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Transactions of the American Mathematical Society

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



Two weight function norm inequalities for the Poisson integral

Author: Benjamin Muckenhoupt
Journal: Trans. Amer. Math. Soc. 210 (1975), 225-231
MSC: Primary 42A40
MathSciNet review: 0374790
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Abstract: Let $ f(x)$ denote a complex valued function with period $ 2\pi $, let

$\displaystyle {P_r}(f,x) = \frac{1}{{2\pi }}\int_{ - \pi }^\pi {\frac{{(1 - {r^2})f(y)dy}}{{1 - 2r\cos (x - y) + {r^2}}}} $

be the Poisson integral of $ f(x)$ and let $ \vert I\vert$ denote the length of an interval $ I$. For $ 1 \leqslant p < \infty $ and nonnegative $ U(x)$ and $ V(x)$ with period $ 2\pi $ it is shown that there is a $ C$, independent of $ f$, such that

$\displaystyle \mathop {\sup }\limits_{0 \leqslant r < 1} \int_{ - \pi }^\pi {\v... ...,x){\vert^p}U(x)dx \leqslant C\int_{ - \pi }^\pi {\vert f(x){\vert^p}V(x)dx} } $

if and only if there is a $ B$ such that for all intervals $ I$

$\displaystyle \left[ {\frac{1}{{\vert I\vert}}\int_I {U(x)dx} } \right]{\left[ ... ...t I\vert}}\int_I {{{[V(x)]}^{ - 1/(p - 1)}}dx} } \right]^{p - 1.}} \leqslant B.$

Similar results are obtained for the nonperiodic case and in the case where $ U(x)dx$ and $ V(x)dx$ are replaced by measures.

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Article copyright: © Copyright 1975 American Mathematical Society