On maximal rearrangement inequalities for the Fourier transform

Authors:
W. B. Jurkat and G. Sampson

Journal:
Trans. Amer. Math. Soc. **282** (1984), 625-643

MSC:
Primary 42B10; Secondary 26D15

DOI:
https://doi.org/10.1090/S0002-9947-1984-0732111-0

MathSciNet review:
732111

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Abstract | References | Similar Articles | Additional Information

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)

**[1]**A. Baernstein,*Integral means, univalent functions and circular symmetrization*, Acta Math.**133**(1974), 139-169. MR**0417406 (54:5456)****[2]**R. M. Gabriel,*A "star inequality" for harmonic functions*, Proc. London Math. Soc.**34**(1932), 305-313.**[3]**G. H. Hardy,*Orders of infinity, the "infinitärcalcül" of Paul DuBois-Reymond*, Cambridge Univ. Press, London and New York, 1910.**[4]**G. H. Hardy and J. E. Littlewood,*Some new properties of Fourier constants*, J. London Math. Soc.**6**(1931), 3-9.**[5]**G. H. Hardy, J. E. Littlewood and G. Pólya,*Inequalities*, Cambridge Univ. Press, London and New York, 1934.**[6]**W. B Jurkat and G. Sampson,*On rearrangement and weight inequalities for the Fourier transform*, Indiana Math. J. (to appear). MR**733899 (85k:42040)****[7]**J. E. Littlewood,*On a theorem of Paley*, J. London Math. Soc.**29**(1954), 387-395. MR**0063473 (16:126e)****[8]**-,*On inequalities between*and , J. London Math. Soc.**35**(1960), 352-365. MR**0130945 (24:A799)****[9]**H. L. Montgomery,*A note on rearrangements of Fourier coefficients*, Ann. Inst. Fourier (Grenoble)**26**(1976), 29-34. MR**0407517 (53:11292)****[10]**B. Muckenhoupt,*Weighted norm inequalities for the Fourier transform*, Trans. Amer. Math. Soc.**276**(1983), 729-742. MR**688974 (84m:42019)****[11]**E. M. Stein and G. Weiss,*Introduction to Fourier analysis on Euclidean spaces*, Princeton Univ. Press, Princeton, N. J., 1971. MR**0304972 (46:4102)****[12]**A. Zygmund,*Trigonometric series*, Vol. 2, Cambridge Univ. Press, London and New York, 1959. MR**0107776 (21:6498)**

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DOI:
https://doi.org/10.1090/S0002-9947-1984-0732111-0

Article copyright:
© Copyright 1984
American Mathematical Society