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In this article, we establish a characterization of the set$M(B^{0,b}_{p,\infty}(\mathbb{R}^n))$ of all pointwise multipliers of Besovspaces $B^{0,b}_{p,\infty}(\mathbb{R}^n)$ with only logarithmic smoothness$b\in\mathbb{R}$ in the special cases $p=1$ and $p=\infty$. As applications ofthese two characterizations, we clarify whether or not the three concreteexamples, namely characteristic functions of open sets, continuous functionsdefined 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$B^{0,b}_{1,\infty}(\mathbb{R}^n)$ and $B^{0,b}_{\infty,\infty}(\mathbb{R}^n)$,respectively; furthermore, we obtain the explicit estimates of $\|e^{ik \cdotx}\|_{M(B^{0,b}_{1,\infty}(\mathbb{R}^n))}$ and $\|e^{ik \cdotx}\|_{M(B^{0,b}_{\infty,\infty}(\mathbb{R}^n))}$. In the case that$p\in(1,\infty)$, we give some sufficient conditions and some necessaryconditions of the pointwise multipliers of $B^{0,b}_{p,\infty}(\mathbb{R}^n)$and a complete characterization of $M(B^{0,b}_{p,\infty}(\mathbb{R}^n))$ isstill open. However, via a different method, we are still able to accuratelycalculate $\|e^{ik \cdot x}\|_{M(B^{0,b}_{p,\infty}(\mathbb{R}^n))}$,$k\in\mathbb{Z}^n$, in this situation. The novelty of this article is that mostof the proofs are constructive and these constructions strongly depend on thelogarithmic structure of Besov spaces under consideration.

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