Maximal Domains for Fractional Derivatives and Integrals

A list of six desiderata was recently proposed in [1] for calling families of operators $\{D^{\alpha },I^{\alpha }\}$ with family index $\alpha \in \mathbb{I}$ from some index set $\mathbb{I}\subseteq \mathbb{C}$ fractional derivatives ($D^{\alpha }$) and fractional integrals ($I^{\alpha }$) of order $\alpha \notin \mathbb{N}$. Distributional domains for $\{D^{\alpha },I^{\alpha }\}$ seem to require a minor modification of these desiderata.

Multiplication of distributions is ill-defined so that for distributions desideratum (f) (Leibniz rule) requires generalization. A slightly modified list of desiderata might read as follows:

Given these modified desiderata, the objective in this short note is to introduce fractional calculi for distributions. Let us stress that the distributional domains $\mathrm{D}\left(I_{}^{\alpha }\right)$, $\mathrm{D}\left(D_{}^{\alpha }\right)$ given in Theorem 1 below are maximal in a precise mathematical sense. One cannot enlarge them without violating either the desiderata or the interpretation of fractional derivatives and integrals as convolution operators. Recall that numerous other mathematical interpretations exist [2], that may have different maximal domains. In this paper fractional operators are interpreted as convolutions with power law kernels (cf. [2], Equation (28)). A comprehensive analysis of convolutions with power law kernels on weighted spaces of continuous functions was recently given in [3].

Define the spaces of continuously differentiable functions, test functions, and smooth functions with bounded derivates in the usual way [4]. The spaces ${\mathcal{C}}^m, \mathcal{D}$ are endowed with the norm ${\parallel f\parallel }_{\infty }=sup\left|f\right|$. The topology on $\mathcal{B}$ is induced by the seminorms ${\parallel f\parallel }_{N,g}=sup\{\parallel g{\partial }^{n_1}...{\partial }^{n_d}f{\parallel }_{\infty }:n_i\in \mathbb{N},{\sum }_i^d{n}_i\le N\}$ with $N\in \mathbb{N}$ and $g\in {\mathcal{C}}_{\mathrm{v}}$, where ${\mathcal{C}}_{\mathrm{v}}$ is the space of continuous functions vanishing at infinity.

The space of distributions ${\mathcal{D}}^{\prime }$ is the topological dual of $\mathcal{D}$. The dual space ${\mathcal{B}}^{\prime }$ is the space of integrable distributions. The pairing $\mathcal{D}\times {\mathcal{D}}^{\prime }\to \mathbb{C}$ is denoted $⟨·,·⟩$, the pairing $\mathcal{B}\times {\mathcal{B}}^{\prime }\to \mathbb{C}$ as ${⟨·,·⟩}_{\mathcal{B}}$.

Let ${\mathcal{D}}_{+}^{\prime }$ denote the space of causal distributions defined as elements $f\in {\mathcal{D}}^{\prime }\left(\mathbb{R}\right)$ whose support is bounded on the left.

The operators $I_{+}^{\alpha }$ and $D_{+}^{\alpha }$ are linear and continuous on ${\mathcal{D}}_{+}^{\prime }$. The kernels $\{K_{\alpha }:\alpha \in \mathbb{C}\}$ form a convolution group for all $\alpha ,\beta \in \mathbb{C}$. This entails the index law $I_{+}^{\alpha }\left(I_{+}^{\beta }f\right)=I_{+}^{\alpha +\beta }f$ for all $f\in {\mathcal{D}}_{+}^{\prime }$ and $\alpha ,\beta \in \mathbb{C}$. Clearly, all desiderata are fulfilled for $\{I_{+}^{\alpha },D_{+}^{\alpha }\}$ with $\mathrm{D}\left(I_{+}^{\alpha }\right)=\mathrm{D}\left(D_{+}^{\alpha }\right)={\mathrm{G}}_{\left(\mathrm{b}\right)}={\mathrm{G}}_{\left(\mathrm{c}\right)}={\mathrm{G}}_{\left(\mathrm{d}\right)}={\mathcal{D}}_{+}^{\prime }$ and ${\mathrm{G}}_{\left(\mathrm{f}\right)}={\mathcal{C}}^{\infty }$.

The domain ${\mathcal{D}}_{+}^{\prime }$ of causal distributions will now be enlarged using certain sets of lower semicontinuous functions as convolution weights. A function $f:\mathbb{R}\to {\overline{\mathbb{R}}}_{+}$, where ${\overline{\mathbb{R}}}_{+}:=[0,\infty ]$, is called lower semicontinuous, if the set $\{f\le a\}$ is closed for every $a\in {\overline{\mathbb{R}}}_{+}$. The set of all lower semicontinuous functions is denoted $\mathcal{I}$, the set of lower semicontinuous functions whose support is bounded on the left is denoted ${\mathcal{I}}_{+}$. For $(p,k)\in \mathbb{R}\times \mathbb{N}$ let be the set of lower semicontinuous functions of power-logarithmic growth of order $(p,k)$. Then are the sets of interest.

The proof of Theorem 1 and its corollary will be published elsewhere, because it is lengthy and giving it here would distract attention from the main message. The domains $\mathrm{D}\left(I_{+}^{\alpha }\right)$, $\mathrm{D}\left(D_{+}^{\alpha }\right)$, ${\mathrm{G}}_{\left(\mathrm{b}\right)},{\mathrm{G}}_{\left(\mathrm{c}\right)}$ are maximal with respect to convolvability in both cases. The second case ${\{I_{+}^{\alpha },D_{+}^{\alpha }\}}_{\alpha \in \mathrm{i}\mathbb{R}}$ yields a (purely imaginary) “fractional calculus of order zero” in the sense that $\mathrm{Re}\alpha =0$ for all operators in that subset.