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 | References | Similar Articles | Additional Information

Abstract: The relation between approach regions and singularities of nonnegative kernels is studied, where , , , and is a homogeneous space. For , a sufficient condition on approach regions () is given so that the maximal function

is weak-type with respect to a pair of measures and . 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**

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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