holomorphic in ${\mathbb{H}}_{max(D/\alpha ,{\sigma }_R)}$, where $D={dim}_S(\Phi )$. Further, ${\zeta }_f(s;\delta )$ is meromorphic in ${\mathbb{H}}_{{\sigma }_R}$, it has poles in a subset of ${\alpha }^{-1}{\mathcal{D}}_{\Phi }\left({\mathbb{H}}_{{\sigma }_R}\right)$, and these poles are independent of δ.

We note that the assumptions on f and R are sufficient to ensure that ${\zeta }_f$ and ${\zeta }_R$ are tamed Dirichlet-type integrals (viz. as a corollary of Lemma A1), as they are holomorphic in the half-planes ${\mathbb{H}}_{{\sigma }_f}$ and ${\mathbb{H}}_{{\sigma }_R}$, respectively. These values of ${\sigma }_f$ and ${\sigma }_R$ give upper bounds for the abscissa of absolute convergence of ${\zeta }_f$ and ${\zeta }_R$, respectively. So long as ${\sigma }_f$ and ${\sigma }_R$ are chosen to be the minimal such exponents for which these estimates hold, they will equal their respective abscissae of absolute convergence. (See Appendix B for more detail.)

When $L=L_{\Phi }^{\alpha }$, we note that (20) is explicitly the entire function

$${\xi }_{L_{\Phi }^{\alpha },f}(s;\delta )=\sum _{\phi \in \Phi }{\lambda }_{\phi }^{\alpha s}{\mathcal{M}}_{\delta }^{\delta /{\lambda }_{\phi }^{\alpha }}\left[f\right]\left(s\right).$$

(23)

In this setting, it will be convenient to state the estimate for R as $t\to 0^{+}$ in the form $R\left(t\right)=O\left(t^{-{\sigma }_0/\alpha }\right)$, where ${\sigma }_R={\sigma }_0/\alpha$. Under the change of variables $s^{\prime }=s/\alpha$, a pole ${\omega }^{\prime }=\omega /\alpha \in {\alpha }^{-1}{\mathcal{D}}_{\Phi }\left({\mathbb{H}}_{{\sigma }_R}\right)$ of ${\zeta }_f(s^{\prime };\delta )$ corresponds to the pole $\omega \in {\mathcal{D}}_{\Phi }\left({\mathbb{H}}_{{\sigma }_0}\right)$ of ${\zeta }_f(s/\alpha ;\delta )$. Thus, in Section 3.3 when we consider poles of ${\zeta }_f$ in ${\mathbb{H}}_{{\sigma }_R}$, under these conventions, we will be considering the poles of ${\zeta }_{\Phi }$ in ${\mathbb{H}}_{{\sigma }_0}$. In Section 4.4, we will see that this change in variables ultimately clarifies the direct connection of spectral complex dimensions (regarding heat content and heat zeta functions) to the geometric complex dimensions (defined by fractal zeta functions such as the tube zeta function).

The next step for solving the scaling functional equation $f=L_{\Phi }^{\alpha }\left[f\right]+R$ is to apply the Mellin inversion theorem to (22), which is the content of ([21], Theorem 4.9). Note the convention regarding the contour integral, in that

$${\int }_{c-i\infty }^{c+i\infty }f\left(z\right)dz:=\underset{T\to +\infty }{lim}{\int }_{c-iT}^{c+iT}f\left(z\right)dz,$$

which may also be considered a principal value integral (e.g., as in [43]). We state the formula using the notation of Corollary 4 in Section 3.3.3 (i.e., in terms of ${\zeta }_{\Phi }$, ${\zeta }_R$, and ${\xi }_{L_{\Phi }^{\alpha },f}$) here as a proposition for later reference.

Proposition 5

(Mellin Inversion Formula). Let $f,R\in C^0\left({\mathbb{R}}^{+}\right)$. Let Φ be a self-similar system, let $\alpha >0$, and let $L_{\Phi }^{\alpha }:={\sum }_{\phi \in \Phi }{M}_{{\lambda }_{\phi }^{\alpha }}$. Suppose that f satisfies the scaling functional equation $f=L_{\Phi }^{\alpha }\left[f\right]+R$ for all $t\in [0,\delta ]$ and that ${\zeta }_R(s;\delta )={\mathcal{M}}^{\delta }\left[R\right]\left(s\right)$ is holomorphic in ${\mathbb{H}}_{{\sigma }_R}$. Lastly, let $D={dim}_S(\Phi )$ and $c>max(D/\alpha ,{\sigma }_R)$. Then, for any $t\in (0,\delta )$, f is given by:

$$\begin{array}{cc}\hfill f\left(t\right)& ={\mathcal{M}}^{-1}\left[{\zeta }_{\Phi }\left(\alpha s\right)({\xi }_{L_{\Phi }^{\alpha },f}(s;\delta )+{\zeta }_R(s;\delta ))\right]\left(t\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi i}}{\int }_{c-i\infty }^{c+i\infty }{t}^{-s}{\zeta }_{\Phi }\left(\alpha s\right)({\xi }_{L_{\Phi }^{\alpha },f}(s;\delta )+{\zeta }_R(s;\delta ))ds.\hfill \end{array}$$

It remains to evaluate this contour integral by means of the residue theorem. Strictly speaking, it requires a sort of unbounded residue theorem, using a sequence of regions for which the integrand has finitely many poles (on which the residue theorem applies) that converge to the region containing all of the countably many singularities. The conditions of languidity are necessary in this process, used to estimate contour integrals over pieces of the boundaries of regions involved in this process. This method developed to establish explicit formulae for fractal tube functions in ([18], Chapters 5 and 8) and then, with refinements to the proof, used to prove explicit formulae for relative tube functions in ([19], Chapter 5).

3.3.4. Languidity from Scaling Functional Equations

We will need growth estimates for ${\zeta }_f(s;\delta )={\mathcal{M}}^{\delta }\left[f\right]\left(s\right)$ when f is the solution of a scaling functional equation. In particular, using the formula in Corollary 4 in Section 3.3.3, we show that ${\zeta }_f$ is languid provided some control over the remainder term itself. The definition of languidity and its relevant hypotheses are provided in Appendix C.

Theorem 6

(Languidity of Zeta Functions from Scaling Functional Equations). Let Φ be a self-similar system, let $\alpha >0$, and $f\in C^0\left({\mathbb{R}}^{+}\right)$. Suppose that Φ induces the scaling functional equation $f=L_{\Phi }^{\alpha }\left[f\right]+R$ on $[0,\delta ]$ with admissible remainder term R for $L_{\Phi }^{\alpha }$ (in the sense of Definition 10) and let S be an admissible screen.

Then the zeta function ${\zeta }_f(s;\delta )={\mathcal{M}}^{\delta }\left[f\right]\left(s\right)$ is also languid with exponent $\kappa =0$ with respect to S.

Proof. 

Because R is an admissible remainder for $L_{\Phi }^{\alpha }$ with admissible screen S, we have that ${\zeta }_{\Phi }\left(\alpha s\right)$ and ${\zeta }_R(s;\delta )$ are jointly languid (with exponent $\kappa =0$) with respect to this screen S. We have that ${\xi }_{L_{\Phi }^{\alpha },f}$ is an entire function by Corollary 4 and explicitly we show that ${\xi }_{L_{\Phi }^{\alpha },f}$ is bounded on any vertical strip of the form ${\mathbb{H}}_a^b=\left\{s\in \mathbb{C}|a<\Re (s)<b\right\}$.

In order to do so, it suffices to find positive constants $C_{\xi },A_{\xi }$ such that

$$|{\xi }_{L_{\Phi }^{\alpha },f}(s;\delta )|\le C_{\xi }{A}_{\xi }^{|\Re (s\left)\right|},$$

as this function is bounded when $|\Re (s\left)\right|$ is bounded as is the case with vertical strips.

To that end, let $\sigma =\Re \left(s\right)$ and define

$$\begin{array}{cc}\hfill {\Lambda }_{+}=& max\left({\left\{{\lambda }_{\phi }^{\alpha }\right\}}_{\phi \in \Phi }\right);\hfill \\ \hfill {\Lambda }_{-}=& min\left({\left\{{\lambda }_{\phi }^{\alpha }\right\}}_{\phi \in \Phi }\right).\hfill \end{array}$$

Then, we have that

$$\begin{array}{cc}\hfill |{\xi }_{L_{\Phi }^{\alpha },f}(s;\delta )|& \le \sum _{\phi \in \Phi }{\lambda }_{\phi }^{\alpha \sigma }\left|{\mathcal{M}}_{\delta }^{\delta /{\lambda }_{\phi }^{\alpha }}\left[f\right]\left(s\right)\right|\hfill \\ \hfill & \le max({\Lambda }_{+}^{\sigma },{\Lambda }_{-}^{\sigma })\sum _{\phi \in \Phi }\left|{\mathcal{M}}_{\delta }^{\delta /{\lambda }_{\phi }^{\alpha }}\left[f\right]\left(s\right)\right|.\hfill \end{array}$$

By Corollary A1, we know that each Mellin transform ${\mathcal{M}}_{\delta }^{\delta /{\lambda }_{\phi }^{\alpha }}\left[f\right]$ is bounded on any vertical strip of the form ${\mathbb{H}}_a^b$. It follows that a finite sum of such functions is bounded on any such vertical strip by the sum of these constants, which we denote by $C_{\xi }$. Choosing $A_{\xi }=max({\Lambda }_{-},{\Lambda }_{+},{\Lambda }_{-}^{-1},{\Lambda }_{+}^{-1})$ allows us to write that $A_{\xi }^{\left|\sigma \right|}\ge max({\Lambda }_{+}^{\sigma },{\Lambda }_{-}^{\sigma })$, thus obtaining the desired bound. It follows that $|{\xi }_{L_{\Phi }^{\alpha },f}|$ is bounded in any vertical strip.

Now, let ${\left\{{\tau }_n\right\}}_{n\in \mathbb{Z}}$ be a sequence of admissible heights shared jointly by ${\zeta }_{\Phi }\left(\alpha s\right)$ and ${\zeta }_R(s;\delta )$, which exists by fiat. We have that each of these respective functions is uniformly bounded on all of the horizontal contours $I_n=[S\left(\tau \right)+i{\tau }_n,c+i{\tau }_n]$, for some particular c sufficiently large so that ${\zeta }_{\Phi }\left(\alpha s\right)$ and ${\zeta }_R(s;\delta )$ are holomorphic in ${\mathbb{H}}_c$. Since ${\xi }_{L_{\Phi }^{\alpha },f}$ is bounded on the strip ${\mathbb{H}}_{inf\left(S\right)}^c$, it follows that it is uniformly bounded on all of these intervals $I_n\subset {\mathbb{H}}_{inf\left(S\right)}^c$. It follows that ${\zeta }_f(s;\delta )={\zeta }_{\Phi }\left(\alpha s\right)({\xi }_{\Phi ,f}(s;\delta )+{\zeta }_R(s;\delta ))$ is uniformly bounded on these intervals. This establishes hypothesis L1 for the screen S.

For hypothesis L2, we note that ${\zeta }_R(s;\delta )$ and ${\zeta }_{\Phi }\left(\alpha s\right)$ are bounded on S by assumption of their joint languidity. Because S is contained in the strip ${\mathbb{H}}_{inf\left(S\right)}^{sup\left(S\right)}$ (noting that screens by definition correspond to the image of a bounded, Lipschitz continuous function), we have that ${\xi }_{L_{\Phi }^{\alpha },f}$ is bounded on S. The boundedness of ${\zeta }_f$ on this screen thus follows. Thus, ${\zeta }_f$ is languid with respect to the same screen S (with exponent $\kappa =0$). □

3.3.5. Proof of the Explicit Formulae

With the languid growth conditions of ${\zeta }_f$ having been established, we may now complete the proofs of the main theorems, Theorems 3 and 4 in Section 3.3. We shall treat the proofs of these theorems together as they require similar setup and estimates. The main differences come in the application of different theorems to provide the relevant result. Starting from the result of Proposition 5, we will compute the resulting contour integral by the methods in ([19], Theorem 5.1.11) and ([19], Theorem 5.2.2) for the pointwise and distributional explicit formula, respectively.

Proof. 

Fix c such that $(\beta /\alpha )+1>c>max(D/\alpha ,{\sigma }_R)$. By Proposition 5, we have that for any $t\in (0,\delta )$,

$$\begin{array}{cc}\hfill f\left(t\right)& ={\displaystyle \frac{1}{2\pi i}}{\int }_{c-i\infty }^{c+i\infty }{t}^{-s}{\zeta }_{\Phi }\left(\alpha s\right)({\xi }_{L_{\Phi }^{\alpha },f}(s;\delta )+{\zeta }_R(s;\delta ))ds,\hfill \end{array}$$

where ${\zeta }_f$, ${\xi }_{L_{\Phi }^{\alpha },f}$, and ${\zeta }_R$ are as in Corollary 4 in Section 3.3.3. By Theorem 6 in Section 3.3.4, we have that ${\zeta }_f$ is languid with exponent $\kappa =0$ on the admissible screen S.

Noting that $F\left(t\right)=t^{\beta /\alpha }f\left(t\right)$, we have that

$$\begin{array}{cc}\hfill F\left(t\right)& ={\displaystyle \frac{1}{2\pi i}}{\int }_{c-i\infty }^{c+i\infty }{t}^{(\beta /\alpha )-s}{\zeta }_{\Phi }\left(\alpha s\right)({\xi }_{L_{\Phi }^{\alpha },f}(s;\delta )+{\zeta }_R(s;\delta ))ds.\hfill \end{array}$$

It is this expression that is analogous to ([19], Equation (5.1.18)), but with $\beta$ in place of N, $\alpha =1$, and ${\zeta }_f(s;\delta )$ in place of the relative tube zeta function ${\tilde{\zeta }}_{A,\Omega }(s;\delta )$. Under these conditions, ([19], Theorem 5.1.11) and ([19], Theorem 5.2.2), along with their proofs, are essentially applicable with only minor modifications and justifications required in the general case.

Firstly, ([19], Proposition 5.1.8) is applicable, where the interchange of the order of integration may be justified by the explicit formula for ${\zeta }_f$ (cf. Corollary 4 in Section 3.3.3) and explicit bounds for the constituent functions. The uniform estimates for ${\zeta }_R$ and ${\xi }_{L_{\Phi }^{\alpha },f}$ follow from Corollary A1.

For an estimate of ${\zeta }_{\Phi }$, first define $p\left(t\right):={\sum }_{\phi \in \Phi }{\lambda }_{\phi }^t$, for all $t\in \mathbb{R}$. Then, when $\sigma =\Re \left(s\right)>D/\alpha$,

$$\begin{array}{c}\hfill \left|\sum _{\phi \in \Phi }{\lambda }_{\phi }^{\alpha s}\right|\le p\left(\alpha \sigma \right)=\sum _{\phi \in \Phi }{\lambda }_{\phi }^{\alpha \sigma }<p\left(D\right)=1,\end{array}$$

using the fact that the function $p\left(t\right)$ is strictly decreasing, since each ${\lambda }_{\phi }\in (0,1)$. It follows that there is a uniform bound (based on $c>D/\alpha$) for ${\zeta }_{\Phi }\left(\alpha s\right)$ when $\Re \left(s\right)=c$. Put together, we obtain that the function $t^{(\beta /\alpha )-s}{\zeta }_f(s;\delta )$, viewed as a multivariate function with variables t and s, is integrable for $(t,s)\in [0,\delta ]\times (c-i\infty ,c+i\infty )$ provided that $(\beta /\alpha )+1>c>D/\alpha$. The assumption that $(\beta /\alpha )+1>c$ ensures that $t^{(\beta /\alpha )-s}$ is integrable as $t\to 0^{+}$. Next, ([19], Lemma 5.1.10) (with $\kappa =0$) is directly applicable based on the languidity of the function ${\zeta }_f$ (see Theorem 6) followed by ([19], Theorem 5.1.11). This establishes the proof of Theorem 3.

Next, this integrability argument is also necessary to prove Theorem 4, namely, to justify the use of the Fubini–Tonelli theorem in ([19], Equation (5.2.10)) (in the proof of ([19], Theorem 5.5.5)). In this case, because $\psi \in \mathcal{S}(0,\delta )$, we may write the corresponding integral in question as

$$\begin{array}{cc}\hfill & {\int }_{c-i\infty }^{c+i\infty }{\int }_0^{\infty }\psi \left(t\right){\displaystyle \frac{t^{(\beta /\alpha )-s+k}}{{((\beta /\alpha )-s+1)}_k}}{\zeta }_f(s;\delta )dt\hfill \\ \hfill & ={\int }_{c-i\infty }^{c+i\infty }{\int }_0^{\delta }\psi \left(t\right){\displaystyle \frac{t^{(\beta /\alpha )-s+k}}{{((\beta /\alpha )-s+1)}_k}}{\zeta }_f(s;\delta )dt.\hfill \end{array}$$

We claim that once again, the integrand is integrable over the product space, that is, for $(t,s)\in [0,\delta ]\times (c-i\infty ,c+i\infty )$. Firstly, we have the same estimates for ${\zeta }_f$ as before. For the Pochhammer symbol, we may write that

$$\begin{array}{cc}\hfill {|((\beta /\alpha )-s+1)}_k|& \ge \left|\right(\beta /\alpha )-c+1|·\left|\right(\beta /\alpha )-c+2|\cdots \left|\right(\beta /\alpha )-c+k-1|\hfill \\ \hfill & \ge {|(\beta /\alpha )-c+1|}^k,\hfill \end{array}$$

noting that on the vertical lines $z=(\beta /\alpha )-c+k+i\mathbb{R}$, the closest point to the origin occurs exactly on the real axis. The second estimate follows from using the fact that $(\beta /\alpha )-c+1>0$ is the smallest term in the product. This gives a uniform upper bound for ${|((\beta /\alpha )-s+1)}_k^{-1}|$. Lastly, the power of t may be arbitrary since $\psi$ is a function of rapid decrease: For any $\gamma \in \mathbb{R}$, $t^{\gamma }\psi \left(t\right)\to 0$ as $t\to 0^{+}$.

The final step is a minor change in variables. Rather than indexing the sum by the poles of ${\zeta }_f$ in $W_S$, we note that as long as $\Re \left(s\right)>{\sigma }_R$, the abscissa of absolute convergence of ${\zeta }_R$, a pole of ${\zeta }_f$ only occurs at s if $\omega =\alpha s$ is a pole of ${\zeta }_{\Phi }$. In other words, we have that

$$\sum _{s\in {\mathcal{D}}_f\left({\mathbb{H}}_{{\sigma }_R}\right)}g\left(s\right)=\sum _{\omega /\alpha \in {\mathcal{D}}_f\left({\mathbb{H}}_{{\sigma }_R}\right)}g(\omega /\alpha )=\sum _{\omega \in {\mathcal{D}}_{\Phi }\left({\mathbb{H}}_{\alpha {\sigma }_R}\right)}g(\omega /\alpha ).$$

Since we assume that S lies in the half-plane ${\mathbb{H}}_{{\sigma }_R}$, this same argument applies with respect to the window $W_S$ and its scaled transformation $\alpha W_S$. This establishes the proof of Theorem 4. □

4. Heat Content Explicit Formulae

In this section, we apply the general results obtained in Section 3 to produce explicit formulae for the heat content of bounded open subsets of ${\mathbb{R}}^N$ with self-similar fractal boundaries, with the formulae expressed in terms of the possible underlying complex dimensions of the boundary (relative to its interior). To achieve this, we introduce a heat zeta function in Section 4.1, prove that its poles are connected to those of an associated scaling zeta function (and, thus, the possible complex dimensions in [21]), and utilize this zeta function to express the short-time asymptotic expansions for the heat content. Section 4.2 contains the main results in the general setting, and we illustrate our results in the case of generalized von Koch fractal domains in Section 4.3.

4.1. Heat Zeta Functions

Suppose that $\Omega \subset {\mathbb{R}}^N$ is a bounded, self-similar set and let $\Phi$ be a self-similar system with $\Omega$ as its attractor. Suppose that $\Omega \subset {\mathbb{R}}^N$ is an osculating set for $\Phi$ so that $(X,\Omega )$ is an osculant fractal drum (see Definition 3 in Section 2.2.3). Let

$$X=\left(\underset{\phi \in \Phi }{\uplus }\phi [\Omega ]\right)\uplus {\Omega }_R$$

(24)

be a partition of X, where ${\Omega }_R$ is defined as ${\Omega }_R:=X\setminus \left({\cup }_{\phi \in \Phi }\phi [\Omega ]\right)$. Here, ⊎ denotes a disjoint union, and that this union is disjoint follows from the definition of an osculating set.

Our goal is to use the methods of Section 3 to study Problem 1 in Section 1.2 on the region $\Omega$. If the self-similar system $\Phi$ induces a decomposition of the total heat contents (in the sense of Definition 6 in Section 3.1), we can use Corollary 1 in Section 3.1.3 to obtain a scaling functional equation. To this end, we now establish some preliminaries and notation for the application of this theory.

We begin with the notion of a heat zeta function, defined in analogy with the definition for tube zeta functions and their normalized quantities which satisfy a scaling law. We define it for the general problem on a cylindrical set and, in the case of Problem 1, we say that the heat zeta function is a parabolic (heat) zeta function owing to the nature of the solution $u_{\Omega }$ as a parabolic average at the points of the given set $\Omega$.

Definition 11

(Heat Zeta Function). Let $u_{\Omega }(x,t)$ be the PWB solution to Problem 2 in Section 2.1 on the open set $\Omega \subset {\mathbb{R}}^{N+1}$ with a resolutive boundary function $F=f{\mathbb{1}}_{t=0}+g{\mathbb{1}}_{x\in \partial \Omega }$. Then, we define the heat zeta function of Ω with boundary conditions F to be

$${\widehat{\zeta }}_{\Omega ,F}(s;\delta ):={\mathcal{M}}^{\delta }\left[t^{-N/2}{E}_{\Omega }\left(t\right)\right]\left(s\right),$$

(25)

for all $s\in \mathbb{C}$ for which the integral converges and extended by analytic continuation to some open connected subset of $\mathbb{C}$. If $F={\mathbb{1}}_{\partial \Omega }$ as in Problem 1 in Section 1.2, we say that ${\widehat{\zeta }}_{\Omega }:={\widehat{\zeta }}_{\Omega ,F}$ is a parabolic (heat) zeta function.

Note that if $E_{\Omega }\left(t\right)$ is integrable, bounded, and satisfies $E_{\Omega }\left(t\right)=O\left(t^{-{\sigma }_0}\right)$ as $t\to 0^{+}$ for some ${\sigma }_0\in \mathbb{R}$, then, by Lemma A1, we have that ${\widehat{\zeta }}_{\Omega ,F}$ is holomorphic in ${\mathbb{H}}_{{\sigma }_0}$, the open right half-plane in $\mathbb{C}$ defined by $\Re \left(s\right)>{\sigma }_0$. (Note the inversion with respect to the exponent of the asymptotic estimate, hence the negative sign convention.) When we establish functional equations for the heat content induced by a self-similar system $\Phi$, we also note that the relevant scaling operator will be $L_{\Phi }^2:={\sum }_{\phi \in \Phi }{M}_{{\lambda }_{\phi }^2}$, but the associated scaling zeta function ${\zeta }_{\Phi }$ is still the primary function of interest since ${\zeta }_{\Phi }\left(2s\right)={\zeta }_{L_{\Phi }^2}\left(s\right)$. Owing to the effect of this quadratic scaling, viz. $\alpha =2$ as in Section 3, we will also state our estimates for the heat content remainders in the form $R\left(t\right)=O\left(t^{-{\sigma }_0/2}\right)$ (as $t\to 0^{+}$). We state our results provided some control over the decomposition remainder, as in Definition 8 in Section 3.1.

Theorem 7

(Formula for the Heat Zeta Function). Let $E_{\Omega }$ be the heat content of the PWB solution to Problem 1 in Section 1.2. Let Φ be a self-similar system and suppose that $E_{\Omega }$ decomposes according to Φ with decomposition remainder $R_{\Omega }$ (as in Definition 8) which satisfies $R_{\Omega }\left(t\right)=O\left(t^{(N-{\sigma }_0)/2}\right)$ as $t\to 0^{+}$, for some ${\sigma }_0\in \mathbb{R}$. Define ${\sigma }_R:={\sigma }_0/2$ and let ${\zeta }_{\Phi }$ be the scaling zeta function associated to Φ. Then, for any $\delta >0$ and for all $s\in {\mathbb{H}}_{{\sigma }_R}\setminus {\mathcal{D}}_{\Phi }$,

$${\widehat{\zeta }}_{\Omega }(s;\delta )={\zeta }_{\Phi }\left(2s\right)({\widehat{\xi }}_{\Omega }(s;\delta )+{\zeta }_R(s;\delta )),$$

(26)

with ${\widehat{\zeta }}_{\Omega }(s;\delta )$ meromorphic in ${\mathbb{H}}_{{\sigma }_R}$, having poles contained in a subset of ${\mathcal{D}}_{\Phi }$, the set of poles of ${\zeta }_{\Phi }$. Here, ${\zeta }_R(s;\delta ):={\mathcal{M}}^{\delta }\left[t^{-N/2}{R}_{\Omega }\left(t\right)\right]\left(s\right)$ is a holomorphic function in ${\mathbb{H}}_{{\sigma }_R}$ and

$${\widehat{\xi }}_{\Omega }(s;\delta ):=\sum _{\phi \in \Phi }{\lambda }_{\phi }^{2s}{\mathcal{M}}_{\delta }^{\delta /{\lambda }_{\phi }^2}\left[t^{-N/2}{E}_{\Omega }\left(t\right)\right]\left(s\right)$$

(27)

is an entire function.

Proof. 

By Corollary 1 in Section 3.1.3, we have that $E_{\Omega }$ satisfies a 2-scaling law. Together with the presupposed induced decomposition, by Proposition 4 in Section 3.1, we have that the normalized $E_{\Omega }$ satisfies the scaling functional equation

$$t^{-N/2}{E}_{\Omega }\left(t\right)=L_{\Phi }^2\left[t^{-N/2}{E}_{\Omega }\left(t\right)\right]+t^{-N/2}{R}_{\Omega }\left(t\right)$$

(28)

for any $t>0$. The result is then a corollary of Theorem 5 in Section 3.3.3, noting that the normalized remainder satisfies $R\left(t\right)=t^{-N/2}{R}_{\Omega }\left(t\right)=O\left(t^{-{\sigma }_R}\right)$ at $t\to 0^{+}$, with ${\sigma }_R={\sigma }_0/2$. □

Note that this formula implies that the poles of the heat zeta function ${\widehat{\zeta }}_{\Omega }$ in the half-plane ${\mathbb{H}}_{{\sigma }_R}={\mathbb{H}}_{{\sigma }_0/2}$ occur exactly at the points $\omega /2$, where $\omega \in {\mathcal{D}}_{\Phi }\left({\mathbb{H}}_{{\sigma }_0}\right)$. Here, we see that the scaling zeta function ${\zeta }_{\Phi }$, determined from the scaling ratios and their multiplicities alone, also governs the nature of the zeta functions for self-similar fractals.

As we will use this scaling functional equation in order to obtain explicit formulae in what follows, we record this result as a proposition to which we will refer later.

Proposition 6

(Heat Scaling Functional Equation). Let $E_{\Omega }$ be the heat content of the PWB solution to Problem 1. Let Φ be a self-similar system and suppose that $E_{\Omega }$ decomposes according to Φ with decomposition remainder $R_{\Omega }$ (as in Definition 8) which satisfies $R_{\Omega }\left(t\right)=O\left(t^{(N-{\sigma }_0)/2}\right)$ as $t\to 0^{+}$, for some ${\sigma }_0\in \mathbb{R}$. Then for all $t\ge 0$, the heat content $E_{\Omega }$ satisfies the scaling functional equation given by (28). Note that the normalized remainder $R\left(t\right)=t^{-N/2}{R}_{\Omega }\left(t\right)$ satisfies the corresponding estimate $R\left(t\right)=O\left(t^{-{\sigma }_R}\right)$ as $t\to 0^{+}$, where ${\sigma }_R:={\sigma }_0/2$.

4.2. General Heat Content Results for Self-Similar Sets

We now state some general results regarding pointwise and distributional explicit formulae for the heat contents of open bounded regions $\Omega \subset {\mathbb{R}}^N$ whose boundary $\partial \Omega$ is a self-similar set and for which $(\partial \Omega ,\Omega )$ is an osculant fractal drum. The pointwise expansions require integrating the heat content sufficiently many times to improve the regularity of the function to be obtained. Meanwhile, the distributional identities (while less direct) remove this technical hurdle.

First, we state the pointwise identity, given in terms of antiderivatives of the heat content. We denote by $E_{\Omega }^{\left[k\right]}$ the $k^{\mathrm{th}}$ antiderivative of the heat content defined so that $E_{\Omega }^{\left[k\right]}\left(0\right)=0$. Explicitly, it may be defined by recursion with $E_{\Omega }^{\left[0\right]}:=E_{\Omega }$ and for $k>0$ by

$$E_{\Omega }^{\left[k\right]}\left(t\right):={\int }_0^t{E}_{\Omega }^{[k-1]}\left(\tau \right)d\tau .$$

As an additional preliminary to stating the result, we define the Pochhammer symbol ${\left(z\right)}_w:=\Gamma (z+w)/\Gamma \left(w\right)$ for $z,w\in \mathbb{C}$. Note that when $w=k$ is a positive integer, this simplifies to ${\left(z\right)}_k=z(z+1)\cdots (z+k-1)$ or, when $w=0$, to ${\left(z\right)}_0=1$. Lastly, we note that the summations to follow are defined as symmetric limits of the sums taken over values of $\omega$ with bounded imaginary parts for increasingly large upper bounds.

Theorem 8

(Heat Content, Pointwise Expansion). Let $E_{\Omega }$ be the heat content of the PWB solution to Dirichlet Problem 1 in Section 1.2. Let Φ be a self-similar system and suppose that $E_{\Omega }$ decomposes according to Φ with decomposition remainder $R_{\Omega }$ satisfying $R_{\Omega }\left(t\right)=O\left(t^{(N-{\sigma }_0)/2}\right)$ as $t\to 0^{+}$, for some ${\sigma }_0\in \mathbb{R}$. Suppose either that ${\sigma }_0<D_{\ell }\le {\underline{dim}}_S(\Phi )$, the bound for ${\underline{dim}}_S(\Phi )$ from Proposition 3, or that the (distinct) scaling ratios of Φ are arithmetically related (see Definition 1). Let $k\in \mathbb{Z}$ with $k\ge 2$ and let $\delta >0$. Then, for all $t\in (0,\delta )$, we have that

$$E_{\Omega }^{\left[k\right]}\left(t\right)=\sum _{\omega \in {\mathcal{D}}_{\Phi }\left({\mathbb{H}}_{{\sigma }_0}\right)}Res\left({\displaystyle \frac{t^{(N-s)/2+k}}{{((N-s)/2+1)}_k}}{\widehat{\zeta }}_{\Omega }(s/2;\delta );\omega \right)+{\mathcal{R}}^k\left(t\right),$$

where ${\widehat{\zeta }}_{\Omega }$ is as in Theorem 7. For any $\epsilon >0$ sufficiently small, the distributional remainder term ${\mathcal{R}}^k$ satisfies ${\mathcal{R}}^k\left(t\right)=O\left(t^{(N-{\sigma }_0)/2-\epsilon +k}\right)$ as $t\to 0^{+}$.

Proof. 

By Proposition 6 in Section 4.1, we have that the normalized heat content satisfies a scaling functional equation with respect to the operator $L_{\Phi }^2$ and remainder $R\left(t\right):=t^{-N/2}{R}_{\Omega }\left(t\right)$. Letting ${\sigma }_R={\sigma }_0/2$, we note that $R\left(t\right)=O\left(t^{-{\sigma }_R}\right)$ as $t\to 0^{+}$ and that ${\sigma }_0<D_{\ell }\le {\underline{dim}}_S(\Phi )$ implies that ${\sigma }_R<D_{\ell }/2$. So, in the first case, we apply Corollary 2 in Section 3.2.2 to obtain that R is an admissible remainder with respect to screens of the form $S_{\epsilon }\left(\tau \right)\equiv {\sigma }_R+\epsilon$ when $0<\epsilon <D_{\ell }/2-{\sigma }_R$. Meanwhile, in the lattice case, we apply Theorem 2 in Section 3.2.3 to obtain that any screen of the form $S_{\epsilon }\left(\tau \right)={\sigma }_R+\epsilon$ is admissible for all but finitely many $\epsilon >0$. In either case, $S_{\epsilon }$ is admissible for any sufficiently small $\epsilon >0$.

We may now apply Theorem 3 in Section 3.3.1. Here, $\beta =N$, $\alpha =2$, and $\delta >0$ is arbitrary. Note that $N/2\le {dim}_S(\Phi )/2$ as a corollary of Moran’s theorem and the open set condition and that $N/2\ge {\sigma }_R$ since Proposition 1 in Section 2.1 implies that ${\zeta }_R$ is holomorphic in the half-plane ${\mathbb{H}}_{N/2}$ as noted in the discussion following Definition 8. Here, with the estimate $R_{\Omega }\left(t\right)=O\left(t^{(N-{\sigma }_0)/2}\right)$ as $t\to 0^{+}$, we obtain by Lemma A1 that ${\zeta }_R$ is holomorphic in ${\mathbb{H}}_{{\sigma }_R}$. Note also that $S_{\epsilon }$ is chosen with $S_{\epsilon }\left(\tau \right)\equiv {\sigma }_R+\epsilon >{\sigma }_R$, so $W_{S_{\epsilon }}={\mathbb{H}}_{{\sigma }_R+\epsilon }$ and is contained in ${\mathbb{H}}_{{\sigma }_R}$. Further, we note that for the estimate of the remainder, $(\beta /\alpha )-sup\left(S\right)+k$ in the theorem is $(N-{\sigma }_0)/2-\epsilon +k$ since $sup\left(S\right)={\sigma }_R+\epsilon ={\sigma }_0/2+\epsilon$ and $\beta /\alpha =N/2$.

Lastly, we simplify the sum over ${\mathcal{D}}_{\Phi }\left(\alpha W_S\right)={\mathcal{D}}_{\Phi }\left(2{\mathbb{H}}_{{\sigma }_R+\epsilon }\right)$. Firstly, note that $2{\mathbb{H}}_{{\sigma }_R+\epsilon }={\mathbb{H}}_{2{\sigma }_R+2\epsilon }$. Since ${\zeta }_R$ is holomorphic in ${\mathbb{H}}_{{\sigma }_R}$, ${\zeta }_R(s/2)$ is holomorphic when $s\in {\mathbb{H}}_{2{\sigma }_R}={\mathbb{H}}_{{\sigma }_0}$. In both cases, we may choose $\epsilon >0$ sufficiently small so that ${\zeta }_{\Phi }\left(2s\right)$ has no poles of the form $s=\omega$ with $\Re \left(\omega \right)\in ({\sigma }_0,{\sigma }_0+2\epsilon )$. In the case when ${\sigma }_0<D_{\ell }$, take $\epsilon <(D_{\ell }-{\sigma }_0)/2$ and in the lattice case choose $2\epsilon$ smaller than the distance between ${\sigma }_0$ and the closest exceptional point. Then, with the above change of variables, we see that when $\Re \left(s\right)\in ({\sigma }_0,{\sigma }_0+2\epsilon )$, the function ${\widehat{\zeta }}_{\Omega }(s/2)$ has no poles because ${\zeta }_{\Phi }\left(s\right)$ has no poles with $\Re \left(s\right)\in ({\sigma }_0,{\sigma }_0+2\epsilon )$ (using Theorem 7). Thus, the sum over ${\mathcal{D}}_{\Phi }\left({\mathbb{H}}_{{\sigma }_0+2\epsilon }\right)$ is the same as over ${\mathcal{D}}_{\Phi }\left({\mathbb{H}}_{{\sigma }_0}\right)$. □

The leading order term occurs at the pole $\omega =D:={dim}_S(\Phi )$, as this is a pole of ${\zeta }_{\Phi }$ (and, hence, of ${\widehat{\zeta }}_{\Omega }$) having the largest real part, provided that the residue is nonvanishing. If D is a simple pole and the only pole with real part D, then the leading order term is explicitly

$${\displaystyle \frac{1}{{((N-D)/2+1)}_k}}Res({\widehat{\zeta }}_{\Omega }(s/2;\delta );D)t^{(N-D)/2+k}.$$

(29)

See Figure 4 for a series of plots of (29) with parameters corresponding to a $(4,{\displaystyle \frac{1}{4}})$-von Koch snowflake (whose definition is recalled in Section 4.3).

When the distinct scaling ratios are non-arithmetically related, that is, in the nonlattice case, D will be the unique pole with real part D; when the distinct scaling ratios are arithmetically related, that is, in the lattice case, this will not be true. Instead, in the lattice case, the leading order terms will form the Fourier expansion of a periodic function (when viewed additively after a change of variable) owing to the nature of the structure of the poles of ${\zeta }_{\Phi }$ in this case: They lie on finitely many vertical lines and are periodically spaced, with the same period on each vertical line. See, for instance, the proof of Theorem 2 in this work for an explicit proof of this and, more generally, we refer the reader to the discussion in ([18], Chapters 2 and 3) regarding the structure of the complex dimensions of self-similar fractal harps/strings (and their natural generalizations), noting that these zeta functions have the same structure as ${\zeta }_{\Phi }$, which controls the poles of ${\widehat{\zeta }}_{\Omega }$.

For the distributional setting, the restriction of $k\ge 2$ may be relaxed. For these formulae, recall from Section 3 that (just as in ([18], Chapter 5) and ([19], Chapter 5)) we will use as the space of test functions the set of Schwartz functions. These are the functions of rapid decrease near the boundary, given explicitly by

$$\mathcal{S}(0,\delta ):=\left\{\psi \in C^{\infty }(0,\delta )|\begin{array}{c}\forall m\in \mathbb{Z},\forall q\in \mathbb{N},t^m{\psi }^{\left(q\right)}\left(t\right)\to 0\hfill \\ \mathrm{and}{(t-\delta )}^m{\psi }^{\left(q\right)}\left(t\right)\to 0\mathrm{as}t\to 0^{+}\hfill \end{array}\right\}.$$

(30)

The tempered distributions on $(0,\delta )$ are the elements of the dual space, ${\mathcal{S}}^{\prime }(0,\delta )$.

Theorem 9

(Heat Content, Distributional Expansion). Let $E_{\Omega }$ be the heat content of the PWB solution to Dirichlet Problem 1 in Section 1.2. Let Φ be a self-similar system and suppose that $E_{\Omega }$ decomposes according to Φ with decomposition remainder $R_{\Omega }$ satisfying $R_{\Omega }\left(t\right)=O\left(t^{(N-{\sigma }_0)/2}\right)$ as $t\to 0^{+}$, for some ${\sigma }_0\in \mathbb{R}$. Suppose either that ${\sigma }_0<D_{\ell }\le {\underline{dim}}_S(\Phi )$, the bound for ${\underline{dim}}_S(\Phi )$ from Proposition 3 in Section 3.1, or that the (distinct) scaling ratios of Φ are arithmetically related (see Definition 1 in Section 2.2.2). Let $k\in \mathbb{Z}$ and let $\delta >0$. Then, the heat content, viewed as a tempered distribution on $(0,\delta )$, satisfies the identity

$$\begin{array}{cc}\hfill E_{\Omega }^{\left[k\right]}\left(t\right)& =\sum _{\omega \in {\mathcal{D}}_{\Phi }\left({\mathbb{H}}_{{\sigma }_0}\right)}Res\left({\displaystyle \frac{t^{(N-s)/2+k}}{{((N-s)/2+1)}_k}}{\widehat{\zeta }}_{\Omega }(s/2;\delta );\omega \right)+{\mathcal{R}}^{\left[k\right]}\left(t\right),\hfill \end{array}$$

(31)

as $t\to 0^{+}$, where two distributions are equal if their action on an arbitrary test function agrees. See (32) for the explicit identity of action on test functions. Here, ${\widehat{\zeta }}_{\Omega }$ is as in Theorem 7. The remainder term, as a distribution, satisfies the estimate that for any $\epsilon >0$ sufficiently small, ${\mathcal{R}}^{\left[k\right]}\left(t\right)=O\left(t^{(N-{\sigma }_0)/2-\epsilon +k}\right)$ as $t\to 0^{+}$, in the sense of (33).

Proof. 

The proof is the same as in Theorem 8, but through the application of Theorem 4, the distributional explicit formula result in Section 3.3.2, in place of Theorem 3, the pointwise explicit formula result. □

Recall that to say (31) is an identity of distributions means the following. For any test function $\psi \in \mathcal{S}(0,\delta )$,

$$\begin{array}{cc}\hfill \langle E_{\Omega }^{\left[k\right]},\psi \rangle & =\sum _{\omega \in {\mathcal{D}}_{\Phi }\left({\mathbb{H}}_{{\sigma }_0}\right)}Res\left({\displaystyle \frac{\mathcal{M}\left[\psi \right]\left((N-s)/2+k+1\right)}{{((N-s)/2+1)}_k}}{\widehat{\zeta }}_{\Omega }(s/2);\omega \right)+\langle {\mathcal{R}}^{\left[k\right]},\psi \rangle .\hfill \end{array}$$

(32)

The distributional remainder estimate is equivalent to the statement that for all $a>0$ and for all $\psi \in \mathcal{S}(0,\delta )$,

$$〈{\mathcal{R}}^{\left[k\right]}\left(t\right),{\displaystyle \frac{1}{a}}\psi (t/a)〉=O\left(a^{(N-{\sigma }_0)/2-\epsilon +k}\right),$$

(33)

as $t\to 0^{+}$. While this formulation is less direct compared to the pointwise expansion, it does have the advantage of requiring less regularity to leverage the expansion. Explicitly, one may set $k=0$ in (31)–(33) and obtain a distributional identity for the heat content itself. Note that in this case, the Pochhammer symbol simplifies to ${((N-s)/2+1)}_0=1$.

4.3. Heat Content Results for Generalized von Koch Snowflakes

We apply here the general theory developed in Section 4.2 to the family of generalized von Koch snowflake domains, a family of self-similar fractal drums including the classic von Koch snowflake and generalizations of its construction. In particular, we provide explicit formulae for the heat content of these domains with self-similar boundaries, with the formulae expressed in terms of the complex dimensions of the generalized von Koch fractals. In doing so, we will recover the results of [3,6,7] in the lattice case and extend them to the nonlattice case when $n\ge 5$, extending the leading order estimates of [9]. Furthermore, we explicitly show the role of the complex dimensions in the explicit formulae for the heat content. Results in the nonlattice case for $n=3$ or 4 may be deduced provided a priori knowledge of pole-free regions and estimates for the explicit function ${\zeta }_{{\Phi }_{n,r}}$ with the methods established in this work as well.

4.3.1. Generalized von Koch Fractals

Generalized von Koch fractals are a class of domains with self-similar fractal boundary obtained by generalizing the construction of its namesake, the von Koch snowflake. The von Koch curve (depicted in Figure 5) was introduced and studied by von Koch as a more geometric/elementary alternative to Weierstrass’ continuous but nowhere differentiable function, being a planar curve with nowhere-defined tangents [1,2].

In general, an $(n,r)$-von Koch curve may be defined as the invariant set of an explicit self-similar system ${\Phi }_{n,r}$ consisting of two mappings with scaling ratio $\ell =(1-r)/2$ and $n-1$ mappings of scaling ratio r; see (34) below for an explicit definition. Here, $n\ge 3$ refers to the type of regular polygon used in the construction and r refers to the proportion of the line segments removed from the middle at each step (where a regular n-gon of that same length is attached, with the bottom edge removed). When r is sufficiently small, the resulting curve is topologically simple [10]. The following bound is a sufficient condition.

Proposition 7

(Self-Avoidance of Generalized von Koch Fractals [10]). An $(n,r)$-von Koch curve has no self-intersections (i.e., it is a simple curve) if the scaling ratio $r>0$ satisfies

$$\begin{array}{cccc}\hfill r& <{\displaystyle \frac{{sin}^2(\pi /n)}{{cos}^2(\pi /n)+1}},\hfill & \hfill & \mathit{if}n\mathit{is}\mathit{even},\mathit{and}\hfill \\ \hfill r& <1-cos(\pi /n),\hfill & \hfill & \mathit{if}n\mathit{is}\mathit{odd}.\hfill \end{array}$$

The boundary of a corresponding $(n,r)$-von Koch snowflake is topologically simple under these conditions.

An $(n,r)$-von Koch snowflake is a union of n copies of an $(n,r)$-von Koch curve arranged along the edges of a regular n-gon. Figure 1 depicts the von Koch snowflake (with $n=3$, $r={\displaystyle \frac{1}{3}}$) as well as two generalizations both corresponding to simple curves, with $n=4$ and $r={\displaystyle \frac{1}{4}}$ and with $n=5$ and $r={\displaystyle \frac{1}{5}}$, respectively. To be precise, under the hypotheses of Proposition 7, the union of curves (when r is sufficiently small) defines a topologically simple, closed curve which we call an $(n,r)$-von Koch snowflake boundary. By the Jordan curve theorem, such a curve partitions its complement into two domains; we call the bounded component an $(n,r)$-von Koch snowflake domain or snowflake interior.

To formally define an $(n,r)$-von Koch curve, where $n\ge 3$ is an integer and $r\in (0,1)$, we explicitly specify the self-similar system ${\Phi }_{n,r}$ whose invariant set is the curve. To that end, we define the mappings $T_{(a,b)}:{\mathbb{R}}^2\to {\mathbb{R}}^2$ which translates by the point $(a,b)\in {\mathbb{R}}^2$, $R_{\theta }:{\mathbb{R}}^2\to {\mathbb{R}}^2$ which rotates a point counterclockwise about the origin by the angle $\theta$, and $S_{\lambda }:{\mathbb{R}}^2\to {\mathbb{R}}^2$ which uniformly scales in both coordinates by the factor of $\lambda$. Next, let $\ell =(1-r)/2$ be the conjugate scaling ratio, let ${\theta }_n=2\pi /n$ be the central angle of a regular n-gon, and let ${\alpha }_n=\pi -2\pi /n$ be the interior angle of a regular n-gon. Then, ${\Phi }_{n,r}$ may be defined by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\Phi }_{n,r}& :=\left\{{\phi }_L,{\phi }_R,{\psi }_k:{\mathbb{R}}^2\to {\mathbb{R}}^2,k=1,...,n-1\right\},\hfill \\ \hfill {\phi }_L& :=S_{\ell },\hfill \\ \hfill {\phi }_R& :=T_{(\ell +r,0)}\circ S_{\ell },\hfill \\ \hfill {\psi }_1& :=T_{(\ell ,0)}\circ R_{{\alpha }_n}\circ S_r,\hfill \\ \hfill {\psi }_k& :=T_{{\psi }_{k-1}(1,0)}\circ R_{{\alpha }_n-(k-1){\theta }_n}\circ S_r,k>1.\hfill \end{array}\end{array}$$

(34)

The maps ${\phi }_L$ and ${\phi }_R$ correspond to the left and right pieces of the curve and the mappings ${\psi }_k$, for $k=1,\dots ,n-1$, correspond to the $n-1$ pieces attached to the edges of a regular n-gon adjoined about the middle gap.

Next, we impose that $r\in (0,{\displaystyle \frac{1}{3}}]$ and satisfies the bound in Proposition 7 based on n. When $r=\ell ={\displaystyle \frac{1}{3}}$ and $n=3$, this is exactly the standard von Koch snowflake (see Figure 1 in Section 1.2). The scaling zeta function associated to ${\Phi }_{n,r}$ is given explicitly by

$${\zeta }_{{\Phi }_{n,r}}\left(s\right)={\displaystyle \frac{1}{1-(n-1)r^s-2{\ell }^s}},$$

(35)

and the set of its poles is precisely the set

$${\mathcal{D}}_{{\Phi }_{n,r}}\left(\mathbb{C}\right):=\left\{\omega \in \mathbb{C}|1=(n-1)r^{\omega }+2{\ell }^{\omega }\right\}.$$

(36)

It is known that the poles of ${\zeta }_{{\Phi }_{n,r}}$ are simple in both the lattice and nonlattice case (see Definition 1 in Section 2.2.2). For example, simplicity of poles in the lattice case follows from ([6], Proposition 1.3); the Dirichlet polynomial therein, under a change of variables made possible by the lattice constraint that $log\ell /logr=p/q$, is the denominator of ${\zeta }_{{\Phi }_{n,r}}$. Explicitly, $z=r_0^s$, where $r_0$ is the multiplicative generator such that $r=r_0^q$ and $\ell =r_0^p$. In the nonlattice case, this follows from ([18], Theorem 3.30).

In order to determine the admissibility of the remainder terms in the nonlattice case, we establish the following bound regarding the lower similarity dimension of ${\Phi }_{n,r}$.

Lemma 1

(Sufficient Lower Dimension Bounds). Let $n\ge 3$, $r\in (0,{\displaystyle \frac{1}{3}}]$, and define $\ell =(1-r)/2$. Let Φ a self similar system having scaling ratios r with multiplicity $n-1$ and ℓ with multiplicity 2. Denote by $D_{\ell }\le {\underline{dim}}_S(\Phi )$ the bound from Proposition 3. Then, we have the following statements:

$$\begin{array}{ccccc}\hfill \mathit{If}n=3,D_{\ell }<0;& \hfill & \hfill \mathit{If}n=4,D_{\ell }=0;& \hfill & \hfill \mathit{If}n\ge 5,D_{\ell }>0.\end{array}$$

Proof. 

Let ${\zeta }_{\Phi }$ the zeta function of $\Phi$ and let $P\left(s\right)$ denote its denominator. Because $r\le {\displaystyle \frac{1}{3}}$, we have that $r\le \ell$. Suppose first that $r=\ell ={\displaystyle \frac{1}{3}}$. Then, we have that

$$P\left(s\right):=1-2{\ell }^s-(n-1)r^s=1-(n+1)3^{-s}.$$

We may explicitly solve to find that $P\left(\omega \right)=0$ when $(n+1)=3^{\omega }$, or $\omega ={log}_3(n+1)+2\pi ik/log3$ for $k\in \mathbb{Z}$. Since $n\ge 3$, the real part of these poles are always positive and exactly equal to ${log}_3(n+1)$. Thus, $0<D_{\ell }={\underline{dim}}_S(\Phi )={log}_3(n+1)$, noting that $D_{\ell }={log}_3(n+1)$ solves (9).

Now, suppose that $r<{\displaystyle \frac{1}{3}}$, in which case, $r<\ell$. Consider the function

$$p\left(t\right)={\displaystyle \frac{1}{n-1}}{\left(t^{-1}\right)}^t+{\displaystyle \frac{2}{n-1}}{\left({\displaystyle \frac{\ell }{t}}\right)}^t.$$

By definition, $D_{\ell }$ is the unique real solution to $p\left(t\right)=1$. Note that p is strictly increasing, so if $p\left(t\right)<1$, then $t<D_{\ell }$. We have that $p\left(0\right)={\displaystyle \frac{3}{n-1}}$. If $n\ge 5$, then $p\left(0\right)<1$. If $n=4$, $p\left(0\right)=1$ and, thus, $D_{\ell }=0$. If $n=3$, $p\left(0\right)>1$ whence $D_{\ell }<0$. □

It is for this reason that our results for nonlattice $(n,r)$-von Koch snowflakes will be restricted to $n\ge 5$ since we need this condition to apply the criterion of Theorem 1 in Section 3.2.2 with the available estimates.

The last major preliminary is an induced decomposition (in the sense of Definition 6 in Section 3.1) and an estimate of the decomposition remainder term, $R_{\Omega }$. This is needed to find a scaling functional equation for the heat content $E_{\Omega }$. A scaling functional equation for $E_{\Omega }$ in the case of $(n,r)$-von Koch fractals was established in [6] (without this terminology). The decomposition remainder, which we will denote by $R_{\Omega }$, is given explicitly by ([6], Equation (1.12)) and ([6], Proposition 1.2) gives an explicit estimate. Namely, $R_{\Omega }\left(t\right)=O\left(t\right)$ as $t\to 0^{+}$, in which case, we have that ${\sigma }_0=0$ since $N/2=1$ in ${\mathbb{R}}^2$.

4.3.2. Heat Content Explicit Formulae for Generalized von Koch Fractals

Given an $(n,r)$-von Koch snowflake, in what follows, let $K_{n,r}=\partial \Omega$ denote its boundary. Assume that r is sufficiently small (viz. Proposition 7 in Section 4.3.1) so that $K_{n,r}$ is a simple closed curve with a well-defined interior domain, $\Omega$. We will consider Problem 1 in Section 1.2. We let $u_{\Omega }$ be the PWB solution and $E_{\Omega }$ be the associated heat content. Our goal will be to explicitly describe this heat content in the limit as $t\to 0^{+}$ by applying the explicit formulae results. We note that by symmetry, we need only consider the heat content in one portion of $\Omega$ contained in a sector of angle $2\pi /n$ and that the heat content of the total set is n times the heat content in this restricted region.

We will give two types of explicit formulae. The first result is regarding the pointwise-valid formulae for $k\ge 2$, and the second will be the distributional formulae specialized to the case when $k=0$. Note that the general pointwise formulae may be deduced in the same way using the general results for heat contents of large classes of self-similar sets. For the preliminaries regarding the notation and conventions defining the antiderivatives $E_{\Omega }^{\left[k\right]}$, the Pochhammer symbol ${\left(z\right)}_w$, $z,w\in \mathbb{C}$, and the definition of the sums over complex dimensions as symmetric limits, we refer the reader to Section 4.2.

Theorem 10

(Heat Content of Generalized von Koch Fractals, Pointwise). Let $K_{n,r}$ be an $(n,r)$-von Koch snowflake boundary satisfying the self-avoidance criterion in Proposition 7 in Section 4.3.1 and let Ω be the interior region defined by $K_{n,r}$ (i.e., the bounded component of ${\mathbb{R}}^2\setminus K_{n,r}$). Suppose that either $n\ge 5$ or that the scaling ratios r and $\ell =(1-r)/2$ are arithmetically related (i.e., the ratio of their logarithms is rational).

Then, for every integer $k\ge 2$, any $\delta >0$, and every $t\in (0,\delta )$, we have that the antiderivatives $E_{\Omega }^{\left[k\right]}$ of the heat content $E_{\Omega }$ for Problem 1 in Section 1.2 on Ω satisfy

$$E_{\Omega }^{\left[k\right]}\left(t\right)=\sum _{\omega \in {\mathcal{D}}_{{\Phi }_{n,r}}\left({\mathbb{H}}_0\right)}{\displaystyle \frac{1}{{\left(\frac{2-\omega }{2}+1\right)}_k}}Res({\widehat{\zeta }}_{\Omega }(s/2;\delta );\omega )t^{(2-\omega )/2+k}+{\mathcal{R}}^k\left(t\right),$$

where ${\widehat{\zeta }}_{\Omega }$ is as in Theorem 7 in Section 4.1 and where ${\mathcal{D}}_{{\Phi }_{n,r}}\left({\mathbb{H}}_0\right)$ is the set of possible complex dimensions of the generalized von Koch snowflake domain (as given in (36)) belonging to the open right half-plane ${\mathbb{H}}_0=\left\{s\in \mathbb{C}|\Re (s)>0\right\}$. For any $\epsilon >0$ sufficiently small, the error term satisfies ${\mathcal{R}}^k\left(t\right)=O\left(t^{1-\epsilon +k}\right)$ as $t\to 0^{+}$.

Proof. 

We have that $E_{\Omega }$ satisfies a scaling functional equation induced by ${\Phi }_{n,r}$ with decomposition remainder $R_{\Omega }\left(t\right)=O\left(t\right)$ as $t\to 0^{+}$ by Proposition 1.2 of [6], where ${\Phi }_{n,r}$ is as in (34). When normalized (noting that $N/2=1$ in ${\mathbb{R}}^2$), the heat content scaling functional equation takes the form

$$t^{-1}{E}_{\Omega }\left(t\right)=L_{{\Phi }_{n,r}}\left[t^{-1}{E}_{\Omega }\left(t\right)\right]\left(t\right)+t^{-1}{R}_{\Omega }\left(t\right).$$

Further, the remainder $R\left(t\right):=t^{-1}{R}_{\Omega }\left(t\right)=O\left(t^{-{\sigma }_R}\right)$, with ${\sigma }_R={\sigma }_0/2=0$, as $t\to 0^{+}$, and is continuous on ${\mathbb{R}}^{+}$. Note that we have also multiplied the expression in [6] by n (and distributed by linearity) to deduce a scaling functional equation for the total heat content, rather than the amount of heat content contained in a single symmetric sector.

The result then follows from application of Theorem 8 in Section 4.2 with the following notes. If we assume that $n\ge 5$, then we have that ${\underline{dim}}_S\left({\Phi }_{n,r}\right)\ge D_{\ell }>0={\sigma }_0$ by Lemma 1 in Section 3.2.2. If $n=3$ or $n=4$, we have assumed that r and ℓ are arithmetically related. Lastly, the poles of ${\zeta }_{{\Phi }_{n,r}}$ are simple, as discussed in Section 4.3.1; see, for example, ([6], Proposition 1.3) in the lattice case and ([18], Theorem 3.30) in the nonlattice case, which allows the residue terms to be simplified. □

In order to obtain expansions for the heat content itself when $k=0$, we now move to the distributional formulation of the explicit formulae. We refer the reader to Section 4.2 for the relevant preliminaries and more information about the action of distributions or the estimates of their remainder terms. The space of test functions is the class of Schwartz functions on $(0,\delta )$, $\mathcal{S}(0,\delta )$, and the tempered distributions are the elements of its dual space, ${\mathcal{S}}^{\prime }(0,\delta )$.

Theorem 11

(Heat Content of Generalized von Koch Fractals, Distributionally with $k=0$). Let $K_{n,r}$ be an $(n,r)$-von Koch snowflake boundary satisfying the self-avoidance criterion in Proposition 7 and let Ω be the interior region defined by $K_{n,r}$ (i.e., the bounded component of ${\mathbb{R}}^2\setminus K_{n,r}$). Suppose that either $n\ge 5$ or that the scaling ratios r and $\ell =(1-r)/2$ are arithmetically related (i.e., the ratio of their logarithms is rational).

Then, for any $\delta >0$ and every $t\in (0,\delta )$, we have the following equality of distributions in the Schwartz space ${\mathcal{S}}^{\prime }(0,\delta )$ (the dual of the space $\mathcal{S}(0,\delta )$ defined in (30)):

$$E_{\Omega }\left(t\right)=\sum _{\omega \in {\mathcal{D}}_{{\Phi }_{n,r}}\left({\mathbb{H}}_0\right)}Res({\widehat{\zeta }}_{\Omega }(s/2;\delta );\omega )t^{(2-\omega )/2}+\mathcal{R}\left(t\right),$$

(37)

where ${\widehat{\zeta }}_{\Omega }$ is as in Theorem 7 in Section 4.1 and where ${\mathcal{D}}_{{\Phi }_{n,r}}\left({\mathbb{H}}_0\right)$ is the set of possible complex dimensions of the generalized von Koch snowflake domain (as given in (36)) belonging to the open right half-plane ${\mathbb{H}}_0=\left\{s\in \mathbb{C}|\Re (s)>0\right\}$. For any $\epsilon >0$ sufficiently small, the error term satisfies $\mathcal{R}\left(t\right)=O\left(t^{1-\epsilon }\right)$ as $t\to 0^{+}$, in the sense of (33). The action of $E_{\Omega }$ on a test function is given explicitly by (32) with $k=0$.

Proof. 

The proof is the same as that of Theorem 10 but with application of Theorem 9. Note that we specialize to the case of $k=0$. □

In the nonlattice case, the leading order term is monotonic and given by (29); see the discussion in Section 4.2, where D is the Minkowski dimension of $K_{n,r}$. Additionally, by ([18], Theorem 3.23), it follows that the distributional estimate

$$E_{\Omega }\left(t\right)-Res({\widehat{\zeta }}_{\Omega }(D/2;\delta );D)t^{(2-D)/2}=o\left(t^{(2-D)/2}\right),$$

as $t\to 0^{+}$, is the best possible in the general setting; there exist terms in (37) of the form $r_{\omega }{t}^{(2-\omega )/2}$, where $\omega \in {\mathcal{D}}_{{\Phi }_{n,r}}\left({\mathbb{H}}_0\right)$ is such that $D-\Re \left(\omega \right)<\epsilon$, for any $\epsilon >0$.

Meanwhile, in the lattice case, the leading order terms correspond to a multiplicatively periodic function. The poles of ${\zeta }_{{\Phi }_{n,r}}$ with real part equal to D (the Minkowski dimension of $K_{n,r}$) are exactly of the form $D+ikP$, where $P\in \mathbb{R}$ is the period and $k\in \mathbb{Z}$ is arbitrary. The other poles (if any) occur on finitely many vertical lines (i.e., having finitely many possible real parts), distributed on each with the same period; see the discussion in Section 2.2.2 and the proof of Theorem 2. Thus, we have the distributional estimate that

$$E_{\Omega }\left(t\right)-t^{(2-D)/2}\sum _{k\in \mathbb{Z}}Res({\widehat{\zeta }}_{\Omega }((D+ikP)/2;\delta );D+ikP)t^{ikP/2}=O\left(t^{(2-{\sigma }_1)/2}\right),$$

as $t\to 0^{+}$, where ${\sigma }_1$ is either the real part of the next vertical line of poles of ${\zeta }_{{\Phi }_{n,r}}$ (if they are distributed on more than one vertical line) or zero, whichever is larger. The leading order contribution is of the form $E_{\Omega }\left(t\right)\sim \psi \left(t\right)t^{(2-D)/2}$, where $\psi$ is the multiplicatively periodic function (viewed as a distribution on $(0,\delta )$) given by

$$\psi \left(t\right):=\sum _{k\in \mathbb{Z}}{r}_k{t}^{ikP/2},$$

with coefficients $r_k:=Res({\widehat{\zeta }}_{\Omega }((D+ikP)/2;\delta );D+ikP)$ determined by the residues of the heat zeta function ${\widehat{\zeta }}_{\Omega }$; it is constant if and only if $r_k$ vanishes for all nonzero integers k. Note that by ([18], Theorem 2.16), the residues of ${\zeta }_{{\Phi }_{n,r}}$ at each pole $D+ikP$ are all nonzero and equal to the same constant for $k\in \mathbb{Z}$; only the multiplicative factor in (26) can otherwise affect the residue.

4.4. Connection to Complex Dimensions

Let $\Omega \subset {\mathbb{R}}^N$ be a bounded open set with boundary $\partial \Omega$. Suppose that $\partial \Omega$ is a self-similar set, arising as the attractor of the self-similar system $\Phi$, and suppose that $(\partial \Omega ,\Omega )$ is an osculant fractal drum with respect to $\Phi$. (Per Definition 3 in Section 2.2.3, this means that $\Phi$ satisfies the open set condition with $\Omega$ as a feasible open set and that the osculating condition holds for $\Omega$.) For explicit examples of such fractals, one may consider when $\Omega$ is the interior of an $(n,r)$-von Koch snowflake and $\partial \Omega$ is the snowflake boundary as in Section 4.3.

Letting $R=\Omega \setminus \left({\cup }_{\phi \in \Phi }\phi [\Omega ]\right)$, denote by $V_{\partial \Omega ,R}$ the tube function of $\partial \Omega$ relative to the residual set R. By ([21], Theorem 5.3), it follows that the normalized tube function $t^{-N}{V}_{\partial \Omega ,\Omega }\left(t\right)$ satisfies the scaling function equation

$$t^{-N}{V}_{\partial \Omega ,\Omega }\left(t\right)=L_{\Phi }\left[t^{-N}{V}_{\partial \Omega ,\Omega }\left(t\right)\right]\left(t\right)+t^{-N}{V}_{\partial \Omega ,R}\left(t\right).$$

(38)

Further, let ${\sigma }_0\in \mathbb{R}$ be such that as $t\to 0^{+}$,

$$\begin{array}{cc}\hfill V_{\partial \Omega ,R}\left(t\right)& =O\left(t^{N-{\sigma }_0}\right),\mathrm{and}\hfill \\ \hfill R_{\Omega }\left(t\right)& =O\left(t^{(N-{\sigma }_0)/2}\right).\hfill \end{array}$$

Lastly, we suppose that either ${\sigma }_0<D_{\ell }\le {\underline{dim}}_S(\Phi )$ (as in Proposition 3 in Section 3.1.1) or that the distinct scaling ratios of the similitudes in $\Phi$ are arithmetically related, which are sufficient criteria for the admissibility of remainder terms.

Under these assumptions, explicit formulae for $V_{\partial \Omega ,\Omega }$ and $E_{\Omega }$ may both be written in terms of the possible complex dimensions, as governed by the poles of ${\zeta }_{\Phi }$. For simplicity, suppose that the poles of ${\zeta }_{\Phi }$ are simple, such as in the case of generalized von Koch fractals. In general, poles of higher multiplicities can occur for self-similar fractals; in the lattice case, notably, the principal complex dimensions, those with maximal real parts, are simple by ([18], Theorem 2.16). Note that under the imposition of the open set condition, the maximal real part is exactly the Minkowski dimension of the relative fractal drum and the (upper) similarity dimension. See also [19,41]. We consider the distributional identities specialized to $k=0$. By ([19], Theorem 5.2.2) and Theorem 9, respectively, we have that

$$\begin{array}{cc}\hfill V_{\partial \Omega ,\Omega }\left(t\right)& =\sum _{\omega \in {\mathcal{D}}_{\Phi }\left({\mathbb{H}}_{{\sigma }_0}\right)}{a}_{\omega }{t}^{N-\omega }+{\mathcal{R}}_V\left(t\right);\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill E_{\Omega }\left(t\right)& =\sum _{\omega \in {\mathcal{D}}_{\Phi }\left({\mathbb{H}}_{{\sigma }_0}\right)}{b}_{\omega }{t}^{(N-\omega )/2}+{\mathcal{R}}_E\left(t\right),\hfill \end{array}$$

(40)

where ${\mathcal{R}}_V\left(t\right)=O\left(t^{N-{\sigma }_0-\epsilon }\right)$ and ${\mathcal{R}}_E\left(t^{(N-{\sigma }_0)/2-\epsilon }\right)$ as $t\to 0^{+}$, for sufficiently small $\epsilon >0$. Here, $a_{\omega }=Res({\tilde{\zeta }}_{\partial \Omega ,\Omega }(s;\delta );\omega )$ and $b_{\omega }=Res({\widehat{\zeta }}_{\Omega }(s;\delta );\omega )$ are constants given by residues of the tube or heat zeta functions, respectively, both over the same set of (possible) complex dimensions. The heat content, just as with the tube function, can, therefore, be seen to have an expansion governed by the complex dimensions of the boundary $\partial \Omega$ relative to $\Omega$.