Sharp Hardy-Rellich Type Inequalities Associated with Dunkl Operators

TL;DR

This paper establishes sharp Hardy and Hardy-Rellich inequalities for Dunkl operators, extending classical inequalities with precise constants and broadening their applicability in G-invariant domains.

Contribution

It introduces the Dunkl analogues of classical Hardy and Hardy-Rellich inequalities with sharp constants, expanding the theoretical framework for Dunkl operators.

Findings

Derived the Dunkl Hardy inequality with sharp constant for p > N + 2γ.

Proved two sharp Hardy-Rellich type inequalities for Dunkl operators.

Extended classical inequalities to Dunkl setting with explicit constants.

Abstract

In this paper, we obtained the Dunkl analogy of classical Lp Hardy inequality for $p > N + 2\gamma$ with sharp constant $\left(\frac{p-N-2\gamma}{p}\right)^{p}$, where $2\gamma$ is the degree of weight function associated with Dunkl operators, and $L^p$ Hardy inequalities with distant function in some G-invariant domains. Moreover we proved two sharp Hardy-Rellich type inequalities for Dunkl operators.