Several functional capacities and Carleson type embeddings of fractional Sobolev sapces on stratified Lie groups

TL;DR

This paper explores the functional and geometric properties of fractional Sobolev, Besov, and Riesz capacities on stratified Lie groups, introducing new characterizations and extension methods using heat semigroups.

Contribution

It provides novel Carleson characterizations of fractional Sobolev space extensions and criteria for measure continuity on stratified Lie groups, advancing the understanding of these capacities.

Findings

New Carleson characterization of fractional Sobolev space extensions.

Criteria for measure continuity of fractional Sobolev spaces.

Characterization of measures ensuring Besov space continuity.

Abstract

In this paper, we focus on the functional and geometrical aspects of the fractional Sobolev capacity, the Besov capacity and the Riesz capacity on stratified lie groups, respectively. Firstly, we provide a new Carleson characterization of the extension of fractional Sobolev spaces to $L^{q}(\X\times\mathbb{R}_{+},\mu)$ with $q\in\mathbb{R}_{+}$ using the fractional heat semigroup and the Caffarelli-Silvestre type extension on stratified Lie groups $\X$. Secondly, a characterization of $\nu$ on $\X$ which ensures the continuity of the fractional Sobolev space belonging to $L^{q}(\X,\nu)$ is also obtained via taking $t\rightarrow 0$. Finally, with the help of inequalities related to the Besov capacity and its properties, we also obtain a characterization of $\nu$ on $\X$ which ensures the continuity of the Besov type space belonging to $L^{q}(\X,\nu)$.