The two-weight fractional Poincar\'e-Sobolev sandwich

TL;DR

This paper proves a new two-weight fractional Poincaré-Sobolev inequality and embedding with sharp constants, extending previous results and introducing sparse domination techniques for Triebel-Lizorkin spaces.

Contribution

It introduces a novel two-weight fractional Poincaré-Sobolev sandwich and establishes explicit dependence on weight characteristics, unifying and extending prior methods.

Findings

Established asymptotically sharp constants as fractional parameter approaches 1

Derived explicit quantitative dependence on Muckenhoupt weight characteristics

Introduced a new sparse domination result for Triebel-Lizorkin difference norms

Abstract

We establish a two-weight fractional Poincar\'e-Sobolev sandwich, consisting of a two-weight fractional Poincar\'e-Sobolev inequality and a two-weight embedding from the first-order Sobolev space to a Triebel-Lizorkin space defined via a difference norm. Our constants are asymptotically sharp as the fractional parameter approaches $1$. Our results are new even in the one-weight case. For each inequality we give explicit quantitative dependence on Muckenhoupt weight characteristics and treat both subcritical and critical regimes, the former via elementary methods and the latter via sparse domination. As one of our main tools, we establish a new sparse domination result for Triebel-Lizorkin difference norms. Our methods unify, simplify and significantly extend various earlier approaches.