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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
DOI: https://doi.org/10.1090/S0002-9939-97-03784-2
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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  • 1. P. Ahern and A. Nagel, Strong $L^p$ estimates for maximal functions with respect to singular measures; with applications to exceptional sets, Duke Math. J. 53 (1986), 359-393. MR 88m:42037
  • 2. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645. MR 56:6264
  • 3. I. Genebashvili, A. Gogatishvili, and V. Kokilashvili, Criteria of general weak type inequalities for integral transforms with positive kernels, Proc. Georgian Acad. Sci. (Math.) 1 (1993), 11-34. MR 94j:42030
  • 4. R. Macias and C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. in Math. 33 (1979), 257-270. MR 81c:32017a
  • 5. B. A. Mair and D. Singman, A generalized Fatou theorem, Trans. Amer. Math. Soc. 300 (1987), 705-719. MR 88f:31011
  • 6. A. Nagel and E. M. Stein, On certain maximal functions and approach regions, Adv. in Math. 54 (1984), 83-106. MR 86a:42026
  • 7. W. Pan, Weighted norm inequalities for certain maxmial operators with approach regions, Harmonic Analysis (M.-T. Cheng et al., ed.), Lecture Notes in Math., vol. 1494, Springer-Verlag, Berlin and Heidelberg, 1991, pp. 167-175. MR 94a:42015
  • 8. E. T. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), 813-874. MR 94i:42024
  • 9. J. Sueiro, On maximal functions and Poisson-Szegö intergrals, Trans. Amer. Math. Soc. 298 (1986), 653-669. MR 87m:42017

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Additional Information

Shiying Zhao
Affiliation: Department of Mathematics and Computer Science, University of Missouri-St. Louis, St. Louis, Missouri 63121
Email: zhao@greatwall.cs.umsl.edu

DOI: https://doi.org/10.1090/S0002-9939-97-03784-2
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

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