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A characterization of $ L^p$-improving measures


Author: Kathryn E. Hare
Journal: Proc. Amer. Math. Soc. 102 (1988), 295-299
MSC: Primary 43A22,; Secondary 42A45,42A85
DOI: https://doi.org/10.1090/S0002-9939-1988-0920989-6
MathSciNet review: 920989
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Abstract: A Borel measure $ \mu $ on a compact abelian group $ G$ is $ {L^p}$-improving if $ \mu $ convolves $ {L^p}(G)$ to $ {L^{p + \varepsilon }}(G)$ for some $ \varepsilon > 0$. We characterize $ {L^p}$-improving measures by means of their Fourier transforms.


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  • [1] W. Beckner, S. Janson and D. Jerison, Convolution inequalities on the circle, Conference on Harmonic Analysis in Honor of Antoni Zygmund (W. Beckner et al., eds.), Wadsworth, Belmont, Calif., 1983. MR 730056 (85j:42016)
  • [2] A. Bonami, Etude des coefficients de Fourier des fonctions de $ {L^p}(G)$, Ann. Inst. Fourier (Grenoble) 20 (1970), 335-402. MR 0283496 (44:727)
  • [3] M. Christ, A convolution inequality concerning Cantor-Lebesgue measures, preprint, 1985. MR 850410 (87k:42011)
  • [4] J. Lopez and K. Ross, Sidon sets, Lecture Notes in Pure and Appl. Math., no. 13, Marcel Dekker, New York, 1975. MR 0440298 (55:13173)
  • [5] D. Oberlin, A convolution property of the Cantor-Lebesgue measure, Colloq. Math. 67 (1982), 113-117. MR 679392 (84f:43001)
  • [6] D. Ritter, Most Riesz product measures are $ {L^p}$-improving, Proc. Amer. Math. Soc. 97 (1986), 291-295. MR 835883 (87j:43003)
  • [7] H. P. Rosenthal, On subspaces of $ {L^p}$, Ann. of Math. 97 (1973), 344-373. MR 0312222 (47:784)
  • [8] -, Convolution by a biased coin, vol. II, Altgeld Book, Univ. of Illinois, 1975, pp. 1-17.
  • [9] W. Rudin, Trigonometric series with gaps, J. Math. Mech. 9 (1960), 203-227. MR 0116177 (22:6972)
  • [10] E. M. Stein, Harmonic analysis on $ {R^n}$, Studies in Harmonic Analysis (J. M. Ash, ed.), M.A.A. Studies Math., vol. 13, Math. Assoc. Amer., Washington, D.C., 1976, pp. 97-135. MR 0461002 (57:990)
  • [11] A. Zygmund, Trigonometric series, 2nd ed., vol. II, Cambridge Univ. Press, 1959. MR 0107776 (21:6498)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0920989-6
Keywords: $ {L^p}$-improving measure, $ \Lambda (p)$ set
Article copyright: © Copyright 1988 American Mathematical Society

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