1. bookVolume 19 (2020): Issue 1 (December 2020)
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11 Dec 2014
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access type Open Access

Maximal functions for Weinstein operator

Published Online: 31 Dec 2020
Page range: 105 - 119
Received: 04 Sep 2019
Accepted: 22 Nov 2019
Journal Details
License
Format
Journal
First Published
11 Dec 2014
Publication timeframe
1 time per year
Languages
English
Abstract

In the present paper, we study in the harmonic analysis associated to the Weinstein operator, the boundedness on Lp of the uncentered maximal function. First, we establish estimates for the Weinstein translation of characteristic function of a closed ball with radius ɛ centered at 0 on the upper half space ℝd–1× ]0, +∞ [. Second, we prove weak-type L1-estimates for the uncentered maximal function associated with the Weinstein operator and we obtain the Lp-boundedness of this operator for 1 < p ≤ + ∞. As application, we define a large class of operators such that each operator of this class satisfies these Lp-inequalities. In particular, the maximal operator associated respectively with the Weinstein heat semigroup and the Weinstein-Poisson semigroup belong to this class.

Keywords

[1] Abdelkefi, Chokri. “Dunkl operators on ℝd and uncentered maximal function.” J. Lie Theory 20, no. 1 (2010): 113-125. Cited on 107.Search in Google Scholar

[2] Ben Nahia, Zouhir, and Néjib Ben Salem. “Spherical harmonics and applications associated with the Weinstein operator.” Potential Theory ICPT 94 (Kouty, 1994), 233-241. Berlin: de Gruyter, 1996. Cited on 106.Search in Google Scholar

[3] Ben Nahia, Zouhir, and Néjib Ben Salem. “On a mean value property associated with the Weinstein operator.” Potential Theory ICPT 94 (Kouty, 1994), 243-253. Berlin: de Gruyter, 1996. Cited on 106.Search in Google Scholar

[4] Bloom, Walter R., and Zeng Fu Xu. “The Hardy-Littlewood maximal function for Chébli-Trimèche hypergroups.” Applications of hypergroups and related measure algebras (Seattle, WA, 1993), 45–70. Vol. 183 of Contemp. Math. Providence, RI: Amer. Math. Soc., 1995. Cited on 107.Search in Google Scholar

[5] Brelot, Marcel. “Équation de Weinstein et potentiels de Marcel Riesz” Séminaire de Théorie du Potentiel, no. 3 (Paris, 1976/1977), 18–38. Vol. 681 of Lecture Notes in Math. Berlin: Springer, 1978. Cited on 106.Search in Google Scholar

[6] Clerc, Jean-Louis, and Elias Menachem Stein. “Lp-multipliers for noncompact symmetric spaces.” Proc. Nat. Acad. Sci. U.S.A. 71 (1974): 3911-3912. Cited on 106.Search in Google Scholar

[7] Connett, William C, and Alan L. Schwartz. “The Littlewood-Paley theory for Jacobi expansions.” Trans. Amer. Math. Soc. 251 (1979): 219-234. Cited on 106.Search in Google Scholar

[8] Connett, William C., and Alan L. Schwartz. “A Hardy-Littlewood maximal inequality for Jacobi type hypergroups.” Proc. Amer. Math. Soc. 107, no. 1 (1989): 137-143. Cited on 106.Search in Google Scholar

[9] Gaudry, Garth Ian, et all. “Hardy-Littlewood maximal functions on some solvable Lie groups.” J. Austral. Math. Soc. Ser. A 45, no. 1 (1988): 78-82. Cited on 106.Search in Google Scholar

[10] Hardy, Godfrey Harold, and John Edensor Littlewood. “A maximal theorem with function-theoretic applications.” Acta Math. 54, no. 1 (1930): 81-116. Cited on 106.Search in Google Scholar

[11] Hewitt, Edwin, and Karl Stromberg. Real and abstract analysis. A modern treatment of the theory of functions of a real variable. New-York: Springer-Verlag, 1965. Cited on 116.Search in Google Scholar

[12] Leutwiler, Heinz. “Best constants in the Harnack inequality for the Weinstein equation.” Aequationes Math. 34, no. 2-3 (1987): 304-315. Cited on 106.Search in Google Scholar

[13] Stein, Elias Menachem. Singular integrals and differentiability properties of functions. Vol. 30 of Princeton Mathematical Series. Princeton, New Jersey: Princeton University Press, 1970. Cited on 106, 115 and 116.Search in Google Scholar

[14] Stempak, Krzysztof. “La théorie de Littlewood-Paley pour la transformation de Fourier-Bessel.” C. R. Acad. Sci. Paris Sér. I Math. 303, no. 1 (1986): 15-18. Cited on 106.Search in Google Scholar

[15] Strömberg, Jan-Olov. “Weak type L1 estimates for maximal functions on noncom-pact symmetric spaces.” Ann. of Math. (2) 114, no. 1 (1981): 115-126. Cited on 106.Search in Google Scholar

[16] Thangavelu, Sundaram, and Yuan Xu. “Convolution operator and maximal function for the Dunkl transform.” J. Anal. Math. 97 (2005): 25-55. Cited on 106 and 117.Search in Google Scholar

[17] Torchinsky, Alberto. Real-variable nethods in harmonic analysis. Vol. 123 of Pure and applied mathematics. Orlando: Academic Press, 1986. Cited on 106.Search in Google Scholar

[18] Watson, George Neville. A treatise on the theory of Bessel functions. Cambridge-New York-Oakleigh: Cambridge University Press, 1966. Cited on 108.Search in Google Scholar

[19] Weinstein, Alexander. “Singular partial differential equations and their applications.” Fluid dynamics and applied mathematics, 29-49. New York: Gordon and Breach, 1962. Cited on 106.Search in Google Scholar

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