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Transactions of the American Mathematical Society

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$ L\sp{p}$ multipliers with weight $ x\sp{kp-1}$

Authors: Benjamin Muckenhoupt and Wo Sang Young
Journal: Trans. Amer. Math. Soc. 275 (1983), 623-639
MSC: Primary 42A45
MathSciNet review: 682722
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Abstract: Let $ k$ be a positive integer and $ 1 < p < \infty $. It is shown that if $ T$ is a multiplier operator on $ {L^p}$ of the line with weight $ \vert x{\vert^{kp-1}}$, then $ Tf$ equals a constant times $ f$ almost everywhere. This does not extend to the periodic case since $ m(j) = 1/j, j \ne 0$, is a multiplier sequence for $ {L^p}$ of the circle with weight $ \vert x{\vert^{kp-1}}$. A necessary and sufficient condition is derived for a sequence $ m(j)$ to be a multiplier on $ {L^2}$ of the circle with weight $ \vert x{\vert^{2k - 1}}$.

References [Enhancements On Off] (What's this?)

  • [1] B. Muckenhoupt, Hardy's inequality with weights, Studia Math. 44 (1972), 31-38. MR 0311856 (47:418)
  • [2] B. Muckenhoupt, R. L. Wheeden and W.-S. Young, $ {L^2}$ multipliers with power weights. Adv. in Math. (to appear). MR 714588 (85d:42010)
  • [3] -, Weighted $ {L^p}$ multipliers (to appear).
  • [4] J. Riordan, Combinatorial identities, Wiley, New York, 1968. MR 0231725 (38:53)
  • [5] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N. J., 1970. MR 0290095 (44:7280)
  • [6] A. Zygmund, Trigonometric series, 2nd ed., Cambridge Univ. Press, New York, 1959. MR 0107776 (21:6498)

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Article copyright: © Copyright 1983 American Mathematical Society

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