Product-convolution operators and mixed-norm spaces

Conditions for boundedness and compactness of product-convolution operators $g \to {P_h}{C_f}g = h \cdot (f\ast g)$ on spaces ${L_p}(G)$ are studied. It is necessary for boundedness to define a class of "mixed-norm" spaces ${L_{(p,q)}}(G)$ interpolating the ${L_p}(G)$ spaces in a natural way $({L_{(p,p)}} = {L_p})$. It is then natural to study the operators acting between ${L_{(p,q)}}(G)$ spaces, where $G$ has a compact invariant neighborhood. The theory of ${L_{(p,q)}}(G)$ is developed and boundedness and compactness conditions of a nonclassical type are obtained. It is demonstrated that the results extend easily to a somewhat broader class of integral operators. Several known results are strengthened or extended as incidental consequences of the investigation.