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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Product-convolution operators and mixed-norm spaces


Authors: Robert C. Busby and Harvey A. Smith
Journal: Trans. Amer. Math. Soc. 263 (1981), 309-341
MSC: Primary 43A15; Secondary 44A35, 47B38
DOI: https://doi.org/10.1090/S0002-9947-1981-0594411-4
MathSciNet review: 594411
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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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DOI: https://doi.org/10.1090/S0002-9947-1981-0594411-4
Article copyright: © Copyright 1981 American Mathematical Society