Maximal Domains for Fractional Derivatives and Integrals

R Hilfer,T Kleiner

doi:10.3390/math8071107

Abstract

The purpose of this short communication is to announce the existence of fractional calculi on precisely specified domains of distributions. The calculi satisfy desiderata proposed above in Mathematics 7, 149 (2019). For the desiderata (a)–(c) the examples are optimal in the sense of having maximal domains with respect to convolvability of distributions. The examples suggest to modify desideratum (f) in the original list.

Highlights

The purpose of this short communication is to announce the existence of fractional calculi on precisely specified domains of distributions
Restricted to a suitable subset G(c) ⊆ D(I α ) of the domain of I α the fractional derivatives Dα of order α operate as left inverses
Let us stress that the distributional domains D(I α ), D(Dα ) given in Theorem 1 below are maximal in a precise mathematical sense

Summary

Introduction

The purpose of this short communication is to announce the existence of fractional calculi on precisely specified domains of distributions. For the desiderata (a)–(c) the examples are optimal in the sense of having maximal domains with respect to convolvability of distributions. A list of six desiderata was recently proposed in [1] for calling families of operators {Dα , I α } with family index α ∈ I from some index set I ⊆ C fractional derivatives (Dα ) and fractional integrals (I α ) of order α ∈ Distributional domains for {Dα , I α } seem to require a minor modification of these desiderata.

Results

Conclusion