Boundedness and inversion properties of certain convolution transforms

Abstract

For a fixed function $h$ we deal with a class of convolution transforms $f \to f*h$, where % $$ (f*h)(x) = {1\over 2x}\int_{{\bf R}_+^2}e^{-{1\over 2}\left(x{u^2+y^2\over uy}+{yu\over x}\right)}f(u)h(y)dudy, x \in {\bf R}_+$$ % as integral operators $L_p({\bf R}_+; xdx) \to L_r({\bf R}_+; xdx), \ p, r \ge 1$. The Young type inequality is proved. Boundedness properties are investigated. Certain examples of these operators are considered and inversion formulas in $L_2({\bf R}_+; xdx)$ are obtained.