Pointwise multipliers of inhomogeneous Besov and Triebel–Lizorkin spaces on spaces of homogeneous type

Abstract

Let \(({\mathcal {X}},d,\mu )\) be a space of homogeneous type with the upper dimension \(\omega\). In this work, the authors characterize the sets of all pointwise multipliers of inhomogeneous Besov spaces \(B_{p,q}^{s}( {\mathcal {X}} )\) and inhomogeneous Triebel–Lizorkin spaces \(F_{p,q}^{s}({\mathcal {X}})\). When \(p\in [1,\infty ]\) and \(s>\frac{\omega }{p}\), the authors show that the set of all pointwise multipliers of \(B_{p,q}^{s}({\mathcal {X}})\) equals to \(B_{p,q,\text {unif}}^{s}({\mathcal {X}})\) for \(q\in [p,\infty )\) or \(M_{p,q}^{s}({\mathcal {X}})\) for \(q\in (0,p)\) if and only if \({\mathcal {X}}\) supports the local lower and upper bound. Corresponding results for \(F_{p,q}^{s}({\mathcal {X}})\) with \(p,q\in (1,\infty )\) and \(s>\frac{\omega }{p}\) are also obtained. When \(p\le 1\) (or \(p=\infty\)), the authors establish a characterization of the collection of all pointwise multipliers of \(B_{p,p}^{s}({\mathcal {X}})\) [or \(B_{\infty ,q}^{s}({\mathcal {X}})\)], which does not need any extra assumption on \(\mu\) and is even new when \({\mathcal {X}}\) supports the Ahlfors regular condition.