ON BOUNDEDNESS PROPERTY OF SINGULAR INTEGRAL OPERATORS ASSOCIATED TO A SCHRÖDINGER OPERATOR IN A GENERALIZED MORREY SPACE AND APPLICATIONS

doi:10.1007/s10473-020-0501-2

数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (5): 1171-1184.doi: 10.1007/s10473-020-0501-2

• 论文 • 下一篇

ON BOUNDEDNESS PROPERTY OF SINGULAR INTEGRAL OPERATORS ASSOCIATED TO A SCHRÖDINGER OPERATOR IN A GENERALIZED MORREY SPACE AND APPLICATIONS

Xuan Truong LE^1,2, Thanh Nhan NGUYEN³, Ngoc Trong NGUYEN³

1. Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam;
2. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam;
3. Department of Mathematics, Ho Chi Minh City University of Education, Ho Chi Minh City, Vietnam

收稿日期:2019-07-02 出版日期:2020-10-25 发布日期:2020-11-04
作者简介:Xuan Truong LE,E-mail:lexuantruong@tdtu.edu.vn;Thanh Nhan NGUYEN,E-mail:nhannt@hcmue.edu.vn;Ngoc Trong NGUYEN,E-mail:trongnn@hcmue.edu.vn

ON BOUNDEDNESS PROPERTY OF SINGULAR INTEGRAL OPERATORS ASSOCIATED TO A SCHRÖDINGER OPERATOR IN A GENERALIZED MORREY SPACE AND APPLICATIONS

Xuan Truong LE^1,2, Thanh Nhan NGUYEN³, Ngoc Trong NGUYEN³

1. Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam;
2. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam;
3. Department of Mathematics, Ho Chi Minh City University of Education, Ho Chi Minh City, Vietnam

Received:2019-07-02 Online:2020-10-25 Published:2020-11-04

摘要/Abstract

摘要： In this paper, we provide the boundedness property of the Riesz transforms associated to the Schrödinger operator ?=-△+V in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces. The additional potential V considered in this paper is a non-negative function satisfying the suitable reverse Hölder's inequality. Our results are new and general in many cases of problems. As an application of the boundedness property of these singular integral operators, we obtain some regularity results of solutions to Schrödinger equations in the new Morrey space.

关键词: weighted Morrey spaces, Schödinger operator, riesz transforms, Regularity estimates

Abstract: In this paper, we provide the boundedness property of the Riesz transforms associated to the Schrödinger operator ?=-△+V in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces. The additional potential V considered in this paper is a non-negative function satisfying the suitable reverse Hölder's inequality. Our results are new and general in many cases of problems. As an application of the boundedness property of these singular integral operators, we obtain some regularity results of solutions to Schrödinger equations in the new Morrey space.

Key words: weighted Morrey spaces, Schödinger operator, riesz transforms, Regularity estimates

中图分类号:

42B20

Xuan Truong LE, Thanh Nhan NGUYEN, Ngoc Trong NGUYEN. ON BOUNDEDNESS PROPERTY OF SINGULAR INTEGRAL OPERATORS ASSOCIATED TO A SCHRÖDINGER OPERATOR IN A GENERALIZED MORREY SPACE AND APPLICATIONS[J]. 数学物理学报(英文版), 2020, 40(5): 1171-1184.

Xuan Truong LE, Thanh Nhan NGUYEN, Ngoc Trong NGUYEN. ON BOUNDEDNESS PROPERTY OF SINGULAR INTEGRAL OPERATORS ASSOCIATED TO A SCHRÖDINGER OPERATOR IN A GENERALIZED MORREY SPACE AND APPLICATIONS[J]. Acta mathematica scientia,Series B, 2020, 40(5): 1171-1184.

参考文献 35

[1]	Adams D, Xiao J. Nonlinear potential analysis on Morrey spaces and their capacities. Indiana Univ Math, 2004, 53:1629-1663
[2]	Akbulut1 A, Guliyev V, Omarova M. Marcinkiewicz integrals associated with Schrödinger operator and their commutators on vanishing generalized Morrey spaces. Boundary Value Problems, 2017, 2017:Art 121
[3]	Bui T A. The weighted norm inequalities for Riesz transforms of magnetic Schrödinger operators. Differential Integral Equations, 2010, 23:811-826
[4]	Bui T A. Weighted estimates for commutators of some singular integrals related to Schrödinger operators. Bulletin des Sciences Mathematiques, 2014, 138(2):270-292
[5]	Bongioanni B, Harboure E, Salinas O. Riesz transform related to Schrödinger operators acting on BMO type spaces. J Math Anal Appl, 2009, 357(2):115-131
[6]	Bongioanni B, Harboure E, Salinas O. Classes of weights related to Schrödinger operators. J Math Anal Appl, 2011, 373:563-579
[7]	Coulhon T, Duong X T. Riesz transforms for 1≤ p ≤ 2. Trans Amer Math Soc, 1999, 351(3):1151-1169
[8]	Duong X T, Xiao J, Yan L. Old and new Morrey spaces with heat kernel bounds. J Fourier Anal Appl, 2007, 13:87-111
[9]	Dziubánski J, Garrigos G, Martinez T, Torrea J, Zienkiewicz J. BMO spaces related to Schrödinger operators with potentials satisfying reverse Hölder inequality. Mat Z, 2005, 49(2):329-356
[10]	Dziubánski J, Zienkiewicz J. Hp spaces for Schrödinger operators. Fourier Anal Related Top, 2002, 56:45-53
[11]	Feuto J, Fofana I, Koua K. Espaces de fonctions moyenne fractionnaire intgrables sur les groupes localement compacts. Afrika Mat, 2003, 3(13):73-91
[12]	Feuto J, Fofana I, Koua K. Integrable fractional mean functions on spaces of homogeneous type. Afr Diaspora J Math, 2010, 9(1):8-30
[13]	Feuto J. Norm inequalities in generalized morrey spaces. J Fourier Anal Appl, 2014, 20(4):896-909
[14]	Fofana I. Étude d'une classe d'espaces de fonctions contenant les esp aces de Lorentz. Afrika Mat, 1988, 1(2):29-50
[15]	Grafakos L. Classical Fourier Analysis. Graduate Texts in Mathematics 250. 3rd ed. New York:Springer, 2014
[16]	Komori Y, Shirai S. Weighted Morrey spaces and a singular integral operator. Math Nachr, 2009, 282:219-231
[17]	Kurata K, Sugano S. Estimate of the fundamental solution for magnetic Schrödinger operators and their applications. Tohoku Math J, 2000, 52:367-382
[18]	Komori Y, Shirai S. Weighted Morrey spaces and a singular integral operator. Math Nachr, 2009, 282:219-231
[19]	Li P, Wan X, Zhang C. Schrödinger type operators on generalized Morrey spaces. J Inequal Appl, 2015, 2015:Art 229
[20]	Liu Y. The weighted estimates for the operators Vα(-△G+V)-β and Vα▽(-△G+V)-β on the stratified Lie group G. J Math Anal Appl, 2009, 349:235-244
[21]	Liu Y, Wang L. Boundedness for Riesz transform associated with Schrödinger operators and its commutator on weighted Morrey spaces related to certain nonnegative potentials. J Inequal Appl, 2014, 2014:Art 194
[22]	Ly F K. Second order Riesz transforms associated to the Schrödinger operator for p ≤ 1. J Math Anal Appl, 2014, 410(1):391-402
[23]	Morrey C. On the solutions of quasi-linear elliptic partial differential equations. Trans Amer Math Soc, 1938, 43:126-166
[24]	Muckenhoupt B, Wheeden R L. Weighted norm inequalities for fractional integrals. Trans Amer Math Soc, 1974, 192:261-274
[25]	Peetre J. On the theory of Lp,λ spaces. J Funct Anal, 1969, 4:71-87
[26]	Pan G, Tang L. Solvability for Schrödinger equations with discontinuous coefficients. J Funct Anal, 2016, 270:88-133
[27]	Ren B, Wang H. Boundedness of higher order Riesz transforms associated with Schrödinger type operator on generalized Morrey spaces. J Nonlinear Sci Appl, 2017, 10:2757-2766
[28]	Samko N. Weighted Hardy and singular operators in Morrey spaces. J Math Anal Appl, 2009, 350:56-72
[29]	Shen Z. Lp estimates for Schrödinger operators with certain potentials. Ann Inst Fourier, 1995, 45:513-546
[30]	Shen Z. Estimates in Lp for magnetic Schrödinger operators. Indiana Univ Math J, 1996, 45:817-841
[31]	Song L, Yang L. Riesz transforms associated to Schrödinger operators on weighted Hardy spaces. J Funct Anal, 2010, 259:1466-1490
[32]	Sugano S. Estimates for the operators Vα(-△ + V)-β and Vα▽(-△ + V)-β with certain nonnegative potentials V. Tokyo J Math, 1988, 21:441-452
[33]	Tang L, Dong J. Boundedness for some Schrödinger type operators on Morrey spaces related to certain nonnegative potentials. J Math Anal Appl, 2009, 355:101-109
[34]	Xiao J. Homothetic variant of fractional Sobolev space with application to Navier-Stokes system. Dyn Partial Differ Equ, 2007, 4:227-245
[35]	Zhong J. Harmonic Analysis for Some Schrödinger Type Operators[D]. Princeton University, 1993

ON BOUNDEDNESS PROPERTY OF SINGULAR INTEGRAL OPERATORS ASSOCIATED TO A SCHRÖDINGER OPERATOR IN A GENERALIZED MORREY SPACE AND APPLICATIONS

ON BOUNDEDNESS PROPERTY OF SINGULAR INTEGRAL OPERATORS ASSOCIATED TO A SCHRÖDINGER OPERATOR IN A GENERALIZED MORREY SPACE AND APPLICATIONS

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献 35

相关文章 4

Metrics

本文评价

推荐阅读 1

[1]	王定怀, 周疆, 陈文艺. ANOTHER CHARACTERIZATIONS OF MUCKENHOUPT A_p CLASS[J]. 数学物理学报(英文版), 2017, 37(6): 1761-1774.
[2]	J.J. Betancor, R. Crescimbeni, J.L. Torrea. OSCILLATION AND VARIATION OF THE LAGUERRE HEAT AND POISSON SEMIGROUPS AND RIESZ TRANSFORMS[J]. 数学物理学报(英文版), 2012, 32(3): 907-928.
[3]	李澎涛, 彭立中. L^p BOUNDEDNESS OF COMMUTATOR OPERATOR ASSOCIATED WITH SCHRÖDINGER OPERATORS ON HEISENBERG GROUP[J]. 数学物理学报(英文版), 2012, 32(2): 568-578.
[4]	张震球. HERMITE EXPANSIONS AND FRACTAL MEASURES[J]. 数学物理学报(英文版), 1998, 18(2): 167-173.