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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Weighted norm inequalities for the conjugate function and Hilbert transform


Authors: Richard Hunt, Benjamin Muckenhoupt and Richard Wheeden
Journal: Trans. Amer. Math. Soc. 176 (1973), 227-251
MSC: Primary 42A40; Secondary 44A15, 47G05
MathSciNet review: 0312139
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Abstract: The principal problem considered is the determination of all non-negative functions $ W(x)$ with period $ 2\pi $ such that

$\displaystyle \int_{ - \pi }^\pi {\vert\tilde f(\theta ){\vert^p}W(\theta )\;d\... ...a \leq C} \;\int_{ - \pi }^\pi {\vert f(\theta ){\vert^p}W(\theta )\;d\theta } $

where $ 1 < p < \infty $, f has period $ 2\pi $, C is a constant independent of f, and $ \tilde f$ is the conjugate function defined by

$\displaystyle \tilde f(\theta ) = \mathop {\lim }\limits_{\varepsilon \to {0^ +... ...rt\phi \vert \leq \pi } {\frac{{f(\theta - \phi )\;d\phi }}{{2\tan \phi /2}}.} $

The main result is that $ W(x)$ is such a function if and only if

$\displaystyle \left[ {\frac{1}{{\vert I\vert}}\int_I {W(\theta )\;d\theta } } \... ...ert}}\int_I {{{[W(\theta )]}^{ - 1/(p - 1)}}d\theta } } \right]^{p - 1}} \leq K$

where I is any interval, $ \vert I\vert$ denotes the length of I and K is a constant independent of I.

Various related problems are also considered. These include weak type results, the nonperiodic case, the discrete case, an application to weighted mean convergence of Fourier series, and an estimate for one of the functions in the Fefferman and Stein decomposition of functions of bounded mean oscillation.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0312139-8
PII: S 0002-9947(1973)0312139-8
Keywords: Conjugate function, Hilbert transform, discrete Hilbert transform, weighted norm inequalities, bounded mean oscillation
Article copyright: © Copyright 1973 American Mathematical Society