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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Weighted weak-type inequalities
for the maximal function of nonnegative
integral transforms over approach regions

Author: Shiying Zhao
Journal: Proc. Amer. Math. Soc. 125 (1997), 2013-2020
MSC (1991): Primary 42B20, 42B25
MathSciNet review: 1372047
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Abstract: The relation between approach regions and singularities of nonnegative kernels $K_t(x,y)$ is studied, where $t\in (0,\infty )$, $x$, $y\in X$, and $X$ is a homogeneous space. For $1\le p<q<\infty $, a sufficient condition on approach regions $\varOmega _a$ ($a\in X$) is given so that the maximal function

\begin{equation*}\sup _{(x,t)\in \Omg_{a}} \int _X K_t(x,y)f(y)\,d\sigma (y) \end{equation*}

is weak-type $(p,q)$ with respect to a pair of measures $\sigma $ and $\omega $. It is shown that this condition is also necessary for operators of potential type in the sense of Sawyer and Wheedon (Amer. J. Math. 114 (1992), 813-874).

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Shiying Zhao
Affiliation: Department of Mathematics and Computer Science, University of Missouri-St. Louis, St. Louis, Missouri 63121

Keywords: Weak type inequalities, weights, operators of potential type, maximal functions, approach regions
Received by editor(s): April 13, 1994
Received by editor(s) in revised form: January 19, 1996
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1997 American Mathematical Society