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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On weighted norm inequalities for the Hilbert transform of functions with moments zero

Author: Ernst Adams
Journal: Trans. Amer. Math. Soc. 272 (1982), 487-500
MSC: Primary 44A15; Secondary 42A50
MathSciNet review: 662048
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Abstract: Let $ \tilde f$ denote the Hilbert transform of $ f$, i.e.

$\displaystyle \tilde f(x) = {\rm {p}}{\rm {.v}}{\rm {.}}\int {\frac{{f(t)}}{{x - t}}dt} $

and let $ 1 < p < \infty $. A weight function $ w$ is shown to satisfy

$\displaystyle \int {\vert\tilde f(x)} {\vert^p}w(x)dx \le C{\int {\vert f(x)\vert} ^p}w(x)dx$

for all $ f$ with the first $ N$ moments zero, if and only if it is of the form $ w(x) = \vert q(x){\vert^p}U(x)$, where $ q$ is a polynomial of degree at most $ N$ and $ U \in {A_p}$.

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

  • [1] R. Arocena, M. Cotlar and C. Sadosky, Weighted inequalities in $ L^2$ and lifting properties, Adv. in Math. (to appear). MR 634237 (83e:42016)
  • [2] A. Beurling and P. Malliavin, On Fourier transforms of measures with compact support, Acta Math. 107 (1962), 291-309. MR 0147848 (26:5361)
  • [3] A. P. Calderon and A. Zygmund, Local properties of solutions of elliptic partial differential equations, Studia Math. 20 (1961), 171-225. MR 0136849 (25:310)
  • [4] H. Helson and D. Sarason, Past and future, Math. Scand. 21 (1967), 5-16. MR 0236989 (38:5282)
  • [5] H. Helson and G. Szegö, A problem in prediction theory, Ann. Mat. Pura Appl. (4) 51 (1960), 107-138. MR 0121608 (22:12343)
  • [6] R. Hunt, B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227-251. MR 0312139 (47:701)
  • [7] B. Muckenhoupt, R. Wheeden and W. Young, $ L^2$ multipliers with power weights, Adv. in Math. (to appear). MR 714588 (85d:42010)
  • [8] -, Weighted $ L^p$ multipliers (to appear).
  • [9] J.-O. Strömberg and A. Torchinsky, Weighted Hardy spaces (to appear).
  • [10] J.-O. Strömberg and R. Wheeden, Relations between $ H_u^{p}$ and $ L_u^{p}$ with polynomial weights (to appear).

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Additional Information

Keywords: Hilbert transform, weighted norm inequalities, $ {A_p}$ weights, functions with vanishing moments
Article copyright: © Copyright 1982 American Mathematical Society

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