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
MathSciNet review: 0315071
Full-text PDF Free Access
Abstract: Hardy's well-known inequality relating the norm of a function and the norm of its integral mean in the Lebesgue spaces , is extended to the class of rearrangement invariant spaces . These spaces include, for example, the , 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 -measurable functions.
-  D. W. Boyd, The Hilbert transform on rearrangement-invariant spaces, Canadian J. Math. 19 (1967), 599–616. MR 212512, https://doi.org/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, https://doi.org/10.4153/CJM-1969-137-x
-  David W. Boyd, Indices for the Orlicz spaces, Pacific J. Math. 38 (1971), 315–323. MR 306887
-  G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, New York, 1934.
-  E. C. Titchmarsh, Han-shu lun, Translated from the English by Wu Chin, Science Press, Peking, 1964 (Chinese). MR 0197687
- D. W. Boyd, The Hilbert transformation on rearrangement invariant spaces, Canad. J. Math. 19 (1967), 599-616. MR 35 #3383. MR 0212512 (35:3383)
- -, Indices of function spaces and their relationship to interpolation, Canad. J. Math. 21 (1969), 1245-1254. MR 0412788 (54:909)
- -, Indices for the Orlicz spaces, Pacific J. Math. 38 (1971), 315-323. MR 0306887 (46:6008)
- G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, New York, 1934.
- E. C. Titchmarsh, The theory of functions, 2nd ed., Clarendon Press, Oxford, 1948. MR 0197687 (33:5850)
Keywords: Integral mean, Laplace transform, rearrangement invariant function space, Lorentz space, Orlicz space
Article copyright: © Copyright 1973 American Mathematical Society