Around the Sobolev Inequalities for the Stable Heat Semigroups

TL;DR

This paper develops a distributional theory for fractional Sobolev spaces linked to symmetric alpha-stable measures, enhancing understanding of their properties and inequalities.

Contribution

It introduces a general distributional framework for fractional isotropic Sobolev spaces associated with alpha-stable measures, extending classical Sobolev inequalities.

Findings

Established new Sobolev inequalities for stable heat semigroups

Provided a distributional approach to fractional Sobolev spaces

Extended classical results to non-degenerate symmetric alpha-stable measures

Abstract

We develop a general distributional theory of fractional (an)isotropic Sobolev spaces associated with the non-degenerate symmetric $\alpha$-stable, $\alpha \in (1,2)$, probability measures on $\mathbb{R}^d$.