Sharp embedding between Wiener amalgam and some classical spaces

This paper investigates the embedding relationships between Wiener amalgam spaces and classical spaces, including Sobolev spaces, local Hardy spaces, Besov spaces, and \(\alpha \) $\alpha$ -modulation spaces. By establishing exact conditions, we provide a detailed characterization of the embeddings between Wiener amalgam spaces and these classical spaces, particularly the most general case when \(\alpha =0\) $\alpha =0$ , which extend the main results obtained by Guo–Wu–Yang–Zhao (J Funct Anal 273(1):404–443, 2017). Furthermore, we discuss the embedding relationship between Wiener amalgam spaces and Triebel–Lizorkin spaces \(F_{p,r}^{s}\) $F_{p,r}^s$ when \(0<p\leqslant 1\) $0<p⩽1$ .