On the Continuity of Pseudo-Differential Operators on Multiplier Spaces Associated to Herz-type Triebel–Lizorkin Spaces

In this paper, for a certain range of parameters, we prove that there exist symbols in the Hörmander class S1,00\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{1,0}^{0}$$\end{document} which do not define bounded operators on M(K˙qα,pFβs)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\big ({\dot{K}}_{q}^{\alpha ,p}{F_{\beta }^{s}}\big )$$\end{document}. To do these, we need the characterization of Herz–Besov spaces by ball means of differences and some properties of pointwise multipliers for Herz–Triebel–Lizorkin spaces.