Sharp embedding between Wiener amalgam and some classical spaces

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 $F_{p,r}^{s}$.