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

Let X be a ball quasi-Banach function space on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}<^>{n}$$\end{document} and hX(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{X}({\mathbb {R}}<^>{n})$$\end{document} 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 S1,delta-alpha\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S<^>{-\alpha }_{1,\delta }$$\end{document} from hX(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{X}({\mathbb {R}}<^>{n})$$\end{document} to hY(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{Y}({\mathbb {R}}<^>{n})$$\end{document} 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 omega p(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h<^>{p}_{\omega }({\mathbb {R}}<^>{n})$$\end{document} and obtain the mapping property of the above pseudo-differential operators from h omega pp(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h<^>{p}_{\omega <^>{p}}({\mathbb {R}}<^>{n})$$\end{document} to h omega qq(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h<^>{q}_{\omega <^>{q}}({\mathbb {R}}<^>{n})$$\end{document}. 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.