Interpolation and embeddings of weighted tent spaces

TL;DR

This paper extends the theory of weighted tent spaces on metric measure spaces, identifying their interpolation spaces with new Z-spaces and establishing Hardy-Littlewood-Sobolev embeddings, advancing analysis in geometric and boundary value problems.

Contribution

It introduces Z-spaces as new interpolation spaces for weighted tent spaces on metric measure spaces, linking them to boundary value problem analysis.

Findings

Identification of interpolation spaces as Z-spaces

Establishment of Hardy-Littlewood-Sobolev embeddings

Extension of tent space theory to weighted and geometric settings

Abstract

Given a metric measure space $X$, we consider a scale of function spaces $T^{p,q}_s(X)$, called the weighted tent space scale. This is an extension of the tent space scale of Coifman, Meyer, and Stein. Under various geometric assumptions on $X$ we identify some associated interpolation spaces, in particular certain real interpolation spaces. These are identified with a new scale of function spaces, which we call $Z$-spaces, that have recently appeared in the work of Barton and Mayboroda on elliptic boundary value problems with boundary data in Besov spaces. We also prove Hardy--Littlewood--Sobolev-type embeddings between weighted tent spaces.