Extremals in Hardy-Littlewood-Sobolev inequalities for stable processes

TL;DR

This paper establishes the existence of extremal functions for Hardy-Littlewood-Sobolev inequalities related to stable processes, using a concentration-compactness principle tailored for these non-local operators.

Contribution

It introduces a concentration-compactness principle for stable processes and proves the existence of extremal functions in the associated Hardy-Littlewood-Sobolev inequality.

Findings

Existence of extremal functions for the inequality.

Development of a concentration-compactness principle for stable processes.

Advancement in understanding non-local operator inequalities.

Abstract

We prove the existence of an extremal function in the Hardy-Littlewood-Sobolev inequality for the energy associated to an stable operator. To this aim we obtain a concentration-compactness principle for stable processes in $\mathbb{R}^N$.