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


Introduction and preparations
In Besov spaces and Lizorkin-Triebel spaces, this paper is concerned with proving some embeddings of the form where F and B, with three indices, will denote the Lizorkin-Triebel space F s p,q and the Besov space B s p,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 R n . If f ∈ , the Fourier transform is defined by the formula 2 Multiplication in Besov and Lizorkin spaces and Ᏺ −1 f denotes the inverse Fourier transform of f ; as usual Ᏺ and Ᏺ −1 are extended from to . Consider a partition of unity where ϕ,ψ ∈ C ∞ are positive functions such that supp ϕ ⊂ {ξ ∈ R n : 1 ≤ |ξ| ≤ 3} and suppψ ⊂ {ξ ∈ R n : |ξ| ≤ 3}. We define the convolution operators Q j and Δ k by the following: , and we set Q 0 = Δ 0 . Thus we obtain the Littlewood-Paley decomposition f = ∞ j=0 Δ j f (convergence in ).
resp., f k k∈N | s q L γ p = 2 ks f k k∈N | q L p < ∞ . (1.5) for all x ∈ R n , f ∈ , a > 0, and k = 0,1,.... Then, in Definition 1.2(i) (resp., (ii)), we can replace Δ k f by Δ * ,a k f with a > (n/min(p, q)) (resp., a > n/p), (cf. see [13, Theorem 2.3.2]). D. Drihem and M. Moussai 3 The product f · g is defined by if the limit on the right-hand side exists in (see [10,Section 4.2]), and we have In the below proofs of the different cases of type (1.1), written as G 1 · G 2 G 3 , to see f · g belongs to G 3 , ( f ∈ G 1 , g ∈ G 2 ), it suffices to an estimate of terms of the form . Now we recall some lemmas which are useful for us. (1.20) Then the sequences δ k = k j=0 γ k− j ε j and η k = ∞ j=k γ j−k ε j belong to q , and the estimate holds. The constant c depends only on γ and q.
holds. The constant c depends only on n, p, and γ.

Multiplication of mixed type
The following results give an extension of the sufficient hypotheses used in [5, Theorem 6.1].

Then it holds
Under the hypotheses of Theorem 2.1. If r < n/p 2 (resp., r = n/ p 2 ) then it holds Furthermore, in particular, if 1< p 1 <∞ and r >n/p 1 + n/ p 2 − n, can be taken To prove Theorem 2.1, we need the following lemma.
Lemma 2.4. Let 0 < p < ∞ and a > n/p. Then there exists a constant c > 0 such that

4)
for any g ∈ F 0 p,2 . Proof. First, we define the maximal function of Q j g, of Hardy-Littelewood type, by the formula where B(x,r) is the ball centered at x of radius r and |B(x, r)| denotes its measure. Next, let t > 0 satisfy n/a < t < p. From [13, Theorem 1.3.1], we have Multiplication in Besov and Lizorkin spaces Then we obtain A proof of the last inequality may be found in [13, Theorem 2.2.2, page 89]. Now, it is easy to see that the last member of (2.7) is bounded by where Q * ,a1 k and Δ * ,a2 k are defined as in Remark 1.3, we obtain where a 1 and a 2 are real numbers at our disposal. We set 1/b = 1/ p 2 − r/n. The left-hand side of (2.10), in L p -norm, is bounded by Choose a 1 >n/b and a 2 >n/min(p 1 , q), then both Lemma 2.4 and the embedding B r p2,q2 We set (2.13) D. Drihem and M. Moussai 7 We have (2.14) For the first embedding of (2.14), we can see [4, Section 2.3, Theorem 3]. On the other hand, the Hölder inequality yields We use the notations v, σ, and β from (2.13). Lemma 1.6 provides A similar argument as above yields We set 1/ q 2 = 1/ p − 1/ p 1 . By the Hölder inequality in p -norm, the right-hand side of (2.19) is bounded by c g | B r p2, q2 f | B β u,p1 . Then we conclude the desired estimate by (2.14).
We now study case 1 We employ the notations υ and σ from (2.13). By Lemma 1.6, we obtain where ρ = s − n/ p 1 + n/u and μ = s + r − n/ p 1 − n/ p 2 + n > 0, therefore, B r p2, q2 successively. Since σ > s and v < p, we can finish the proof of this case by applying, in the left-hand side of (2.22), embeddings (2.14). Case 2 (r = n/ p 2 ). We only estimate {Π k,1 ( f ,g)} k∈N . It is sufficient to see that with a > n/min(p 1 , q) and to take the L p1 ( q )-norm.
If either of the following assertions is satisfied: we obtain the desired result.

Multiplication in Besov and Lizorkin spaces
We employ the notations v, σ, and β from (2.13). We have we can finish the proof of this case using (2.14). Subcase 2.3 (n/ p 2 < s < r). We have only case p < b needs to be verified. As in (2.32), we immediately obtain the result. Case 3 (s ≥ n/ p 1 ). We have the following subcases. Subcase 3.1 (p < p 2 ). We set (2.38) Subcase 3.3 (s = n/ p 1 and p ≥ p 2 ). We choose α > 0 such that ε = α−n/ p+n/ p 1 +n/ p 2 − r < 0, then it suffices to apply (2.29) to (2.39) The proof of this case is obtained similarly to the proof of Theorem 2.1 just by replacing (2.17) and (2.20) with respectively.

Some limit cases
We will prove results of independent interest concerning the limit case for the parameters s + r, see [5, Theorems 6.5 and 6.11].
The same method works for the proofs of Theorems 4.3(ii) and 4.5. We omit the details.

Application
We consider S 0 1,0 (E) (E a Banach space), the class of symbols (x,ξ) → a(x,ξ) satisfying and we define the pseudodifferential operator by the formula As mentioned in the introduction, the theorems of this section present an application of the previous results in this paper.