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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
DOI: https://doi.org/10.1090/S0002-9947-1982-0662048-5
MathSciNet review: 662048
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Abstract: Let $\tilde f$ denote the Hilbert transform of $f$, i.e. \[ \tilde f(x) = {\rm {p}}{\rm {.v}}{\rm {.}}\int {\frac {{f(t)}}{{x - t}}dt} \] and let $1 < p < \infty$. A weight function $w$ is shown to satisfy \[ \int {|\tilde f(x)} {|^p}w(x)dx \le C{\int {|f(x)|} ^p}w(x)dx\] for all $f$ with the first $N$ moments zero, if and only if it is of the form $w(x) = |q(x){|^p}U(x)$, where $q$ is a polynomial of degree at most $N$ and $U \in {A_p}$.


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Keywords: Hilbert transform, weighted norm inequalities, <IMG WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${A_p}$"> weights, functions with vanishing moments
Article copyright: © Copyright 1982 American Mathematical Society