This paper is devoted to the study of the embedding relation between the modulation spaces \(M^s_{p,q}\) and the inhomogeneous Triebel-Lizorkin spaces \(F_{p,r}=F^0_{p,r}\) when \(p\leq 1\). Let \(p\in(0,1]\), \(q\in(0,\infty]\) and \(s\in\mathbb{R}\). In [Stud. Math. 192, No. 1, 79–96 (2009; Zbl 1163.42007)], M. Kobayashi et al. obtained the sharp ranges of \(p,q,s\) for the embedding \(M^s_{p,q} \subset h_p\) and also for \(h_p\subset M^s_{p,q}\), where \(h_p\) denotes the local Hardy space. In this paper, via a different proof, the authors obtain the following characterizations: =0.6cm

(a)

\(M^s_{p,q}\subset F_{p,r}\) if and only if one the following conditions holds: cm

(i)

\(1/p\leq 1/q\), \(s\geq 0\) and \(1/r\leq 1/q\);

(ii)

\(1/p>1/q\) and \(s>n(1/p-1/q)\).

(b)

\(F_{p,r}\subset M^s_{p,q}\) if and only if one the following conditions holds:

(i)

\(1/q\leq 1/p\), \(s\leq n(1-1/p-1/q)\);

(ii)

\(1/q>1/p\) and \(s<n(1-1/p-1/q)\).

Since \(h_p=F_{p,r}\), these results improve those obtained by M. Kobayashi et al. [Stud. Math. 192, No. 1, 79–96 (2009; Zbl 1163.42007)].