Pointwise Multiplier Spaces of Logarithmic Besov Spaces: Duality Principle and Fourier-Analytical Characterization in Endpoint Cases

Abstract

Let \(s,b\in \mathbb {R}\). This article is devoted to establishing the Fourier-analytical characterization of the pointwise multiplier space \(M(B^{s,b}_{p,q}(\mathbb {R}^n))\) for the logarithmic Besov space \(B^{s,b}_{p,q}(\mathbb {R}^n)\) in the endpoint cases, that is, \(p,q\in \{1,\infty \}\). The authors first obtain such a characterization for the cases where \(p=1\) and \(q=\infty \) and where \(p=\infty \) and \(q=1\). Applying this, the authors then establish the duality formula \(M(B^{s,b}_{p,q}(\mathbb {R}^n))=M(B^{-s,-b}_{p',q'}(\mathbb {R}^n)),\) where \(s,b\in \mathbb {R}\), \(p,q\in [1,\infty ]\), and \(p'\) and \(q'\) are respectively the conjugate indices of p and q. This duality principle is further applied to establish the Fourier-analytical characterization of \(M(B^{s,b}_{p,q}(\mathbb {R}^n))\) in the cases where \(p=\infty =q\) and where \(p=1=q\).