On maximal rearrangement inequalities for the Fourier transform
Authors:
W. B. Jurkat and G. Sampson
Journal:
Trans. Amer. Math. Soc. 282 (1984), 625643
MSC:
Primary 42B10; Secondary 26D15
MathSciNet review:
732111
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Abstract: Suppose that is a measurable function on and denote by the decreasing rearrangement of (provided that it exists). We show that the dimensional Fourier transform satisfies (1) if and for . We also show that (2) if and is nonnegative and symmetrically decreasing. Inequality (2) implies that (1) is maximal in the sense that the left side reaches the right side if is nonnegative and symmetrically decreasing. Hence, (1) implies all other possible estimates in terms of and . The cases of (1) can be derived from the case (and same ) by a convexity principle which does not involve interpolation. The analogue of (1) for Fourier series is due to H. L. Montgomery if (then the extra condition on is automatically satisfied).
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 A. Baernstein, Integral means, univalent functions and circular symmetrization, Acta Math. 133 (1974), 139169. MR 0417406 (54:5456)
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 R. M. Gabriel, A "star inequality" for harmonic functions, Proc. London Math. Soc. 34 (1932), 305313.
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 G. H. Hardy and J. E. Littlewood, Some new properties of Fourier constants, J. London Math. Soc. 6 (1931), 39.
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 W. B Jurkat and G. Sampson, On rearrangement and weight inequalities for the Fourier transform, Indiana Math. J. (to appear). MR 733899 (85k:42040)
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 J. E. Littlewood, On a theorem of Paley, J. London Math. Soc. 29 (1954), 387395. MR 0063473 (16:126e)
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 , On inequalities between and , J. London Math. Soc. 35 (1960), 352365. MR 0130945 (24:A799)
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 H. L. Montgomery, A note on rearrangements of Fourier coefficients, Ann. Inst. Fourier (Grenoble) 26 (1976), 2934. MR 0407517 (53:11292)
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 B. Muckenhoupt, Weighted norm inequalities for the Fourier transform, Trans. Amer. Math. Soc. 276 (1983), 729742. MR 688974 (84m:42019)
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 E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, N. J., 1971. MR 0304972 (46:4102)
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 A. Zygmund, Trigonometric series, Vol. 2, Cambridge Univ. Press, London and New York, 1959. MR 0107776 (21:6498)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198407321110
PII:
S 00029947(1984)07321110
Article copyright:
© Copyright 1984
American Mathematical Society
