Maximal functions for Weinstein operator

Chokri Abdelkefi

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica (2020)

  • Volume: 19, page 105-119
  • ISSN: 2300-133X

Abstract

top
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 ε 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.

How to cite

top

Chokri Abdelkefi. "Maximal functions for Weinstein operator." Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica 19 (2020): 105-119. <http://eudml.org/doc/296830>.

@article{ChokriAbdelkefi2020,
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 ε 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.},
author = {Chokri Abdelkefi},
journal = {Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica},
keywords = {Weinstein operator; Weinstein transform; Weinstein translation operators; Maximal functions},
language = {eng},
pages = {105-119},
title = {Maximal functions for Weinstein operator},
url = {http://eudml.org/doc/296830},
volume = {19},
year = {2020},
}

TY - JOUR
AU - Chokri Abdelkefi
TI - Maximal functions for Weinstein operator
JO - Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
PY - 2020
VL - 19
SP - 105
EP - 119
AB - 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 ε 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.
LA - eng
KW - Weinstein operator; Weinstein transform; Weinstein translation operators; Maximal functions
UR - http://eudml.org/doc/296830
ER -

References

top
  1. Abdelkefi, Chokri. "Dunkl operators on Rd and uncentered maximal function." J. Lie Theory 20, no. 1 (2010): 113-125. 
  2. Ben Nahia, Zouhir, and Néjib Ben Salem. "Spherical harmonics and applications associated with theWeinstein operator." Potential Theory ICPT 94 (Kouty, 1994), 233-241. Berlin: de Gruyter, 1996. 
  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. 
  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. 
  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. 
  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. 
  7. Connett, William C, and Alan L. Schwartz. "The Littlewood-Paley theory for Jacobi expansions." Trans. Amer. Math. Soc. 251 (1979): 219-234. 
  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. 
  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. 
  10. Hardy, Godfrey Harold, and John Edensor Littlewood. "A maximal theorem with function-theoretic applications." Acta Math. 54, no. 1 (1930): 81-116. 
  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. 
  12. Leutwiler, Heinz. "Best constants in the Harnack inequality for the Weinstein equation." Aequationes Math. 34, no. 2-3 (1987): 304-315. 
  13. Stein, Elias Menachem. Singular integrals and differentiability properties of functions. Vol. 30 of Princeton Mathematical Series. Princeton, New Jersey: Princeton University Press, 1970. 
  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. 
  15. Strömberg, Jan-Olov. "Weak type L1 estimates for maximal functions on noncompact symmetric spaces." Ann. of Math. (2) 114, no. 1 (1981): 115-126. 
  16. Thangavelu, Sundaram, and Yuan Xu. "Convolution operator and maximal function for the Dunkl transform." J. Anal. Math. 97 (2005): 25-55. 
  17. Torchinsky, Alberto. Real-variable nethods in harmonic analysis. Vol. 123 of Pure and applied mathematics. Orlando: Academic Press, 1986. 
  18. Watson, George Neville. A treatise on the theory of Bessel functions. Cambridge-New York-Oakleigh: Cambridge University Press, 1966. 
  19. Weinstein, Alexander. "Singular partial differential equations and their applications." Fluid dynamics and applied mathematics, 29-49. New York: Gordon and Breach, 1962. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.