Proceedings of the American Mathematical Society

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

Author(s): Shiying Zhao
Journal: Proc. Amer. Math. Soc. 125 (1997), 2013-2020.
MSC (1991): Primary 42B20, 42B25
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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).

1.: P. Ahern and A. Nagel, Strong 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

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 42B20, 42B25

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: 10.1090/S0002-9939-97-03784-2
PII: S 0002-9939(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
Copyright of article: Copyright 1997, American Mathematical Society

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