Relation between solvability and a regularity of convolution operators in \(\mathcal{K}'_ p\), \(p>1\) (Q1346369)

Let \({\mathcal K}_ p^ \prime\) be the space of distributions on \(\mathbb{R}^ n\) which grow not faster than \(e^{k| x|^ p}\) for some \(k>0\) and let \({\mathcal O}_ c^ \prime ({\mathcal K}_ p^ \prime; {\mathcal K}_ p^ \prime)\) be the space of convolution operators in \({\mathcal K}_ p^ \prime\). The authors show that, for \(S\in {\mathcal O}_ c^ \prime ({\mathcal K}_ p^ \prime; {\mathcal K}_ p^ \prime)\), \(S*{\mathcal K}_ p^ \prime= {\mathcal K}_ p^ \prime\) is equivalent to the following: Every distribution \(u\in {\mathcal O}_ c^ \prime ({\mathcal K}_ p^ \prime; {\mathcal K}_ p^ \prime)\) with \(S*u\in {\mathcal K}_ p\) is in \({\mathcal K}_ p\).