Sharp embedding results for spaces of smooth functions with power weights

TL;DR

This paper establishes sharp conditions for Sobolev embeddings in weighted function spaces like Besov and Triebel-Lizorkin on , with proofs applicable to vector-valued functions, advancing understanding of weighted smooth function spaces.

Contribution

It provides precise characterizations of when two-weight Sobolev embeddings hold for weighted Besov, Triebel-Lizorkin, Bessel-potential, and Sobolev spaces, including vector-valued cases.

Findings

Sharp embedding conditions for weighted function spaces

Characterization of parameter ranges for embeddings

Extensions to vector-valued functions

Abstract

We consider function spaces of Besov, Triebel-Lizorkin, Bessel-potential and Sobolev type on $\R^d$, equipped with power weights $w(x) = |x|^\gamma$, $\gamma>-d$. We prove two-weight Sobolev embeddings for these spaces. Moreover, we precisely characterize for which parameters the embeddings hold. The proofs are presented in such a way that they also hold for vector-valued functions.