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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Weighted norm inequalities for certain integral operators. II


Author: H. P. Heinig
Journal: Proc. Amer. Math. Soc. 95 (1985), 387-395
MSC: Primary 26D10; Secondary 42A50, 44A10, 47G05
MathSciNet review: 806076
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Abstract: Conditions on nonnegative weight functions $ u(x)$ and $ \upsilon (x)$ are given which ensure that an inequality of the form $ {(\smallint {\left\vert {Tf(x)} \right\vert^q}u(x)\;dx)^{1/q}} \leqslant C{(\smallint {\left\vert {f(x)} \right\vert^p}\upsilon (x)\;dx)^{1/p}}$ holds for $ 1 \leqslant q < p < \infty $, where $ T$ is an integral operator of the form $ \int_{ - \infty }^x {K(x,y)f(y)dy} $ or $ \int_x^\infty {K(y,x)f(y)\;dy}$ and $ C$ a constant independent of $ f$. Specifically a number of inequalities for well-known classical operators are obtained. Inequalities of the above form for $ 1 \leqslant p \leqslant q < \infty $ were obtained in [1].


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DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0806076-3
PII: S 0002-9939(1985)0806076-3
Article copyright: © Copyright 1985 American Mathematical Society