On Hardy’s inequality and Laplace transforms in weighted rearrangement invariant spaces
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- by Kenneth F. Andersen PDF
- Proc. Amer. Math. Soc. 39 (1973), 295-299 Request permission
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.References
- D. W. Boyd, The Hilbert transform on rearrangement-invariant spaces, Canadian J. Math. 19 (1967), 599–616. MR 212512, DOI 10.4153/CJM-1967-053-7
- David W. Boyd, Indices of function spaces and their relationship to interpolation, Canadian J. Math. 21 (1969), 1245–1254. MR 412788, DOI 10.4153/CJM-1969-137-x
- David W. Boyd, Indices for the Orlicz spaces, Pacific J. Math. 38 (1971), 315–323. MR 306887, DOI 10.2140/pjm.1971.38.315 G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, New York, 1934.
- E. C. Titchmarsh, Han-shu lun, Science Press, Peking, 1964 (Chinese). Translated from the English by Wu Chin. MR 0197687
Additional Information
- © Copyright 1973 American Mathematical Society
- 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