Pseudo-differential operators on local Hardy spaces associated with ball quasi-Banach function spaces

Abstract

Let X be a ball quasi-Banach function space on \({\mathbb {R}}^{n}\) and \(h_{X}({\mathbb {R}}^{n})\) the local Hardy space associated with X. In this paper, under some reasonable assumptions on both X and another ball quasi-Banach function space Y, we aim to derive the boundedness of pseudo-differential operators with symbols in \(S^{-\alpha }_{1,\delta }\) from \(h_{X}({\mathbb {R}}^{n})\) to \(h_{Y}({\mathbb {R}}^{n})\) via applying the extrapolation theorem. In order to prove this result, we also establish the infinite and finite atomic decompositions for the weighted local Hardy space \(h^{p}_{\omega }({\mathbb {R}}^{n})\) and obtain the mapping property of the above pseudo-differential operators from \(h^{p}_{\omega ^{p}}({\mathbb {R}}^{n})\) to \(h^{q}_{\omega ^{q}}({\mathbb {R}}^{n})\). Moreover, the above results have a wide range of generality. For example, they can be applied to the variable Lebesgue space, the Lorentz space, the mixed-norm Lebesgue space, the local generalized Herz space and the mixed Herz space.