Extension of Mikhlin Multiplier Theorem to Fractional Derivatives and Stable Processes

TL;DR

This paper extends the classical Mikhlin multiplier theorem to fractional derivatives and explores its connection with stable processes, providing new conditions and showing classical results as special cases.

Contribution

It introduces a generalized Mikhlin theorem involving fractional derivatives and links it to stable processes via the infinitesimal generator, broadening the theorem's applicability.

Findings

Generalized Mikhlin theorem with fractional derivatives

Connection established between fractional derivatives and stable processes

Classical Mikhlin theorem derived as a special case

Abstract

In this paper, we prove a new generalized Mikhlin multiplier theorem whose conditions are given with respect to fractional derivatives in integral forms with two different integration intervals. We also discuss the connection between fractional derivatives and stable processes and prove a version of Mikhlin theorem under a condition given in terms of the infinitesimal generator of symmetric stable process. The classical Mikhlin theorem is shown to be a corollary of this new generalized version in this paper.