Sharp embedding between Wiener amalgam and some classical spaces

TL;DR

This paper determines the precise conditions under which Wiener amalgam spaces embed into classical function spaces, extending previous results and providing full characterizations for various space inclusions.

Contribution

It offers sharp embedding conditions between Wiener amalgam spaces and classical spaces, including Sobolev, Hardy, Besov, and modulation spaces, improving and extending prior work.

Findings

Established sharp embedding conditions for Wiener amalgam and Sobolev spaces.

Provided full characterization of inclusions between Wiener amalgam and $ ext{alpha}$-modulation spaces.

Derived sharp embeddings between Wiener amalgam and Triebel spaces for $0<p extless 1$.

Abstract

We establish the sharp conditions for the embedding between Wiener amalgam spaces $W_{p,q}^s$ and some classical spaces, including Sobolev spaces $L^{s,r}$, local Hardy spaces $h_{r}$, Besov spaces $B_{p,q}^s$, which partially improve and extend the main result obtained by Guo et al. in J. Funct. Anal., 273(1):404-443, 2017. In addition, we give the full characterization of inclusion between Wiener amalgam spaces $W_{p,q}^s$ and $\alpha$-modulation spaces $M_{p,q}^{s,\alpha}$. Especially, in the case of $\alpha=0$ with $M_{p,q}^{s,\alpha} = M_{p,q}^s$, we give the sharp conditions of the most general case of these embedding. When $0<p\leqslant 1$, we also establish the sharp embedding between Wiener amalgam spaces and Triebel spaces $F_{p,r}^{s}$.