Product-convolution operators and mixed-norm spaces

Busby, Robert C.; Smith, Harvey A.

doi:10.1090/S0002-9947-1981-0594411-4

Product-convolution operators and mixed-norm spaces
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by Robert C. Busby and Harvey A. Smith

Trans. Amer. Math. Soc. 263 (1981), 309-341

DOI: https://doi.org/10.1090/S0002-9947-1981-0594411-4

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Abstract:

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.

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Product-convolution operators and mixed-norm spaces
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Abstract: