Spectral Properties of Fractional Differentiation Operators

TL;DR

This paper explores the spectral properties of fractional differentiation operators, establishing new propositions, constructing multidimensional fractional integrals, and analyzing their functional and spectral characteristics.

Contribution

It introduces a multidimensional fractional integral construction, proves new propositions in fractional calculus, and extends the Kipriyanov operator with a conjugate operator, enriching spectral theory.

Findings

Embedded Sobolev spaces into fractional integral classes

Constructed a multidimensional fractional integral in a specific direction

Extended the Kipriyanov operator and identified its conjugate

Abstract

In this paper presents the results obtained in the field of spectral theory operators of fractional differentiation. Proven a number of propositions which represents independent interest in the theory of fractional calculus. Introduced construction of multidimensional fractional integral in the direction. Formulated the sufficient conditions of representability functions by the fractional integral in the direction, in particular proved the embedding of a Sobolev space in classes of functions representable by the fractional integral in direction. Note that the technique of proof borrowed from the one-dimensional case is of particular interest. It should be noted that was constructed extension of Kipriyanov operator, was found a conjugate operator. This is all creates a complete picture reflecting the qualitative properties of fractional differential operators.