Weighted weak-type inequalities for the maximal function of nonnegative integral transforms over approach regions
HTML articles powered by AMS MathViewer
- by Shiying Zhao PDF
- Proc. Amer. Math. Soc. 125 (1997), 2013-2020 Request permission
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 $\Omega _a$ ($a\in X$) is given so that the maximal function \begin{equation*} \sup _{(x,t)\in \Omega _{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).References
- Patrick Ahern and Alexander Nagel, Strong $L^p$ estimates for maximal functions with respect to singular measures; with applications to exceptional sets, Duke Math. J. 53 (1986), no. 2, 359–393. MR 850541, DOI 10.1215/S0012-7094-86-05323-8
- Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. MR 447954, DOI 10.1090/S0002-9904-1977-14325-5
- 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), no. 1, 11–34 (English, with English and Georgian summaries). This paper also appears in Georgian Math. J. 1 (1994), no. 1, 9–29. MR 1251491, DOI 10.1007/bf02315300
- Roberto A. Macías and Carlos Segovia, Lipschitz functions on spaces of homogeneous type, Adv. in Math. 33 (1979), no. 3, 257–270. MR 546295, DOI 10.1016/0001-8708(79)90012-4
- B. A. Mair and David Singman, A generalized Fatou theorem, Trans. Amer. Math. Soc. 300 (1987), no. 2, 705–719. MR 876474, DOI 10.1090/S0002-9947-1987-0876474-7
- Alexander Nagel and Elias M. Stein, On certain maximal functions and approach regions, Adv. in Math. 54 (1984), no. 1, 83–106. MR 761764, DOI 10.1016/0001-8708(84)90038-0
- Wen Jie Pan, Weighted norm inequalities for certain maximal operators with approach regions, Harmonic analysis (Tianjin, 1988) Lecture Notes in Math., vol. 1494, Springer, Berlin, 1991, pp. 169–175. MR 1187076, DOI 10.1007/BFb0087768
- E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), no. 4, 813–874. MR 1175693, DOI 10.2307/2374799
- Juan Sueiro, On maximal functions and Poisson-Szegő integrals, Trans. Amer. Math. Soc. 298 (1986), no. 2, 653–669. MR 860386, DOI 10.1090/S0002-9947-1986-0860386-8
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
- Received by editor(s): April 13, 1994
- Received by editor(s) in revised form: January 19, 1996
- Communicated by: J. Marshall Ash
- © Copyright 1997 American Mathematical Society
- 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