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Transactions of the American Mathematical Society
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 | References | Similar Articles | Additional Information

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}$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0662048-5
PII: S 0002-9947(1982)0662048-5
Keywords: Hilbert transform, weighted norm inequalities, $ {A_p}$ weights, functions with vanishing moments
Article copyright: © Copyright 1982 American Mathematical Society