Remark on the Chain rule of fractional derivative in the Sobolev framework

TL;DR

This paper investigates a new chain rule for fractional derivatives within Sobolev spaces, focusing on operators of order between one and two, using Fourier multipliers to extend classical calculus rules.

Contribution

It introduces a novel chain rule for fractional derivatives of order between one and two in Sobolev spaces, expanding the theoretical understanding of fractional calculus.

Findings

Established a chain rule for fractional derivatives of order between one and two.

Extended classical Leibniz rule to fractional differential operators.

Provided a framework for fractional derivatives using Fourier multipliers.

Abstract

A chain rule for power product is studied with fractional differential operators in the framework of Sobolev spaces. The fractional differential operators are defined by the Fourier multipliers. The chain rule is considered newly in the case where the order of differential operators is between one and two. The study is based on the analogy of the classical chain rule or Leibniz rule.