On Hardy-Littlewood-Sobolev estimates for degenerate Laplacians

TL;DR

This paper proves new norm inequalities for fractional powers of degenerate Laplacians influenced by Muckenhoupt weights, extending classical results using heat kernel estimates.

Contribution

It introduces novel inequalities for weighted degenerate Laplacians, broadening the scope of classical Riesz potential results with new heat kernel techniques.

Findings

Established norm inequalities for fractional degenerate Laplacians.

Extended classical Riesz potential results to weighted degenerate operators.

Used heat kernel size estimates to derive inequalities.

Abstract

We establish norm inequalities for fractional powers of degenerate Laplacians, with degeneracy being determined by weights in the Muckenhoupt class $A_2(\mathbb{R}^n)$, accompanied by specific additional reverse H\"older assumptions. This extends the known results for classical Riesz potentials. The approach is based on size estimates for the degenerate heat kernels. The approach also applies to more general weighted degenerate operators.