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Iterated Littlewood-Paley functions and a multiplier theorem

Author: W. R. Madych
Journal: Proc. Amer. Math. Soc. 45 (1974), 325-331
MSC: Primary 42A92; Secondary 42A18
MathSciNet review: 0355475
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Abstract: A sufficient condition for a bounded function to be a multiplier of Fourier transforms on $ {L^p}({R^n}),1 < p < \infty $, is established. The classical case of Marcinkiewicz is properly included. The main tools used in obtaining this result are iterated variants of the classical LittlewoodPaley functions together with an $ {L^p}$ estimate on certain maximal functions closely related to strong differentiability of multiple integrals.

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  • [1] A. Benedek, A. P. Calderón and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 356-365. MR 24 #A3479. MR 0133653 (24:A3479)
  • [2] L. Hörmander, Estimates for translation invariant operators in $ {L^p}$ spaces, Acta. Math. 104 (1960), 93-140. MR 22 #12389. MR 0121655 (22:12389)
  • [3] B. Jessen, J. Marcinkiewicz and A. Zygmund, Note on the differentiability of multiple integrals, Fund. Math. 25 (1936), 217-234.
  • [4] W. Littman, C. McCarthy and N. Rivière, $ {L^p}$-multiplier theorems, Studia Math. 30 (1968), 193-217. MR 37 #6681. MR 0231126 (37:6681)
  • [5] W. R. Madych, On Littlewood-Paley functions, Studia Math. (to appear). MR 0344797 (49:9536)
  • [6] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Math. Series, no. 30, Princeton Univ. Press, Princeton, N. J., 1970. MR 44 #7280. MR 0290095 (44:7280)

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