Embeddings of $\alpha$-modulation spaces

TL;DR

This paper investigates the relationships between different alpha-modulation spaces by establishing embedding theorems and analyzing their sharpness, enhancing understanding of their structural properties.

Contribution

It provides new embedding results between alpha-modulation spaces for various alpha parameters and explores the optimality of these embeddings.

Findings

Established upper and lower embeddings between alpha-modulation spaces for different alpha values.

Proved partial results concerning the sharpness of these embeddings.

Enhanced understanding of the structural relationships among alpha-modulation spaces.

Abstract

We show upper and lower embeddings of $\alpha_1$-modulation spaces in $\alpha_2$-modulation spaces for $0 \leq \alpha_1 \leq \alpha_2 \leq 1$, and prove partial results on the sharpness of the embeddings.