On the pointwise multiplication in Besov and Lizorkin-Triebel spaces

where F and B, with three indices, will denote the Lizorkin-Triebel space Fs p,q and the Besov space Bp,q, respectively. The different embeddings obtained here are under certain restrictions on the parameters. In this introduction, we will recall the definition of some spaces and some necessary tools. In Sections 2 and 3, we give the first contribution of this work. The theorems of Section 2 will treat the case F · B↩ F where the first theorem is a generalization of the results of Franke [4, Section 3.2, Theorem 1, Section 3.4, Corollary 1] and Marschall [7]. The second theorem is in the sense of Johnsen’s works (see [5]). Section 3 will contain a treatment of the embeddings of the types F · F↩ F and B · B↩ B which presents an improvement of [3]. In the sense of [5, Theorems 6.5, 6.11], some limit cases are considered in Section 4, which constitute the second contribution of this paper. Section 5 is an application of our results to the continuity of pseudodifferential operators on Lizorkin-Triebel spaces. We will work on the Euclidean space Rn. If f ∈ , the Fourier transform is defined by the formula