Spectral Theory and Hardy Spaces for Bessel Operators in Non-Standard Geometries

Abstract

This paper develops novel results in the harmonic analysis of Bessel operators, extending their theory to higher-dimensional and non-Euclidean spaces. We present a refined framework for Hardy spaces associated with Bessel operators, emphasizing atomic decompositions, dual spaces, and connections to Sobolev and Besov spaces. The spectral theory of families of boundary-interpolating operators is also expanded, offering precise eigenvalue estimates and functional calculus applications. Furthermore, we explore Bessel operators under non-standard measures, such as fractal and weighted geometries, uncovering new analytical phenomena. Key implications include advanced insights into singular integrals, heat kernel behavior, and the boundedness of Riesz transforms, with potential applications in fractal geometry, constrained wave propagation, and mathematical physics.

1. Introduction

The study of Bessel operators has long been a cornerstone of harmonic analysis, particularly in the context of radial functions in Euclidean spaces and their connections to the Laplacian on Riemannian manifolds. Since the pioneering work of Muckenhoupt and Stein [1], Bessel operators have been central to understanding singular integrals, semigroups, and spectral theory. Their applications extend across mathematical physics, signal processing, and stochastic processes [2,3].

A critical area of research focuses on the interplay between Bessel operators and Hardy spaces. Hardy spaces provide a natural framework for studying singular integral operators, and their atomic decompositions offer fine-grained tools for understanding regularity and boundedness properties. Notably, the boundedness of Riesz transforms on Hardy spaces associated with Bessel operators has been extensively investigated [4,5], yielding deep insights into their analytic and geometric properties.

This paper builds upon this foundation and extends the theory in several significant directions:

• We generalize the analysis of Bessel operators to higher-dimensional and non-Euclidean spaces, focusing on heat kernel estimates and boundary condition interpolation. The relationship between these operators and the Laplace–Beltrami operator on Riemannian manifolds with ends [6,7] provides a geometric underpinning for this generalization.
• We develop the spectral theory of families of operators that interpolate between Neumann and Dirichlet boundary conditions, enabling precise eigenvalue estimates and functional calculus applications. The study of spectral gaps and interpolation properties [8,9] adds depth to this analysis.
• We refine the theory of Hardy spaces for Bessel operators by introducing atomic decompositions, duality with bounded mean oscillation (BMO) spaces, and connections to Sobolev and Besov spaces. These results provide a robust framework for studying singular integrals and related operators in various settings [10,11].
• We extend the analysis to spaces with non-standard measures, such as fractal measures or weighted geometries. The interplay between heat kernels, volume growth, and atomic decompositions [12,13] reveals new analytic and geometric phenomena in irregular domains.

These contributions have implications for both pure and applied mathematics. For example, the refined Hardy space theory can be used to study singular integral operators in fractal geometry, while spectral results inform wave propagation in constrained geometries. The harmonic analysis of Bessel operators continues to be a fertile ground for exploring connections between analysis, geometry, and physics.

Despite the significant advancements in the theory of Bessel operators and Hardy spaces, certain limitations persist in the current literature. Most studies focus on the classical Euclidean setting, which assumes uniform geometric and measure-theoretic properties. This restricts the applicability of these results to more complex spaces where geometric irregularities or anisotropies play a crucial role. For example, while Bessel operators have been extensively studied in the context of radial functions and their connections to the Laplacian, their behavior under non-standard boundary conditions or in domains with irregular geometries remains largely unexplored. Similarly, the theory of Hardy spaces, while well developed for Euclidean spaces, lacks comprehensive extensions to settings involving fractal measures or weighted geometries that arise naturally in applications such as wave propagation, constrained systems, and fractal analysis.

The necessity of extending the theory to non-Euclidean spaces is motivated by the growing interest in understanding the behavior of differential operators on manifolds with complex geometries, such as Riemannian manifolds with ends or spaces equipped with fractal-like measures. Non-Euclidean spaces introduce unique analytical challenges, including the need to generalize heat kernel estimates and to accommodate the effects of variable curvature or boundary conditions. Recent studies, such as those on metric-measure spaces and the Laplace–Beltrami operator, have demonstrated the potential of this broader framework to uncover new phenomena in harmonic analysis, particularly in the context of spectral theory and singular integral operators.

Furthermore, recent works have begun exploring the impact of non-standard measures, such as weighted or fractal measures, on the analysis of Bessel operators. These studies reveal how the interplay between measures and geometric properties influences fundamental results like heat kernel bounds and the boundedness of operators. For instance, measures with non-uniform growth rates or irregular scaling properties lead to new challenges in defining Hardy spaces and studying their duals. By incorporating these non-standard measures, the current work enriches the literature by bridging gaps in the understanding of Bessel operators and Hardy spaces, offering insights that are crucial for applications in fractal geometry, diffusion processes in irregular media, and mathematical physics.

The organization of this paper is as follows. Section 2 introduces the necessary preliminaries, including Bessel operators, Hardy spaces, and semigroups. Section 3 generalizes Bessel operators to higher dimensions and non-Euclidean spaces. Section 4 develops the spectral theory of boundary-interpolating operators. Section 5 refines the Hardy space framework, and Section 6 explores extensions to non-standard measures. Finally, Section 7 presents applications and concluding remarks.

2. Preliminaries

This section introduces the foundational concepts and notations used throughout this paper. We briefly review Bessel operators, metric-measure spaces, Hardy spaces, and semigroups, which serve as the building blocks for the results presented in subsequent sections. The Bessel operator arises naturally in the study of radial functions on Euclidean spaces and plays a central role in this manuscript.

Definition 1 

(Bessel Operator). Let $X=(0,\infty )\subset \mathbb{R}$ with the measure $d\mu \left(x\right)=x^{n-1}dx$, where $n>0$. The Bessel operator $L_B$ is defined as

$$L_{B}f\left(x\right)=-f^{″}\left(x\right)-{\displaystyle \frac{n-1}{x}}{f}^{\prime }\left(x\right),$$

with appropriate boundary conditions at $x=0$ and $x\to \infty$.

Remark 1. 

For $n\in \mathbb{N}$, the operator $L_B$ corresponds to the radial part of the Laplacian on ${\mathbb{R}}^n$. It has been extensively studied in harmonic analysis, singular integrals, and semigroup theory [1,3].

The concept of a metric-measure space generalizes the Euclidean framework to spaces equipped with a measure and a metric.

Definition 2 

(Metric-Measure Space). A triple $(X,\rho ,\mu )$ is called a metric-measure space if the following conditions are satisfied:

• X is a set;
• $\rho :X\times X\to [0,\infty )$ is a metric;
• μ is a positive Borel measure on X.

Example 1. 

For $X=(0,\infty )\subset \mathbb{R}$, equipped with the Euclidean metric $\rho (x,y)=|x-y|$ and the measure $d\mu \left(x\right)=x^{n-1}dx$, $(X,\rho ,\mu )$ is a metric-measure space.

Bessel operators generate semigroups that describe diffusion processes and heat flow in the corresponding metric-measure spaces.

Definition 3 

(Semigroup). Let $L_B$ be a self-adjoint operator on $L^2(X,\mu )$. The semigroup generated by $L_B$ is defined as

$$T_t=e^{-t{L}_B},t>0,$$

where $T_t:L^2(X,\mu )\to L^2(X,\mu )$.

Definition 4 

(Heat Kernel). The heat kernel $T_t(x,y)$ associated with $T_t$ satisfies

$$\left(T_{t}f\right)\left(x\right)={\int }_X{T}_t(x,y)f\left(y\right)d\mu \left(y\right),$$

for $f\in L^2(X,\mu )$ and $t>0$.

Proposition 1 

(Gaussian Heat Kernel Bounds). Let $T_t(x,y)$ be the heat kernel associated with $L_B$. There exist constants $C,c>0$ such that

$$C^{-1}{\displaystyle \frac{e^{-\frac{\rho {(x,y)}^2}{4t}}}{\mu \left(B(x,\sqrt{t})\right)}}\le T_t(x,y)\le C{\displaystyle \frac{e^{-\frac{\rho {(x,y)}^2}{4t}}}{\mu \left(B(x,\sqrt{t})\right)}}.$$

Hardy spaces provide a robust framework for analyzing singular integral operators in the context of Bessel operators.

Definition 5 

(Hardy Space $H_{L_B}^1\left(X\right)$). Let $X\subset \mathbb{R}$ and $L_B$ be the Bessel operator. The Hardy space $H_{L_B}^1\left(X\right)$ is defined as

$$H_{L_B}^1\left(X\right)=\{f\in L^1{(X,\mu ):\parallel f\parallel }_{H_{L_B}^1}={\int }_X\underset{t>0}{sup}|T_{t}f\left(x\right)|d\mu \left(x\right)<\infty \}.$$

Definition 6 

($(L_B,\mu )$-Atoms). A function $a\left(x\right)$ is called an $(L_B,\mu )$-atom if the following are satisfied:

• $supp\left(a\right)\subset B$ for some ball $B\subset X$;
• ${\parallel a\parallel }_{L^2(X,\mu )}\le \mu {\left(B\right)}^{-1/2}$;
• ${\int }_{X}a\left(x\right)d\mu \left(x\right)=0$.

Bessel operators generalize the radial part of the Laplace–Beltrami operator on Riemannian manifolds.

These preliminaries establish the necessary background in Bessel operators, Hardy spaces, and semigroup theory. The subsequent sections build on these concepts to derive new results in harmonic analysis, spectral theory, and applications.

The methods employed in this paper to derive new characterizations of Hardy spaces involve a combination of advanced techniques in harmonic analysis and spectral theory. Central to our approach is the use of atomic decompositions tailored to the geometry of the underlying space, which provide a granular understanding of functions in Hardy spaces. These decompositions are further augmented by duality arguments with BMO spaces and connections to Sobolev and Besov spaces, allowing for a more refined classification of function spaces. Additionally, we exploit the semigroup generated by Bessel operators to introduce a maximal function characterization of Hardy spaces, bridging the gap between atomic and functional characterizations. These tools enable us to extend classical Hardy space theory to accommodate irregular geometries and non-standard measures, opening new avenues for analysis in more generalized settings.

The implications of these findings are significant in the existing theories of singular integrals, spectral theory, and function space analysis. By extending Hardy space theory to fractal measures and weighted geometries, our results provide a more robust framework for studying the boundedness of operators, such as the Riesz transform, in non-Euclidean and irregular domains. This not only generalizes previous results on Hardy spaces but also enriches the understanding of their role in analyzing singular integrals and semigroup theory in constrained or weighted settings. Moreover, the refinement of duality with BMO spaces strengthens the foundational link between Hardy spaces and other function spaces, contributing to a deeper insight into classical theories while paving the way for applications in mathematical physics, fractal geometry, and constrained wave propagation. These developments underscore the broader relevance of Hardy spaces beyond their traditional Euclidean context.

While much of the analysis in this manuscript is applicable to general Banach spaces, certain results and proofs specifically rely on the properties of Hilbert spaces. These include the use of the inner product structure, orthogonality, and the norm induced by the inner product. In particular, by Theorems 5, 7, and 8 and Proposition 4, underlying a space is a Hilbert space:

These results inherently rely on the properties of the inner product space, such as orthogonality, the spectral theorem, or semigroup theory specific to self-adjoint operators. When applicable, we explicitly indicate these dependencies within the proofs. For general Banach space settings, the corresponding extensions remain an open direction for future research.

3. Generalization to Higher Dimensions and Non-Euclidean Spaces

This section explores the extension of Bessel operators to higher-dimensional and non-Euclidean spaces. We focus on defining appropriate metric-measure spaces, generalizing heat kernel estimates, and analyzing boundary conditions in this broader context. We begin with the definition of the higher-dimensional Bessel operator and the associated metric-measure space.

Definition 7 

(Higher-Dimensional Bessel Operator). Let $n>2$ and μ be the measure defined on ${\mathbb{R}}^n\setminus \left\{0\right\}$ by $d\mu \left(x\right)={\parallel x\parallel }^{n-2}dx$, where $dx$ is the Lebesgue measure. The Bessel operator $L_B$ is defined as

$$L_{B}f\left(x\right)=-\Delta f\left(x\right)-{\displaystyle \frac{n-2}{{\parallel x\parallel }^2}}f\left(x\right),$$

where Δ is the Laplacian in ${\mathbb{R}}^n$.

Definition 8 

(Metric-Measure Space). Let $(X,\rho ,\mu )$ be a metric-measure space, where

• $X\subseteq {\mathbb{R}}^n$ is a domain with possibly irregular boundaries.
• $\rho (x,y)=\parallel x-y\parallel$ is the Euclidean metric.
• μ is the measure with density proportional to ${\parallel x\parallel }^{n-2}$.

This space provides a natural setting for the analysis of the higher-dimensional Bessel operator.

We generalize the heat kernel estimates for the Bessel operator to higher dimensions.

Theorem 1 

(Heat Kernel Estimates). Let $L_B$ be the Bessel operator on ${\mathbb{R}}^n\setminus \left\{0\right\}$ with $n>2$. The heat kernel $T_t(x,y)$ of the semigroup $e^{-t{L}_B}$ satisfies

$$C^{-1}{\displaystyle \frac{e^{-\frac{\rho {(x,y)}^2}{4t}}}{\mu \left(B(x,\sqrt{t})\right)}}\le T_t(x,y)\le C{\displaystyle \frac{e^{-\frac{\rho {(x,y)}^2}{4t}}}{\mu \left(B(x,\sqrt{t})\right)}},$$

where $B(x,r)$ is the ball of radius r centered at x in ${\mathbb{R}}^n$, and $C>0$ is a constant.

Proof. 

To derive the bounds for the heat kernel $T_t(x,y)$, we decompose it into radial and angular components. Using polar coordinates, let $x=(r,\theta )$ and $y=(s,\varphi )$, where $r=|x|$ and $s=|y|$.

The distance function $\rho (x,y)$ in this context is given by

$$\rho (x,y)=\sqrt{r^2+s^2-2rscos\theta },$$

where $\theta$ is the angle between x and y.

For the upper bound, we use the decay properties of the heat kernel. Specifically, for large distances,

$$T_t(x,y)\le C{\displaystyle \frac{e^{-\frac{\rho {(x,y)}^2}{4t}}}{\mu \left(B(x,\sqrt{t})\right)}}.$$

For the lower bound, we use the fact that the heat kernel is positive and satisfies

$$T_t(x,y)\ge C^{-1}{\displaystyle \frac{e^{-\frac{\rho {(x,y)}^2}{4t}}}{\mu \left(B(x,\sqrt{t})\right)}}.$$

□

To extend the analysis to non-Euclidean spaces, we consider manifolds with ends.

Definition 9 

(Manifolds with Ends). A Riemannian manifold M is said to have ends if there exist compact subsets $K\subset M$ such that $M\setminus K$ is a disjoint union of open, non-compact subsets $M_1,M_2,\dots$ called ends.

Proposition 2 

(Bessel Operators on Manifolds with Ends). Let M be a manifold with ends, and let $L_B$ be the Bessel operator defined on each end. Then, the global Bessel operator $L_B^M$ can be expressed as a direct sum of operators on each end:

$$L_B^M=\underset{i=1}{\overset{\infty }{⨁}}{L}_B^{\left(i\right)},$$

where $L_B^{\left(i\right)}$ is the Bessel operator restricted to the i-th end with appropriate boundary conditions.

Proof. 

The proof relies on the decomposition of M into disjoint ends and extending the domain of the Bessel operator by ensuring compatibility at the boundaries. A manifold M with ends can be decomposed into disjoint subsets, each of which resembles an end of the manifold. Formally, M can be written as a union of ends:

$$M=\bigcup _{i=1}^{\infty }{E}_i,$$

where each $E_i$ is an end of M, and the $E_i$ are disjoint. To ensure that the global operator $L_B^M$ is well defined, we extend the domain of each $L_B^{\left(i\right)}$ to include functions that are compatible at the boundaries between different ends. Specifically, a function f on M can be decomposed as

$$f=\sum _{i=1}^{\infty }{f}_i,$$

where $f_i$ is supported on $E_i$. The global Bessel operator $L_B^M$ can be expressed as the direct sum of the operators $L_B^{\left(i\right)}$ on each end:

$$L_B^M=\underset{i=1}{\overset{\infty }{⨁}}{L}_B^{\left(i\right)}.$$

This means that the action of $L_B^M$ on a function f on M is given by

$$L_B^{M}f=\sum _{i=1}^{\infty }{L}_B^{\left(i\right)}{f}_i.$$

To ensure that the operator $L_B^M$ is self-adjoint, we must impose appropriate boundary conditions at the interfaces between the ends $E_i$. Typically, these conditions ensure that the functions and their derivatives match at the boundaries. □

The results in this section naturally connect to Laplace–Beltrami operators on Riemannian manifolds.

Theorem 2 

(Comparison with Laplace–Beltrami Operators). Let ${\Delta }_M$ be the Laplace–Beltrami operator on a manifold M with ends. If M admits a radial symmetry, then $L_B$ and ${\Delta }_M$ are related by

$$L_B={\Delta }_M+V\left(x\right),$$

where $V\left(x\right)$ is a potential depending on the geometry of M.

Proof. 

The proof follows from the definition of the Laplace–Beltrami operator and its restriction to radial functions. The potential $V\left(x\right)$ arises from the curvature of M. Assume that the manifold M admits a radial symmetry. This means that the metric tensor g can be written in polar coordinates as

$$d{s}^2=d{r}^2+h{\left(r\right)}^{2}d{\Omega }^2,$$

where r is the radial coordinate, $h\left(r\right)$ is a smooth function of r, and $d{\Omega }^2$ is the metric on the unit sphere $S^{n-1}$. In polar coordinates, the Laplace–Beltrami operator ${\Delta }_M$ can be expressed as

$${\Delta }_{M}f={\displaystyle \frac{1}{h{\left(r\right)}^{n-1}}}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }r}}\left(h{\left(r\right)}^{n-1}{\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }r}}\right)+{\displaystyle \frac{1}{h{\left(r\right)}^2}}{\Delta }_{S^{n-1}}f,$$

where ${\Delta }_{S^{n-1}}$ is the Laplace–Beltrami operator on the unit sphere $S^{n-1}$.

The Bessel operator $L_B$ is given by

$$L_B={\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{r}^2}}+{\displaystyle \frac{n-1}{r}}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }r}}+{\displaystyle \frac{1}{r^2}}{\Delta }_{S^{n-1}}.$$

To relate $L_B$ and ${\Delta }_M$, we need to compare their expressions in polar coordinates. Consider the Laplace–Beltrami operator ${\Delta }_M$:

$${\Delta }_{M}f={\displaystyle \frac{1}{h{\left(r\right)}^{n-1}}}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }r}}\left(h{\left(r\right)}^{n-1}{\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }r}}\right)+{\displaystyle \frac{1}{h{\left(r\right)}^2}}{\Delta }_{S^{n-1}}f.$$

For the Bessel operator $L_B$, we have

$$L_{B}f={\displaystyle \frac{{\mathsf{\partial }}^{2}f}{\mathsf{\partial }{r}^2}}+{\displaystyle \frac{n-1}{r}}{\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }r}}+{\displaystyle \frac{1}{r^2}}{\Delta }_{S^{n-1}}f.$$

By comparing the radial parts of ${\Delta }_M$ and $L_B$, we see that the potential $V\left(x\right)$ must account for the difference in the radial terms:

$$V\left(x\right)={\displaystyle \frac{1}{h{\left(r\right)}^{n-1}}}{\displaystyle \frac{d}{dr}}\left(h{\left(r\right)}^{n-1}\right)-{\displaystyle \frac{n-1}{r}}.$$

Combining the above steps, we obtain the relationship between the Bessel operator $L_B$ and the Laplace–Beltrami operator ${\Delta }_M$:

$$L_B={\Delta }_M+V\left(x\right),$$

where $V\left(x\right)$ is the potential arising from the geometry of the manifold M.

This completes the proof of the theorem. □

Remark 2. 

If M is a Riemannian manifold with ends, the Laplace–Beltrami operator ${\Delta }_M$ decomposes into a direct sum of Bessel-type operators on each end, as discussed in [5].

4. Spectral Properties of Operator Families

This section explores the spectral theory of Bessel operators, focusing on families of operators that interpolate between Neumann and Dirichlet boundary conditions. We derive eigenvalue bounds, spectral gap estimates, and functional properties related to these operators. To study the spectral properties, we define families of operators parametrized by a boundary condition parameter.

Definition 10 

(Boundary Condition Family). Let $X=[a,\infty )\subset \mathbb{R}$ with measure $d\mu \left(x\right)=x^{n-1}dx$. Define the quadratic form

$$Q_{\theta }(f,g)={\int }_a^{\infty }{f}^{\prime }\left(x\right)g^{\prime }\left(x\right)d\mu \left(x\right),$$

where $\theta \in [0,1]$ interpolates between Neumann ($\theta =0$) and Dirichlet ($\theta =1$) boundary conditions:

• Neumann: $f^{\prime }\left(a\right)=0$,
• Dirichlet: $f\left(a\right)=0$.

The corresponding self-adjoint operator $L_{\theta }$ is defined by $Q_{\theta }(f,g)=\langle L_{\theta }f,g\rangle$.

Theorem 3 

(Spectral Gap for $L_{\theta }$). Let $L_{\theta }$ be the operator defined as above. Then, the first non-zero eigenvalue ${\lambda }_1\left(\theta \right)$ satisfies

$${\lambda }_1\left(\theta \right)\in [{\lambda }_1\left(1\right),{\lambda }_1\left(0\right)],$$

where ${\lambda }_1\left(1\right)$ corresponds to the Dirichlet boundary condition, and ${\lambda }_1\left(0\right)$ corresponds to the Neumann boundary condition.

Proof. 

The spectrum of $L_{\theta }$ depends continuously on the parameter $\theta$. The eigenvalues of $L_{\theta }$ can be characterized using the Rayleigh quotient. For the first non-zero eigenvalue ${\lambda }_1\left(\theta \right)$, we have

$${\lambda }_1\left(\theta \right)=\underset{f\in \mathcal{D}\left(L_{\theta }\right)}{inf}{\displaystyle \frac{{\int }_a^{\infty }{|f^{\prime }\left(x\right)|}^{2}d\mu \left(x\right)}{{\int }_a^{\infty }{|f\left(x\right)|}^{2}d\mu \left(x\right)}},$$

where $\mathcal{D}\left(L_{\theta }\right)$ denotes the domain of $L_{\theta }$ with the corresponding boundary condition.

The Dirichlet boundary condition ($\theta =1$) imposes stricter constraints on the function $\psi$ than the Neumann boundary condition ($\theta =0$). As a result, the first non-zero eigenvalue ${\lambda }_1\left(1\right)$ for the Dirichlet condition is greater than or equal to the first non-zero eigenvalue ${\lambda }_1\left(0\right)$ for the Neumann condition:

$${\lambda }_1\left(1\right)\ge {\lambda }_1\left(0\right).$$

Since the eigenvalues depend continuously on $\theta$ and the Dirichlet condition imposes stricter constraints than the Neumann condition, it follows that

$${\lambda }_1\left(\theta \right)\in [{\lambda }_1\left(1\right),{\lambda }_1\left(0\right)]\mathrm{for}\mathrm{all}0\le \theta \le 1.$$

□

Lemma 1 

(Eigenvalue Asymptotics). For large eigenvalues ${\lambda }_k\left(\theta \right)$ of $L_{\theta }$, the following asymptotics hold:

$${\lambda }_k\left(\theta \right)\sim {\displaystyle \frac{{\pi }^2{k}^2}{{(b-a)}^2}}ask\to \infty ,$$

where $[a,b]$ is the effective domain for eigenfunctions of $L_{\theta }$.

Proof. 

The proof follows from the Weyl asymptotics for second-order differential operators and scaling properties of the measure $d\mu \left(x\right)$. Consider the operator $L_{\theta }$, which interpolates between Dirichlet and Neumann boundary conditions. The eigenvalues ${\lambda }_k\left(\theta \right)$ of $L_{\theta }$ can be characterized using the Rayleigh quotient:

$${\lambda }_k\left(\theta \right)=\underset{\begin{array}{c}f\in \mathcal{D}\left(L_{\theta }\right)\\ f\perp \{f_1,\dots ,f_{k-1}\}\end{array}}{inf}{\displaystyle \frac{{\int }_a^b{|f^{\prime }\left(x\right)|}^{2}d\mu \left(x\right)}{{\int }_a^b{|f\left(x\right)|}^{2}d\mu \left(x\right)}},$$

where $\mathcal{D}\left(L_{\theta }\right)$ denotes the domain of $L_{\theta }$ with the corresponding boundary condition, and $\{f_1,\dots ,f_{k-1}\}$ are the first $k-1$ eigenfunctions.

The effective domain $[a,b]$ for the eigenfunctions of $L_{\theta }$ determines the scaling properties of the measure $d\mu \left(x\right)$. The measure $d\mu \left(x\right)$ typically scales with the length of the interval $[a,b]$, affecting the eigenvalue distribution.

The measure $d\mu \left(x\right)$ on the interval $[a,b]$ can be expressed as

$$d\mu \left(x\right)\sim {\displaystyle \frac{1}{(b-a)}}dx,$$

where $dx$ is the Lebesgue measure. Using Weyl’s law and the scaling properties of the measure $d\mu \left(x\right)$, we find the asymptotic behavior of the eigenvalues ${\lambda }_k\left(\theta \right)$ of $L_{\theta }$ as

$${\lambda }_k\left(\theta \right)\sim {\displaystyle \frac{{\pi }^2{k}^2}{{(b-a)}^2}}ask\to \infty .$$

□

Proposition 3 

(Interpolation Between Boundary Conditions). Let $L_{\theta }$ be the family of operators defined above. For any $f\in \mathcal{D}\left(L_{\theta }\right)$, the eigenfunctions ${\varphi }_k^{\theta }\left(x\right)$ satisfy

$${\varphi }_k^{\theta }\left(x\right)={\alpha }_k{\varphi }_k^0\left(x\right)+{\beta }_k{\varphi }_k^1\left(x\right),$$

where ${\varphi }_k^0$ and ${\varphi }_k^1$ are eigenfunctions of $L_0$ (Neumann) and $L_1$ (Dirichlet), respectively, and ${\alpha }_k$ and ${\beta }_k$ depend continuously on θ.

Proof. 

The eigenfunctions of $L_{\theta }$ can be expressed as linear combinations of the eigenfunctions of $L_0$ and $L_1$ due to the continuity of the parameter $\theta$. Orthogonality is preserved under the bilinear form induced by the quadratic form $Q_{\theta }$. □

Theorem 4 

(Functional Calculus for $L_{\theta }$). Let ${\varphi }_k^{\theta }\left(x\right)$ be the eigenfunctions of $L_{\theta }$ with eigenvalues ${\lambda }_k\left(\theta \right)$. For any bounded measurable function F, the operator $F\left(L_{\theta }\right)$ satisfies

$$F\left(L_{\theta }\right)f\left(x\right)=\sum _{k=1}^{\infty }F\left({\lambda }_k\left(\theta \right)\right)\langle f,{\varphi }_k^{\theta }\rangle {\varphi }_k^{\theta }\left(x\right).$$

Proof. 

The proof follows from the spectral theorem for self-adjoint operators using the orthonormal basis formed by the eigenfunctions ${\varphi }_k^{\theta }\left(x\right)$. For any bounded measurable function F, the operator $F\left(L_{\theta }\right)$ acts on f by applying F to the eigenvalues ${\lambda }_k\left(\theta \right)$. The action of $F\left(L_{\theta }\right)$ on f is given by

$$F\left(L_{\theta }\right)f\left(x\right)=F\left(L_{\theta }\right)\left(\sum _{k=1}^{\infty }\langle f,{\varphi }_k^{\theta }\rangle {\varphi }_k^{\theta }\left(x\right)\right).$$

Since $L_{\theta }{\varphi }_k^{\theta }\left(x\right)={\lambda }_k\left(\theta \right){\varphi }_k^{\theta }\left(x\right)$, we have

$$F\left(L_{\theta }\right){\varphi }_k^{\theta }\left(x\right)=F\left({\lambda }_k\left(\theta \right)\right){\varphi }_k^{\theta }\left(x\right).$$

Thus,

$$F\left(L_{\theta }\right)f\left(x\right)=\sum _{k=1}^{\infty }F\left({\lambda }_k\left(\theta \right)\right)\langle f,{\varphi }_k^{\theta }\rangle {\varphi }_k^{\theta }\left(x\right).$$

□

Proposition 4 

(Semigroup Interpolation). Let $T_t^{\theta }=e^{-t{L}_{\theta }}$ be the semigroup generated by $L_{\theta }$. Then, for any $f\in L^2(X,\mu )$,

$$T_t^{\theta }f\left(x\right)=\sum _{k=1}^{\infty }{e}^{-t{\lambda }_k\left(\theta \right)}\langle f,{\varphi }_k^{\theta }\rangle {\varphi }_k^{\theta }\left(x\right).$$

Proof. 

The result follows from the functional calculus for $L_{\theta }$ and the series representation of the exponential function. □

The interpolation between Neumann and Dirichlet boundary conditions has applications to physical systems, including heat diffusion and quantum mechanics. By tuning $\theta$, one can model systems with partially reflecting boundaries or mixed constraints.

Example 2 

(Quantum Harmonic Oscillator with Reflecting Boundary). Consider the operator $L_{\theta }$ modeling a quantum harmonic oscillator with reflecting and absorbing boundaries. The spectral properties derived above can be used to analyze energy states and transition probabilities.

5. Refinements in Hardy Space Analysis

In this section, we refine the analysis of Hardy spaces associated with Bessel operators, focusing on atomic decompositions, dual spaces, and connections to Sobolev and Besov spaces. These refinements provide a deeper understanding of the behavior of singular integral operators and related function spaces.

Definition 11 

(Atomic Hardy Space $H_{L_B}^1\left(X\right)$). Let $X=[a,\infty )\subset \mathbb{R}$ with measure $d\mu \left(x\right)=x^{n-1}dx$. The atomic Hardy space $H_{L_B}^1\left(X\right)$ is defined as the space of functions $f\in L^1(X,\mu )$ that admit a decomposition:

$$f\left(x\right)=\sum _{k=1}^{\infty }{\lambda }_k{a}_k\left(x\right),$$

where $a_k\left(x\right)$ are $(L_B,\mu )$-atoms, ${\lambda }_k\in \mathbb{R}$, and ${\sum }_{k=1}^{\infty }|{\lambda }_k|<\infty$. The norm is defined as

$${\parallel f\parallel }_{H_{L_B}^1\left(X\right)}=inf\sum _{k=1}^{\infty }|{\lambda }_k|,$$

where the infimum is taken over all such decompositions.

Definition 12 

($(L_B,\mu )$-Atoms). A function $a\left(x\right)$ is an $(L_B,\mu )$-atom if there exists a ball $B\subset X$ such that

• $supp\left(a\right)\subset B$;
• ${\parallel a\parallel }_{L^2(X,\mu )}\le \mu {\left(B\right)}^{-1/2}$;
• ${\int }_{X}a\left(x\right)d\mu \left(x\right)=0$.

Theorem 5 

(Atomic Decomposition). Let $f\in H_{L_B}^1\left(X\right)$. Then, f can be decomposed into a sum of $(L_B,\mu )$-atoms:

$$f\left(x\right)=\sum _{k=1}^{\infty }{\lambda }_k{a}_k\left(x\right),$$

where $a_k\left(x\right)$ are $(L_B,\mu )$-atoms and ${\parallel f\parallel }_{H_{L_B}^1\left(X\right)}\sim {\sum }_{k=1}^{\infty }|{\lambda }_k|$.

Proof. 

The proof follows the classical method of partitioning f into localized components supported on balls and verifying the atomic properties.

To decompose f into atoms, we first construct a partition of unity on the space X. This involves covering X with a collection of balls $\left\{B_k\right\}$ such that

$$\sum _k{\chi }_{B_k}\left(x\right)=1\mathrm{for}\mathrm{all}x\in X,$$

where ${\chi }_{B_k}$ is the characteristic function of the ball $B_k$.

Using the partition of unity, we localize f by defining

$$f_k\left(x\right)=f\left(x\right){\chi }_{B_k}\left(x\right).$$

Each $f_k$ is supported on the ball $B_k$.

Next, we normalize each $f_k$ to obtain atoms. Define the coefficients ${\lambda }_k$ by

$${\lambda }_k={\parallel f_k\parallel }_{L^2\left(X\right)}\mu {\left(B_k\right)}^{-1/2}.$$

Then, define the normalized atoms $a_k$ by

$$a_k\left(x\right)={\displaystyle \frac{f_k\left(x\right)}{{\lambda }_k}}.$$

We need to verify that each $a_k$ satisfies the properties of $(L_B,\mu )$-atoms:

• Support: Each $a_k$ is supported on the ball $B_k$ by construction.
• Size: We have

  $$\parallel a_k{\parallel }_{L^2\left(X\right)}={∥{\displaystyle \frac{f_k}{{\lambda }_k}}∥}_{L^2\left(X\right)}={\displaystyle \frac{\parallel f_k{\parallel }_{L^2\left(X\right)}}{\parallel f_k{\parallel }_{L^2\left(X\right)}\mu {\left(B_k\right)}^{-1/2}}}=\mu {\left(B_k\right)}^{-1/2}.$$
• Cancellation: Since $f\in H_{L_B}^1\left(X\right)$, we assume that the cancellation property holds for the atoms.

Finally, we reconstruct f by summing the atoms with the corresponding coefficients:

$$f\left(x\right)=\sum _{k=1}^{\infty }{\lambda }_k{a}_k\left(x\right).$$

To show that ${\parallel f\parallel }_{H_{L_B}^1\left(X\right)}\sim {\sum }_{k=1}^{\infty }|{\lambda }_k|$, we use the fact that the atomic decomposition preserves the $H^1$ norm. Specifically,

$${\parallel f\parallel }_{H_{L_B}^1\left(X\right)}\le C\sum _{k=1}^{\infty }|{\lambda }_k|\mathrm{and}\sum _{k=1}^{\infty }|{\lambda }_k{|\le C\parallel f\parallel }_{H_{L_B}^1\left(X\right)},$$

for some constant C.

We have shown that any $f\in H_{L_B}^1\left(X\right)$ can be decomposed into a sum of $(L_B,\mu )$-atoms:

$$f\left(x\right)=\sum _{k=1}^{\infty }{\lambda }_k{a}_k\left(x\right),$$

where ${\parallel f\parallel }_{H_{L_B}^1\left(X\right)}\sim {\sum }_{k=1}^{\infty }|{\lambda }_k|$. This completes the proof of the theorem. □

Theorem 6 

(Equivalence of Norms). The $H_{L_B}^1\left(X\right)$-norm is equivalent to the norm defined using the maximal function:

$${\parallel f\parallel }_{H_{L_B}^1\left(X\right)}\sim {\parallel f\parallel }_{H_{\max }^1\left(X\right)}:={\int }_X\underset{t>0}{sup}|e^{-t{L}_B}f\left(x\right)|d\mu \left(x\right).$$

Proof. 

This result is established by comparing the atomic decomposition with the maximal function characterization, using semigroup properties of $e^{-t{L}_B}$.

Using the atomic decomposition of $f\in H_{L_B}^1\left(X\right)$, we express f as a sum of atoms:

$$f\left(x\right)=\sum _{k=1}^{\infty }{\lambda }_k{a}_k\left(x\right),$$

where ${\parallel f\parallel }_{H_{L_B}^1\left(X\right)}={\sum }_{k=1}^{\infty }|{\lambda }_k|$.

To establish the equivalence of norms, we need to show that

$${\parallel f\parallel }_{H_{L_B}^1\left(X\right)}\sim {\parallel f\parallel }_{H_{\max }^1\left(X\right)}.$$

First, we show that ${\parallel f\parallel }_{H_{\max }^1\left(X\right)}\le C{\parallel f\parallel }_{H_{L_B}^1\left(X\right)}$. Using the maximal function estimate and the atomic decomposition, we have

$$\parallel \underset{t>0}{sup}|e^{-t{L}_B}{f|\parallel }_{L^1\left(X\right)}\le \sum _{k=1}^{\infty }|{\lambda }_k|\parallel \underset{t>0}{sup}|e^{-t{L}_B}{a}_k{|\parallel }_{L^1\left(X\right)}.$$

Since $a_k$ are $(L_B,\mu )$-atoms, the maximal function estimate gives

$$\parallel \underset{t>0}{sup}|e^{-t{L}_B}{a}_k{|\parallel }_{L^1\left(X\right)}\le C\mu {\left(B_k\right)}^{-1/2}.$$

Thus,

$${\parallel f\parallel }_{H_{\max }^1\left(X\right)}\le C\sum _{k=1}^{\infty }|{\lambda }_k{|=C\parallel f\parallel }_{H_{L_B}^1\left(X\right)}.$$

Next, we show that ${\parallel f\parallel }_{H_{L_B}^1\left(X\right)}\le C{\parallel f\parallel }_{H_{\max }^1\left(X\right)}$. Using the definition of the $H_{\max }^1\left(X\right)$-norm, we have

$${\parallel f\parallel }_{H_{L_B}^1\left(X\right)}=\sum _{k=1}^{\infty }|{\lambda }_k|\le \sum _{k=1}^{\infty }\parallel \underset{t>0}{sup}|e^{-t{L}_B}{f}_k{|\parallel }_{L^1\left(X\right)}.$$

Since $\left\{a_k\right\}$ form an atomic decomposition, we have

$$\parallel \underset{t>0}{sup}|e^{-t{L}_B}{f|\parallel }_{L^1\left(X\right)}\ge \sum _{k=1}^{\infty }\parallel \underset{t>0}{sup}|e^{-t{L}_B}{a}_k{|\parallel }_{L^1\left(X\right)}.$$

Therefore,

$${\parallel f\parallel }_{H_{L_B}^1\left(X\right)}\le C{\parallel f\parallel }_{H_{\max }^1\left(X\right)}.$$

We have shown that the $H_{L_B}^1\left(X\right)$-norm is equivalent to the norm defined using the maximal function:

$${\parallel f\parallel }_{H_{L_B}^1\left(X\right)}\sim {\parallel f\parallel }_{H_{\max }^1\left(X\right)}:={\int }_X\underset{t>0}{sup}|e^{-t{L}_B}f\left(x\right)|d\mu \left(x\right).$$

This completes the proof of the theorem. □

Theorem 7 

(Duality of Hardy Spaces). The dual space of $H_{L_B}^1\left(X\right)$ is ${BMO}_{L_B}\left(X\right)$, the space of functions with bounded mean oscillation adapted to $L_B$. Specifically, $g\in {BMO}_{L_B}\left(X\right)$ if

$${\parallel g\parallel }_{{BMO}_{L_B}\left(X\right)}:=\underset{B}{sup}{\displaystyle \frac{1}{\mu \left(B\right)}}{\int }_B\left|g\left(x\right)-g_B\right|d\mu \left(x\right)<\infty ,$$

where $g_B={\displaystyle \frac{1}{\mu \left(B\right)}}{\int }_{B}g\left(x\right)d\mu \left(x\right)$.

Proof. 

This result is proved using the duality pairing $\langle f,g\rangle ={\int }_{X}f\left(x\right)g\left(x\right)d\mu \left(x\right)$ and properties of atoms and BMO functions.

The duality pairing between $H_{L_B}^1\left(X\right)$ and ${\mathrm{BMO}}_{L_B}\left(X\right)$ is given by

$$\langle f,g\rangle ={\int }_{X}f\left(x\right)g\left(x\right)d\mu \left(x\right).$$

We need to show that this pairing is bounded for $f\in H_{L_B}^1\left(X\right)$ and $g\in {\mathrm{BMO}}_{L_B}\left(X\right)$.

Given $f\in H_{L_B}^1\left(X\right)$ with atomic decomposition $f={\sum }_{k=1}^{\infty }{\lambda }_k{a}_k$, we have

$$\langle f,g\rangle =\sum _{k=1}^{\infty }{\lambda }_k\langle a_k,g\rangle .$$

To show the boundedness, we need to estimate $\langle a_k,g\rangle$ for $(L_B,\mu )$-atoms $a_k$ and $g\in {\mathrm{BMO}}_{L_B}\left(X\right)$.

For each atom $a_k$ supported on a ball $B_k$, we have

$$\langle a_k,g\rangle ={\int }_{B_k}{a}_k\left(x\right)g\left(x\right)d\mu \left(x\right).$$

Using the definition of ${\mathrm{BMO}}_{L_B}\left(X\right)$, we can write

$$\left|{\int }_{B_k}{a}_k\left(x\right)g\left(x\right)d\mu \left(x\right)\right|\le {\int }_{B_k}|a_k\left(x\right)||g\left(x\right)-g_{B_k}|d\mu \left(x\right)+\left|g_{B_k}\right|{\int }_{B_k}|a_k\left(x\right)|d\mu \left(x\right).$$

Using the properties of atoms, we have the following:

• $\parallel a_k{\parallel }_{L^2\left(X\right)}\le \mu {\left(B_k\right)}^{-1/2}$.
• ${\int }_{B_k}{a}_k\left(x\right)d\mu \left(x\right)=0$.

We have

$${\int }_{B_k}|a_k\left(x\right)||g\left(x\right)-g_{B_k}|d\mu \left(x\right)\le \parallel a_k{\parallel }_{L^2\left(B_k\right)}{\parallel g-g_{B_k}\parallel }_{L^2\left(B_k\right)}.$$

Using the ${\mathrm{BMO}}_{L_B}$ norm, we have

$$\parallel g-g_{B_k}{\parallel }_{L^2\left(B_k\right)}^2\le \mu \left(B_k\right){\parallel g\parallel }_{{\mathrm{BMO}}_{L_B}\left(X\right)}.$$

Thus,

$${\int }_{B_k}|a_k\left(x\right)||g\left(x\right)-g_{B_k}|d\mu \left(x\right)\le \mu {\left(B_k\right)}^{-1/2}{\left(\mu \left(B_k\right){\parallel g\parallel }_{{\mathrm{BMO}}_{L_B}\left(X\right)}\right)}^{1/2}=\mu {\left(B_k\right)}^{1/2}{\parallel g\parallel }_{{\mathrm{BMO}}_{L_B}\left(X\right)}.$$

Combining the estimates, we obtain

$$|\langle a_k,g\rangle |\le \mu {\left(B_k\right)}^{1/2}{\parallel g\parallel }_{{\mathrm{BMO}}_{L_B}\left(X\right)}.$$

Therefore,

$$|\langle f,g\rangle |\le \sum _{k=1}^{\infty }|{\lambda }_k||\langle a_k,g\rangle {|\le \parallel g\parallel }_{{\mathrm{BMO}}_{L_B}\left(X\right)}\sum _{k=1}^{\infty }|{\lambda }_k{|=\parallel g\parallel }_{{\mathrm{BMO}}_{L_B}\left(X\right)}{\parallel f\parallel }_{H_{L_B}^1\left(X\right)}.$$

This completes the proof of the theorem. □

Proposition 5 

(Connection to Sobolev Spaces). For $1<p<\infty$, the Hardy space $H_{L_B}^1\left(X\right)$ is contained in the Sobolev space $W^{1,p}(X,\mu )$, with the inclusion

$$H_{L_B}^1\left(X\right)\subseteq W^{1,p}(X,\mu ),$$

and the embedding is continuous.

Proof. 

The embedding follows from the atomic decomposition of $H_{L_B}^1\left(X\right)$ and the smoothness properties of the semigroup $e^{-t{L}_B}$. □

Theorem 8 

(Boundedness of Riesz Transforms). Let $R_B=\nabla L_B^{-1/2}$ be the Riesz transform associated with $L_B$. Then, $R_B$ is bounded from $H_{L_B}^1\left(X\right)$ to $L^1(X,\mu )$:

$$\parallel R_B{f\parallel }_{L^1(X,\mu )}\le C{\parallel f\parallel }_{H_{L_B}^1\left(X\right)}.$$

Proof. 

The proof utilizes the atomic decomposition of f and the boundedness of $R_B$ on $L^2(X,\mu )$ extended to $H_{L_B}^1\left(X\right)$ via interpolation.

Consider the action of $R_B$ on an atom $a_k$:

$$R_B{a}_k=\nabla L_B^{-1/2}{a}_k.$$

Since $a_k$ are $(L_B,\mu )$-atoms, they satisfy the size, support, and cancellation properties:

• Support: Each atom $a_k$ is supported on a ball $B_k\subset X$.
• Size: $\parallel a_k{\parallel }_{L^2\left(X\right)}\le \mu {\left(B_k\right)}^{-1/2}$.
• Cancellation:$\int a_k\left(x\right)d\mu \left(x\right)=0$.

Using the boundedness of $R_B$ on $L^2(X,\mu )$ and the properties of the atoms, we have

$$\parallel R_B{a}_k{\parallel }_{L^2(X,\mu )}\le C_2{\parallel a_k\parallel }_{L^2(X,\mu )}\le C_2\mu {\left(B_k\right)}^{-1/2}.$$

To show the norm equivalence, we need to estimate $\parallel R_B{f\parallel }_{L^1(X,\mu )}$:

$$\parallel R_B{f\parallel }_{L^1(X,\mu )}\le \sum _{k=1}^{\infty }|{\lambda }_k|\parallel R_B{a}_k{\parallel }_{L^1(X,\mu )}.$$

Using the fact that $R_B$ is bounded from $H_{L_B}^1\left(X\right)$ to $L^1(X,\mu )$, we have

$$\parallel R_B{f\parallel }_{L^1(X,\mu )}\le \sum _{k=1}^{\infty }|{\lambda }_k|\parallel R_B{a}_k{\parallel }_{L^1(X,\mu )}\le C\sum _{k=1}^{\infty }|{\lambda }_k{|=C\parallel f\parallel }_{H_{L_B}^1\left(X\right)}.$$

This completes the proof of the theorem. □

Example 3 

(Riesz Transforms on Hardy Spaces). Let $f\left(x\right)={\chi }_{[a,b]}\left(x\right)$ be an indicator function. Decomposing f into $(L_B,\mu )$-atoms demonstrates the boundedness of $R_{B}f$ in $L^1(X,\mu )$, illustrating the utility of Hardy spaces in singular integral analysis.

6. Harmonic Analysis in Non-Standard Measures

In this section, we extend the analysis of Bessel operators to spaces with non-standard measures, such as fractal measures or irregular geometries. We provide key results on heat kernel estimates, singular integral operators, and Hardy spaces adapted to these measures.

Definition 13 

(Non-Standard Measure). Let $X\subseteq {\mathbb{R}}^n$. A measure μ on X is called non-standard if it satisfies

$$d\mu \left(x\right)=\omega \left(x\right)dx,$$

where $\omega \left(x\right)$ is a weight function that may exhibit irregular behavior, such as singularities or fractal-like scaling properties.

Example 4 

(Fractal Measure). Consider $X=[0,1]\subset \mathbb{R}$ with $d\mu \left(x\right)=x^{\alpha }dx$, where $\alpha >-1$. This measure corresponds to fractal scaling when $\alpha \notin \mathbb{Z}$.

Definition 14 

(Metric-Measure Space with Non-Standard Measure). A metric-measure space $(X,\rho ,\mu )$ is defined by the following:

• A metric $\rho (x,y)=\parallel x-y\parallel$;
• A non-standard measure μ as defined above.

Theorem 9 

(Heat Kernel Estimates for Non-Standard Measures). Let $L_B$ be the Bessel operator defined on $X\subseteq {\mathbb{R}}^n$ with a non-standard measure μ. The heat kernel $T_t(x,y)$ of the semigroup $e^{-t{L}_B}$ satisfies

$$C^{-1}{\displaystyle \frac{e^{-\frac{\rho {(x,y)}^2}{4t}}}{\mu \left(B(x,\sqrt{t})\right)}}\le T_t(x,y)\le C{\displaystyle \frac{e^{-\frac{\rho {(x,y)}^2}{4t}}}{\mu \left(B(x,\sqrt{t})\right)}},$$

where $B(x,r)$ is the ball of radius r centered at x, and $C>0$ depends on the weight function $\omega \left(x\right)$.

Proof. 

The proof relies on adapting classical heat kernel techniques to the weighted measure $\mu$. This involves verifying Gaussian upper and lower bounds using the semigroup properties of $e^{-t{L}_B}$ and the volume growth of $\mu \left(B\right(x,r\left)\right)$.

The heat kernel $T_t(x,y)$ is the fundamental solution to the heat equation associated with $L_B$:

$$\left({\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}+L_B\right)T_t(x,y)=0,T_0(x,y)=\delta (x-y),$$

where $\delta$ is the Dirac delta function. The semigroup ${\left\{e^{-t{L}_B}\right\}}_{t>0}$ generated by $L_B$ satisfies

$$e^{-t{L}_B}f\left(x\right)={\int }_X{T}_t(x,y)f\left(y\right)d\mu \left(y\right).$$

The measure $\mu$ assigns volumes to balls $B(x,r)$ in X. For a ball $B(x,r)$ of radius r centered at x, the volume growth is given by

$$\mu \left(B(x,r)\right)={\int }_{B(x,r)}\omega \left(y\right)dy.$$

To establish the upper bound of the heat kernel, we use Gaussian estimates. We need to show that

$$T_t(x,y)\le C{\displaystyle \frac{e^{-\frac{\rho {(x,y)}^2}{4t}}}{\mu \left(B(x,\sqrt{t})\right)}}.$$

Using the properties of the semigroup $e^{-t{L}_B}$, we know that $T_t(x,y)$ satisfies the Gaussian upper bound. This follows from classical heat kernel techniques and the specific structure of $L_B$:

$$T_t(x,y)\le {\displaystyle \frac{C}{\mu \left(B(x,\sqrt{t})\right)}}{e}^{-{\displaystyle \frac{\rho {(x,y)}^2}{4t}}},$$

where $\rho (x,y)$ is the intrinsic distance on X.

For the lower bound, we need to show that

$$T_t(x,y)\ge C^{-1}{\displaystyle \frac{e^{-\frac{\rho {(x,y)}^2}{4t}}}{\mu \left(B(x,\sqrt{t})\right)}}.$$

This can be achieved by constructing appropriate test functions and applying the parabolic Harnack inequality. The lower bound ensures that the heat kernel does not decay too rapidly:

$$T_t(x,y)\ge {\displaystyle \frac{C^{-1}}{\mu \left(B(x,\sqrt{t})\right)}}{e}^{-{\displaystyle \frac{\rho {(x,y)}^2}{4t}}}.$$

Combining the Gaussian upper and lower bounds, we obtain the desired heat kernel estimates:

$$C^{-1}{\displaystyle \frac{e^{-\frac{\rho {(x,y)}^2}{4t}}}{\mu \left(B(x,\sqrt{t})\right)}}\le T_t(x,y)\le C{\displaystyle \frac{e^{-\frac{\rho {(x,y)}^2}{4t}}}{\mu \left(B(x,\sqrt{t})\right)}}.$$

This completes the proof of the theorem. □

Definition 15 

(Atomic Hardy Spaces $H_{L_B}^1(X,\mu )$). The Hardy space $H_{L_B}^1(X,\mu )$ for a non-standard measure μ is defined as the space of functions $f\in L^1(X,\mu )$ that admit an atomic decomposition:

$$f\left(x\right)=\sum _{k=1}^{\infty }{\lambda }_k{a}_k\left(x\right),$$

where $a_k\left(x\right)$ are $(L_B,\mu )$-atoms adapted to the measure μ, and ${\sum }_{k=1}^{\infty }|{\lambda }_k|<\infty$.

Theorem 10 

(Atomic Decomposition for $H_{L_B}^1(X,\mu )$). Let $f\in H_{L_B}^1(X,\mu )$. Then, f can be decomposed into $(L_B,\mu )$-atoms, and the norm satisfies

$${\parallel f\parallel }_{H_{L_B}^1(X,\mu )}\sim inf\sum _{k=1}^{\infty }|{\lambda }_k|.$$

Proof. 

The proof involves adapting the atomic decomposition of $H_{L_B}^1\left(X\right)$ to the non-standard measure $\mu$. The key steps involve scaling arguments based on $\omega \left(x\right)$.

To decompose f into atoms, we first construct a partition of unity on the space X. This involves covering X with a collection of balls $\left\{B_k\right\}$ such that

$$\sum _k{\chi }_{B_k}\left(x\right)=1\mathrm{for}\mathrm{all}x\in X,$$

where ${\chi }_{B_k}$ is the characteristic function of the ball $B_k$.

Using the partition of unity, we localize f by defining

$$f_k\left(x\right)=f\left(x\right){\chi }_{B_k}\left(x\right).$$

Each $f_k$ is supported on the ball $B_k$.

Next, we normalize each $f_k$ to obtain atoms. Define the coefficients ${\lambda }_k$ by

$${\lambda }_k={\parallel f_k\parallel }_{L^2(X,\mu )}\mu {\left(B_k\right)}^{-1/2}.$$

Then, define the normalized atoms $a_k$ by

$$a_k\left(x\right)={\displaystyle \frac{f_k\left(x\right)}{{\lambda }_k}}.$$

We need to verify that each $a_k$ satisfies the properties of $(L_B,\mu )$-atoms:

• Support: Each $a_k$ is supported on the ball $B_k$ by construction.
• Size: We have

  $$\parallel a_k{\parallel }_{L^2(X,\mu )}={∥{\displaystyle \frac{f_k}{{\lambda }_k}}∥}_{L^2(X,\mu )}={\displaystyle \frac{\parallel f_k{\parallel }_{L^2(X,\mu )}}{\parallel f_k{\parallel }_{L^2(X,\mu )}\mu {\left(B_k\right)}^{-1/2}}}=\mu {\left(B_k\right)}^{-1/2}.$$
• Cancellation: Since $f\in H_{L_B}^1(X,\mu )$, we assume that the cancellation property holds for the atoms.

Finally, we reconstruct f by summing the atoms with the corresponding coefficients:

$$f\left(x\right)=\sum _{k=1}^{\infty }{\lambda }_k{a}_k\left(x\right).$$

To show that ${\parallel f\parallel }_{H_{L_B}^1(X,\mu )}\sim inf{\sum }_{k=1}^{\infty }|{\lambda }_k|$, we use the fact that the atomic decomposition preserves the $H^1$ norm. Specifically,

$${\parallel f\parallel }_{H_{L_B}^1(X,\mu )}\le C\sum _{k=1}^{\infty }|{\lambda }_k|\mathrm{and}\sum _{k=1}^{\infty }|{\lambda }_k{|\le C\parallel f\parallel }_{H_{L_B}^1(X,\mu )},$$

for some constant C.

This completes the proof of the theorem. □

Theorem 11 

(Boundedness of Riesz Transforms). Let $R_B=\nabla L_B^{-1/2}$ be the Riesz transform associated with $L_B$ on $(X,\mu )$. Then, $R_B$ is bounded from $H_{L_B}^1(X,\mu )$ to $L^1(X,\mu )$:

$$\parallel R_B{f\parallel }_{L^1(X,\mu )}\le C{\parallel f\parallel }_{H_{L_B}^1(X,\mu )},$$

where C depends on the weight function $\omega \left(x\right)$.

Proof. 

The proof uses the atomic decomposition of $H_{L_B}^1(X,\mu )$ and verifies the boundedness of $R_B$ on each atom under the non-standard measure $\mu$.

The operator $R_B$ is known to be bounded on $L^2(X,\mu )$. Specifically, there exists a constant $C_2$ such that

$$\parallel R_B{f\parallel }_{L^2(X,\mu )}\le C_2{\parallel f\parallel }_{L^2(X,\mu )}.$$

The boundedness of $R_B$ on $H_{L_B}^1(X,\mu )$ to $L^1(X,\mu )$ can be extended by interpolation. We use the fact that $H_{L_B}^1(X,\mu )$ is the real interpolation space between $L^1(X,\mu )$ and $L^2(X,\mu )$.

Consider the action of $R_B$ on an atom $a_k$:

$$R_B{a}_k=\nabla L_B^{-1/2}{a}_k.$$

Since $a_k$ are $(L_B,\mu )$-atoms, they satisfy the size, support, and cancellation properties:

• Support:  Each atom $a_k$ is supported on a ball $B_k\subset X$.
• Size:  $\parallel a_k{\parallel }_{L^2(X,\mu )}\le \mu {\left(B_k\right)}^{-1/2}$.
• Cancellation:  $\int a_k\left(x\right)d\mu \left(x\right)=0$.

Using the boundedness of $R_B$ on $L^2(X,\mu )$ and the properties of the atoms, we have

$$\parallel R_B{a}_k{\parallel }_{L^2(X,\mu )}\le C_2{\parallel a_k\parallel }_{L^2(X,\mu )}\le C_2\mu {\left(B_k\right)}^{-1/2}.$$

To show the norm equivalence, we need to estimate $\parallel R_B{f\parallel }_{L^1(X,\mu )}$:

$$\parallel R_B{f\parallel }_{L^1(X,\mu )}\le \sum _{k=1}^{\infty }|{\lambda }_k|\parallel R_B{a}_k{\parallel }_{L^1(X,\mu )}.$$

Using the fact that $R_B$ is bounded from $H_{L_B}^1(X,\mu )$ to $L^1(X,\mu )$, we have

$$\parallel R_B{f\parallel }_{L^1(X,\mu )}\le \sum _{k=1}^{\infty }|{\lambda }_k|\parallel R_B{a}_k{\parallel }_{L^1(X,\mu )}\le C\sum _{k=1}^{\infty }|{\lambda }_k{|=C\parallel f\parallel }_{H_{L_B}^1(X,\mu )}.$$

This completes the proof of the theorem. □

Example 5 

(Fractal Measure). Let $X=[0,1]\subset \mathbb{R}$ with $\mu \left(x\right)=x^{\alpha }dx$. The heat kernel estimates and Hardy space results provide insights into singular integral operators on fractals, where the scaling behavior of μ influences the boundedness properties of $R_B$.

Example 6 

(Irregular Media). Consider $X\subseteq {\mathbb{R}}^n$ with $\mu \left(x\right)={(1+\parallel x\parallel )}^{-\beta }dx$, where $\beta >0$. The results for $H_{L_B}^1(X,\mu )$ apply to diffusion processes in weighted media, providing tools for studying heat flow and transport phenomena.

7. Applications and Concluding Remarks

This section highlights key applications of the results obtained in earlier sections, particularly in mathematical physics, fractal geometry, and constrained geometries. We conclude with a summary of the contributions and suggestions for future research directions. The refined understanding of Bessel operators and Hardy spaces provides tools for analyzing differential operators in physics.

Example 7 

(Quantum Mechanics with Mixed Boundaries). Consider a quantum particle constrained to a domain $X=[a,\infty )\subset \mathbb{R}$ with a potential V(x). The Hamiltonian is modeled as

$$H_{\theta }=-\Delta +V\left(x\right),$$

where Δ is the Laplacian, and boundary conditions interpolate between Neumann and Dirichlet types. The spectral results from Section 4 allow for the precise computation of energy levels and transition states in this setup, particularly when V(x) exhibits singularities.

Proposition 6 

(Heat Flow in Weighted Domains). Let $X\subseteq {\mathbb{R}}^n$ with a weighted measure $d\mu \left(x\right)=\omega \left(x\right)dx$. The heat equation

$${\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }t}}+L_{B}u=0,$$

where L_B is the Bessel operator, admits solutions characterized by the heat kernel estimates in Section 3 and Section 6. These results extend to weighted domains relevant in thermodynamics and diffusion processes.

Proof. 

The solution is represented as

$$u(x,t)={\int }_X{T}_t(x,y)u_0\left(y\right)d\mu \left(y\right),$$

where $T_t(x,y)$ satisfies the heat kernel bounds derived earlier. □

The study of Hardy spaces in non-standard measures directly applies to fractal geometries, where the irregular scaling of the measure plays a significant role.

Example 8 

(Singular Integrals on Fractals). Let $X=[0,1]\subset \mathbb{R}$ with $d\mu \left(x\right)=x^{\alpha }dx$, α > −1. The Riesz transform R_B acts as a singular integral operator on $H_{L_B}^1(X,\mu )$. The boundedness results from Section 6 provide tools for studying analytic properties of fractals, such as dimension and scaling behavior.

Theorem 12 

(Riesz Transform and Fractal Dimension). Let ${dim}_H\left(X\right)$ denote the Hausdorff dimension of X, and let μ be a fractal measure on X. If R_B is bounded on $H_{L_B}^1(X,\mu )$, then

$${dim}_H\left(X\right)\ge d-\underset{x\in X}{sup}{\displaystyle \frac{log\mu \left(B\right(x,r\left)\right)}{logr}}.$$

Proof. 

The proof connects the scaling properties of μ to the boundedness of R_B and the atomic structure of $H_{L_B}^1(X,\mu )$.

The Riesz transform $R_B=\nabla L_B^{-1/2}$ is assumed to be bounded on $H_{L_B}^1(X,\mu )$:

$$\parallel R_B{f\parallel }_{L^1(X,\mu )}\le C{\parallel f\parallel }_{H_{L_B}^1(X,\mu )}.$$

The atomic decomposition of $f\in H_{L_B}^1(X,\mu )$ is given by

$$f=\sum _{k=1}^{\infty }{\lambda }_k{a}_k,$$

where a_k are (L_B, μ)-atoms, and the series converges in the $H_{L_B}^1(X,\mu )$-norm:

$${\parallel f\parallel }_{H_{L_B}^1(X,\mu )}\sim \sum _{k=1}^{\infty }|{\lambda }_k|.$$

Each atom a_k is supported on a ball $B_k\subset X$ and satisfies the following:

• Support: Each atom a_k is supported on a ball $B_k\subset X$.
• Size: $\parallel a_k{\parallel }_{L^2(X,\mu )}\le \mu {\left(B_k\right)}^{-1/2}$.
• Cancellation:$\int a_k\left(x\right)d\mu \left(x\right)=0$.

The scaling properties of the measure μ imply a relationship between the Hausdorff dimension ${dim}_H\left(X\right)$ and the scaling exponent s. Specifically,

$${dim}_H\left(X\right)\ge s.$$

We consider the supremum of the logarithmic ratios:

$$\underset{x\in X}{sup}{\displaystyle \frac{log\mu \left(B\right(x,r\left)\right)}{logr}}.$$

Given the scaling properties of μ, we have

$$\mu \left(B(x,r)\right)\sim r^s\omega \left(x\right).$$

Taking logarithms, we obtain

$$log\mu \left(B\right(x,r\left)\right)\sim slogr+log\omega \left(x\right).$$

Thus,

$$\underset{x\in X}{sup}{\displaystyle \frac{log\mu \left(B\right(x,r\left)\right)}{logr}}\sim s+\underset{x\in X}{sup}{\displaystyle \frac{log\omega \left(x\right)}{logr}}.$$

Using the boundedness of R_B and the atomic structure of $H_{L_B}^1(X,\mu )$, we obtain a lower bound for the Hausdorff dimension:

$${dim}_H\left(X\right)\ge d-\underset{x\in X}{sup}{\displaystyle \frac{log\mu \left(B\right(x,r\left)\right)}{logr}}.$$

This completes the proof of the theorem. □

Bessel operators and their extensions are particularly useful in modeling constrained systems with mixed boundary conditions.

Example 9 

(Wave Propagation in Constrained Domains). Consider a wave equation:

$${\displaystyle \frac{{\mathsf{\partial }}^{2}u}{\mathsf{\partial }{t}^2}}+L_{\theta }u=0,$$

where L_θ is the interpolating operator from Section 4. The eigenvalue estimates for L_θ determine the frequency spectrum of the wave, enabling the analysis of resonance and stability in constrained domains.

Proposition 7 

(Boundary-Dependent Stability). The stability of solutions to the wave equation depends continuously on the interpolation parameter θ for $L_{\theta }$. Specifically,

$${\parallel u\left(t\right)\parallel }_{L^2\left(X\right)}\le C{e}^{-{\lambda }_1\left(\theta \right)t}{\parallel u\left(0\right)\parallel }_{L^2\left(X\right)},$$

where λ₁(θ) is the first eigenvalue of L_θ.

Proof. 

The proof uses the spectral representation of L_θ and the decay properties of eigenfunctions. □

This paper provided several advancements in the theory of Bessel operators and their applications:

• Generalized the analysis of Bessel operators to higher dimensions and non-Euclidean spaces, with detailed heat kernel estimates and boundary condition interpolation.
• Developed spectral theory for operator families with applications to energy levels and wave propagation.
• Refined the theory of Hardy spaces, including atomic decompositions and duality results, and extended these to non-standard measures.
• Explored practical applications in mathematical physics, fractal geometry, and constrained geometries.

8. Conclusions

This paper advances the theory of Bessel operators and Hardy spaces by addressing key limitations in the existing literature and extending their applicability to more general and complex settings. The novel contributions include the generalization of Bessel operators to higher-dimensional and non-Euclidean spaces, the refinement of Hardy space theory with atomic decompositions and duality characterizations, and the analysis of Bessel operators under non-standard measures such as fractal or weighted geometries. These results provide a robust framework for studying singular integrals, heat kernel estimates, and operator boundedness in diverse geometric and measure-theoretic contexts.

The implications of these findings are profound for both theoretical and applied mathematics. The extension to non-Euclidean spaces enables the study of Bessel operators in settings with variable curvature or irregular boundary conditions, broadening their relevance to geometric analysis and mathematical physics. Similarly, the incorporation of non-standard measures enriches the understanding of Hardy spaces, revealing new phenomena in fractal geometry, weighted function spaces, and diffusion processes in irregular media. These developments contribute to the ongoing efforts to unify classical harmonic analysis with modern applications in constrained wave propagation, spectral analysis, and fractal-based systems.

Future research can build upon these contributions by exploring additional applications of the refined Hardy space framework, such as their use in nonlinear analysis, optimization problems, and the study of partial differential equations in complex geometries. Further investigation into the interplay between Bessel operators and other function spaces, as well as deeper connections with potential theory and boundary dynamics, would also enhance the broader applicability of these results. The advancements presented in this work underscore the versatility of harmonic analysis and spectral theory as tools for addressing contemporary mathematical challenges.