Product-convolution operators and mixed-norm spaces

Abstract

Conditions for boundedness and compactness of product-convolution operators g → P h C f g = h ⋅ ( f ∗ g ) g \to {P_h}{C_f}g = h \cdot (f\ast g) on spaces L p ( G ) {L_p}(G) are studied. It is necessary for boundedness to define a class of "mixed-norm" spaces L ( p , q ) ( G ) {L_{(p,q)}}(G) interpolating the L p ( G ) {L_p}(G) spaces in a natural way ( L ( p , p ) = L p ) ({L_{(p,p)}} = {L_p}) . It is then natural to study the operators acting between L ( p , q ) ( G ) {L_{(p,q)}}(G) spaces, where G G has a compact invariant neighborhood. The theory of L ( p , q ) ( G ) {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.