Pointwise multipliers for Besov spaces $$B^{0,b}_{p,\infty }({\mathbb {R}}^n)$$ with only logarithmic smoothness

Abstract

AbstractIn this article, we establish a characterization of the set $$M(B^{0,b}_{p,\infty }({\mathbb {R}}^n))$$ M ( B p , ∞ 0 , b ( R n ) ) of all pointwise multipliers of Besov spaces $$B^{0,b}_{p,\infty }({\mathbb {R}}^n)$$ B p , ∞ 0 , b ( R n ) with only logarithmic smoothness $$b\in {\mathbb {R}}$$ b ∈ R in the special cases $$p=1$$ p = 1 and $$p=\infty $$ p = ∞ . As applications of these two characterizations, we clarify whether or not the three concrete examples, namely characteristic functions of open sets, continuous functions defined by differences, and the functions $$e^{ik\cdot x}$$ e i k · x with $$k\in {\mathbb {Z}}^n$$ k ∈ Z n and $$x\in {\mathbb {R}}^n$$ x ∈ R n , are pointwise multipliers of $$B^{0,b}_{1,\infty }({\mathbb {R}}^n)$$ B 1 , ∞ 0 , b ( R n ) and $$B^{0,b}_{\infty ,\infty }({\mathbb {R}}^n)$$ B ∞ , ∞ 0 , b ( R n ) , respectively; furthermore, we obtain the explicit estimates of $$\Vert e^{ik \cdot x}\Vert _{M(B^{0,b}_{1,\infty }({\mathbb {R}}^n))}$$ ‖ e i k · x ‖ M ( B 1 , ∞ 0 , b ( R n ) ) and $$\Vert e^{ik \cdot x}\Vert _{M(B^{0,b}_{\infty ,\infty }({\mathbb {R}}^n))}$$ ‖ e i k · x ‖ M ( B ∞ , ∞ 0 , b ( R n ) ) . In the case where $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) , we give some sufficient conditions and some necessary conditions of the pointwise multipliers of $$B^{0,b}_{p,\infty }({\mathbb {R}}^n)$$ B p , ∞ 0 , b ( R n ) and a complete characterization of $$M(B^{0,b}_{p,\infty }({\mathbb {R}}^n))$$ M ( B p , ∞ 0 , b ( R n ) ) is still open. However, via a different method, we are still able to accurately calculate $$\Vert e^{ik \cdot x}\Vert _{M(B^{0,b}_{p,\infty }({\mathbb {R}}^n))}$$ ‖ e i k · x ‖ M ( B p , ∞ 0 , b ( R n ) ) , $$k\in {\mathbb {Z}}^n$$ k ∈ Z n , in this situation. The novelty of this article is that most of the proofs are constructive and these constructions strongly depend on the logarithmic structure of Besov spaces under consideration.