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Most Riesz product measures are $ L\sp p$-improving


Author: David L. Ritter
Journal: Proc. Amer. Math. Soc. 97 (1986), 291-295
MSC: Primary 43A15; Secondary 43A25
MathSciNet review: 835883
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Abstract: A Borel measure $ \mu $ on a compact abelian group $ G$ is $ {L^p}$-improving if, given $ p > 1$, there is a $ q = q(p,\mu ) > p$ and $ {\text{a}}\;K = K(p,q,\mu ) > 0$ such that $ {\left\Vert {\mu * f} \right\Vert _q} \leq K{\left\Vert f \right\Vert _p}$ for each $ f$ in $ {L^p}(G)$. Here the $ {L^p}$-improving Riesz product measures on infinite compact abelian groups are characterized by means of their Fourier transforms.


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  • [1] William Beckner, Svante Janson, and David Jerison, Convolution inequalities on the circle, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 32–43. MR 730056
  • [2] Aline Bonami, Étude des coefficients de Fourier des fonctions de 𝐿^{𝑝}(𝐺), Ann. Inst. Fourier (Grenoble) 20 (1970), no. fasc. 2, 335–402 (1971) (French, with English summary). MR 0283496
  • [3] Gavin Brown, Riesz products and generalized characters, Proc. London Math. Soc. (3) 30 (1975), 209–238. MR 0372530
  • [4] R. E. Edwards, Fourier series. Vol. 2, 2nd ed., Graduate Texts in Mathematics, vol. 85, Springer-Verlag, New York-Berlin, 1982. A modern introduction. MR 667519
  • [5] Colin C. Graham and O. Carruth McGehee, Essays in commutative harmonic analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 238, Springer-Verlag, New York-Berlin, 1979. MR 550606
  • [6] Edwin Hewitt and Herbert S. Zuckerman, Singular measures with absolutely continuous convolution squares, Proc. Cambridge Philos. Soc. 62 (1966), 399–420. MR 0193435
  • [7] Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley and Sons), New York-London, 1962. MR 0152834
  • [8] Elias M. Stein, Harmonic analysis on 𝑅ⁿ, Studies in harmonic analysis (Proc. Conf., DePaul Univ., Chicago, Ill., 1974), Math. Assoc. Amer., Washington, D.C., 1976, pp. 97–135. MAA Stud. Math., Vol. 13. MR 0461002
  • [9] Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972

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DOI: https://doi.org/10.1090/S0002-9939-1986-0835883-7
Article copyright: © Copyright 1986 American Mathematical Society