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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On Hardy's inequality and Laplace transforms in weighted rearrangement invariant spaces


Author: Kenneth F. Andersen
Journal: Proc. Amer. Math. Soc. 39 (1973), 295-299
MSC: Primary 26A86; Secondary 44A10, 46E30
DOI: https://doi.org/10.1090/S0002-9939-1973-0315071-4
MathSciNet review: 0315071
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Abstract: Hardy's well-known inequality relating the norm of a function and the norm of its integral mean in the Lebesgue spaces $ {L^p}(\mu ),d\mu (t) = {t^{\sigma - 1}}dt$, is extended to the class of rearrangement invariant spaces $ X(\mu )$. These spaces include, for example, the $ {L^p}(\mu )$, the Lorentz and the Orlicz spaces. As an application, necessary and sufficient conditions are obtained for an operator related to the Laplace transform to be bounded as a linear operator between rearrangement invariant spaces of $ \mu $-measurable functions.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0315071-4
Keywords: Integral mean, Laplace transform, rearrangement invariant function space, Lorentz space, Orlicz space
Article copyright: © Copyright 1973 American Mathematical Society