Characterizations of multipliers in the distribution spaces with restricted growth

Abstract

Let $\mathscr{K'}_{M}$ be the space of distributions on ${\mathbf R}^{n}$ which grow no faster than $e^{M(kx)}$ for some $k>0$ and an index function $M(x)$ and let ${K'}_{M}$ be the Fourier transform of $\mathscr{K'}_{M}$. We establish the characterizations of the space $\mathscr{O}_{M}(\mathscr{K'}_{M} ;\mathscr{{'}_M)$ of multipliers in $\mathscr{K'}_{M}$ and prove various types of the continuity from or into $\mathscr{O}_{M}(\mathscr{K'}_{M} ; \mathscr{K'}_{M})$. Also we define the space $\mathscr{O}_M({K'}_{M} ; {K'}_{M})$ of multipliers in ${K'}_{M}$ and find the relation between $\mathscr{O}_{M}({K'}_{M} ; {K'}_{M})$ and the space $\mathscr{O'}_{C}(\mathscr{K'}_{{M};\mathscr{K'}_M)$ of convolution operators in $\mathscr{K'}_{M}$ by the Fourier transformation.