Pointwise Multipliers for Besov Spaces $B_{p,\infty }^{0,b}({\mathbb{R}}^n)$ with Only Logarithmic Smoothness

Abstract

In this article, we establish a characterization of the set $M(B_{p,\infty }^{0,b}({\mathbb{R}}^n))$ of all pointwise multipliers of Besov spaces $B_{p,\infty }^{0,b}({\mathbb{R}}^n)$ with only logarithmic smoothness $b\in \mathbb{R}$ in the special cases $p=1$ and $p=\infty$. 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 $e^{ik\cdot x}$ with $k\in {\mathbb{Z}}^n$ and $x\in {\mathbb{R}}^n$, are pointwise multipliers of $B_{1,\infty }^{0,b}({\mathbb{R}}^n)$ and $B_{\infty ,\infty }^{0,b}({\mathbb{R}}^n)$, respectively; furthermore, we obtain the explicit estimates of $\Vert e^{ik\cdot x}{\Vert }_{M(B_{1,\infty }^{0,b}({\mathbb{R}}^n))}$ and $\Vert e^{ik\cdot x}{\Vert }_{M(B_{\infty ,\infty }^{0,b}({\mathbb{R}}^n))}$. In the case that $p\in (1,\infty )$, we give some sufficient conditions and some necessary conditions of the pointwise multipliers of $B_{p,\infty }^{0,b}({\mathbb{R}}^n)$ and a complete characterization of $M(B_{p,\infty }^{0,b}({\mathbb{R}}^n))$ is still open. However, via a different method, we are still able to accurately calculate $\Vert e^{ik\cdot x}{\Vert }_{M(B_{p,\infty }^{0,b}({\mathbb{R}}^n))}$, $k\in {\mathbb{Z}}^n$, 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.

64 pages; Submitted