Sobolev, Hardy, Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenberg type inequalities for some fractional derivatives

TL;DR

This paper establishes various fractional derivative inequalities, including Sobolev, Hardy, Gagliardo-Nirenberg, and Caffarelli-Kohn-Nirenberg types, for Caputo, Riemann-Liouville, and Hadamard derivatives, with some applications.

Contribution

It introduces new fractional inequalities for different derivatives and demonstrates their applications, expanding the theoretical framework of fractional calculus.

Findings

Derived inequalities for Caputo, Riemann-Liouville, and Hadamard derivatives

Established connections between fractional inequalities and classical results

Presented applications demonstrating the utility of these inequalities

Abstract

In this paper we show different inequalities for fractional order differential operators. In particular, the Sobolev, Hardy, Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenberg type inequalities for the Caputo, Riemann-Liouville and Hadamard derivatives are obtained. In addition, we show some applications of these inequalities.