BOUNDEDNESS OF VARIATION OPERATORS ASSOCIATED WITH THE HEAT SEMIGROUP GENERATED BY HIGH ORDER SCHRÖDINGER TYPE OPERATORS

doi:10.1007/s10473-020-0504-z

数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (5): 1215-1228.doi: 10.1007/s10473-020-0504-z

BOUNDEDNESS OF VARIATION OPERATORS ASSOCIATED WITH THE HEAT SEMIGROUP GENERATED BY HIGH ORDER SCHRÖDINGER TYPE OPERATORS

刘素英¹, 张超²

1. Department of Applied Mathematics, Northwest Polytechnical University, Xi'an 710072, China;
2. School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China

收稿日期:2019-09-19 修回日期:2020-01-15 出版日期:2020-10-25 发布日期:2020-11-04
通讯作者: Chao ZHANG E-mail:zaoyangzhangchao@163.com
作者简介:Suying LIU,E-mail:suyingliu@nwpu.edu.cn
基金资助:
The first author was supported by the National Natural Science Foundation of China (11701453) and Fundamental Research Funds for the Central Universities (31020180QD05). The second author was supported by the National Natural Science Foundation of China (11971431, 11401525), the Natural Science Foundation of Zhejiang Province (LY18A010006), and the first Class Discipline of Zhejiang-A (Zhejiang Gongshang University-Statistics).

BOUNDEDNESS OF VARIATION OPERATORS ASSOCIATED WITH THE HEAT SEMIGROUP GENERATED BY HIGH ORDER SCHRÖDINGER TYPE OPERATORS

Suying LIU¹, Chao ZHANG²

1. Department of Applied Mathematics, Northwest Polytechnical University, Xi'an 710072, China;
2. School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China

Received:2019-09-19 Revised:2020-01-15 Online:2020-10-25 Published:2020-11-04
Contact: Chao ZHANG E-mail:zaoyangzhangchao@163.com
Supported by:
The first author was supported by the National Natural Science Foundation of China (11701453) and Fundamental Research Funds for the Central Universities (31020180QD05). The second author was supported by the National Natural Science Foundation of China (11971431, 11401525), the Natural Science Foundation of Zhejiang Province (LY18A010006), and the first Class Discipline of Zhejiang-A (Zhejiang Gongshang University-Statistics).

摘要/Abstract

摘要： In this article, we derive the $L^p$-boundedness of the variation operators associated with the heat semigroup which is generated by the high order Schrödinger type operator $(-\Delta)^2+V^2$ in $\mathbb R^n(n\ge 5)$ with $V$ being a nonnegative potential satisfying the reverse Hölder inequality. Furthermore, we prove the boundedness of the variation operators on associated Morrey spaces. In the proof of the main results, we always make use of the variation inequalities associated with the heat semigroup generated by the biharmonic operator $(-\Delta)^2.$

关键词: Variation operators, high order Schrödinger type operators, heat semigroup

Abstract: In this article, we derive the $L^p$-boundedness of the variation operators associated with the heat semigroup which is generated by the high order Schrödinger type operator $(-\Delta)^2+V^2$ in $\mathbb R^n(n\ge 5)$ with $V$ being a nonnegative potential satisfying the reverse Hölder inequality. Furthermore, we prove the boundedness of the variation operators on associated Morrey spaces. In the proof of the main results, we always make use of the variation inequalities associated with the heat semigroup generated by the biharmonic operator $(-\Delta)^2.$

Key words: Variation operators, high order Schrödinger type operators, heat semigroup

中图分类号:

42B35

刘素英, 张超. BOUNDEDNESS OF VARIATION OPERATORS ASSOCIATED WITH THE HEAT SEMIGROUP GENERATED BY HIGH ORDER SCHRÖDINGER TYPE OPERATORS[J]. 数学物理学报(英文版), 2020, 40(5): 1215-1228.

Suying LIU, Chao ZHANG. BOUNDEDNESS OF VARIATION OPERATORS ASSOCIATED WITH THE HEAT SEMIGROUP GENERATED BY HIGH ORDER SCHRÖDINGER TYPE OPERATORS[J]. Acta mathematica scientia,Series B, 2020, 40(5): 1215-1228.

参考文献 29

[1]	Barbatis G, Davies E. Sharp bounds on heat kernels of higher order uniformly elliptic operators. J Operator Theory, 1996, 36:179-198
[2]	Betancor J J, Fariña J C, Harboure E, et al. Lp-boundedness properties of variation operators in the Schrödinger setting. Rev Mat Complut, 2013, 26:485-534
[3]	Bourgain J. Pointwise ergodic theorems for arithmetic sets. Publ Math IHES, 1989, 69:5-41
[4]	Bui T A. Boundedness of variation operators and oscillation operators for certain semigroups. Nonlinear Anal, 2014, 106:124-137
[5]	Campbell J T, Jones R L, Reinhold K, et al. Oscillation and variation for the Hilbert trans-form. Duke Math J, 2000, 105:59-83
[6]	Campbell J T, Jones R L, Reinhold K, et al. Oscillation and variation for singular integrals in higher dimensions. Trans Amer Math Soc, 2003, 355(5):2115-2137
[7]	Cao J, Liu Y, Yang D. Hardy spaces $H^1_{\mathcal{L}}(\mathbb{R}^n)$ associated to Schrödinger type opertors (-△)2 +V2. Houston Journal of Mathematics, 2010, 36(4):1067-1095
[8]	Dziubánski J, Zienkiewicz J. Hp spaces associated with Schrödinger operators with potentials from reverse Hölder classes. Colloquium Mathematicum, 2003, 98:5-37
[9]	Gillespie A T, Torrea J L. Dimension free estimates for the oscillation of Riesz transforms. Israel J Math, 2004, 141:125-144
[10]	Huang Q, Zhang C. Characterization of temperatures associated to Schrödinger operators with initial data in Morrey spaces. Taiwanese J Math, 2019, 23(5):1133-1151
[11]	Jones R L, Reinhold K. Oscillation and variation inequalities for convolution powers. Ergodic Theory Dynam Systems, 2001, 21:1809-1829
[12]	Jones R L, Seeger A, Wright J. Strong variational and jump inequalities in harmonic analysis. Trans Amer Math Soc, 2008, 360:6711-6742
[13]	Jones R L, Wang G. Variation inequalities for the Fejér and Poisson kernels. Trans Amer Math Soc, 2004, 356:4493-4518
[14]	Koch H, Lamm T. Geometric flows with rough initial data. Asian J Math, 2012, 16:209-236
[15]	Le Merdy C, Xu Q. Strong q-variation inequalities for analytic semigroups. Ann Inst Fourier (Grenoble), 2012, 62:2069-2097
[16]	Lépingle D. La variation d'ordre pdes semi-martingales. Z Wahrscheinlichkeitstheor Verw Geb, 1976, 36:295-316
[17]	Liu Y, Dong J. Some estimates of higher order Riesz transform related to Schrödinger operator. Potential Anal, 2010, 32:41-55
[18]	Liu Y, Zhang J, Sheng J, et al. Some estimates for commutators of Riesz transform associated with Schrödinger type operators. Czechoslovak Math J, 2016, 66(141):169-191
[19]	Ma T, Torrea J L, Xu Q. Weighted variation inequalities for differential operators and singular integrals. J Funct Anal, 2015, 268:376-416
[20]	Morrey C B. On the solutions of quasi-linear elliptic partial differential equations. Trans Amer Math Soc, 1938, 43:126-166
[21]	Shen Z. Lp estimates for Schrödinger operators with certain potentials. Ann Inst Fourier (Grenoble), 1995, 45:513-546
[22]	Stein E M. Harmonic Analysis:Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Monographs in Harmonic Analysis, Ⅲ. Princeton:Princeton Univ Press, 1993
[23]	Song L, Tian X, Yan L. On characterization of Poisson integrals of Schrödinger operators with Morrey traces. Acta Math Sin (Engl Ser), 2018, 34:787-800
[24]	Stein E M, Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. Princeton:Princeton Univ Press, 1970
[25]	Sugano S. Lp estimates for some Schrödinger type operators and a Calderón-Zygmund operator of Schrödinger type. Tokyo J Math, 2007, 30:179-197
[26]	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
[27]	Yuan W, Sickel W, Yang D. Morrey and Campanato meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, 2005. Berlin:Springer-Verlag, 2010
[28]	Zhang J, Wu H. Variation inequalities related to Schrödinger operators om Morrey spaces. Chin Ann Math Ser B, 2018, 39(6):973-988
[29]	Zhong J. Harmonic Analysis for Some Schröinger Type Operators[D]. Princeton University, 1993

BOUNDEDNESS OF VARIATION OPERATORS ASSOCIATED WITH THE HEAT SEMIGROUP GENERATED BY HIGH ORDER SCHRÖDINGER TYPE OPERATORS

BOUNDEDNESS OF VARIATION OPERATORS ASSOCIATED WITH THE HEAT SEMIGROUP GENERATED BY HIGH ORDER SCHRÖDINGER TYPE OPERATORS

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献 29

相关文章 15

Metrics

本文评价

推荐阅读 1

[1]	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.
[2]	丁卫, 朱月萍. MIXED HARDY SPACES AND THEIR APPLICATIONS[J]. 数学物理学报(英文版), 2020, 40(4): 945-969.
[3]	李金霞, 李宝德, 何建勋. THE BOUNDEDNESS FOR COMMUTATORS OF ANISOTROPIC CALDERÓN-ZYGMUND OPERATORS[J]. 数学物理学报(英文版), 2020, 40(1): 45-58.
[4]	Azzeddine EL BARAKA, Mohamed TOUMLILIN. WELL-POSEDNESS AND STABILITY FOR THE GENERALIZED INCOMPRESSIBLE MAGNETO-HYDRODYNAMIC EQUATIONS IN CRITICAL FOURIER-BESOV-MORREY SPACES[J]. 数学物理学报(英文版), 2019, 39(6): 1551-1567.
[5]	Ha Duy HUNG, Luong Dang KY. NEW WEIGHTED MULTILINEAR OPERATORS AND COMMUTATORS OF HARDY-CESÁRO TYPE[J]. 数学物理学报(英文版), 2015, 35(6): 1411-1425.
[6]	A. Turan GüRKANLI, Ismail AYDIN. ON THE WEIGHTED VARIABLE EXPONENT AMALGAM SPACE W(L^p^(x), L^q_m)[J]. 数学物理学报(英文版), 2014, 34(4): 1098-1110.
[7]	Vagif GULIYEV, Ali AKBULUT, Yagub MAMMADOV. BOUNDEDNESS OF FRACTIONAL MAXIMAL OPERATOR AND THEIR HIGHER ORDER COMMUTATORS IN GENERALIZED MORREY SPACES ON CARNOT GROUPS[J]. 数学物理学报(英文版), 2013, 33(5): 1329-1346.
[8]	廖建全, 李兴民. AN IMPROVEMENT OF THE OCTONIONIC TAYLOR TYPE THEOREM[J]. 数学物理学报(英文版), 2011, 31(2): 561-566.
[9]	杨占英, 杨奇祥. A NOTE ON CONVOLUTION-TYPE CALDERÓN-ZYGMUND OPERATORS[J]. 数学物理学报(英文版), 2009, 29(5): 1341-1350.
[10]	周伟军; 马柏林; 徐景实. THE BOUNDEDNESS OF MULTILINEAR COMMUTATORS ON WEIGHTED SPACES#br# AND HERZ-TYPE SPACES[J]. 数学物理学报(英文版), 2007, 27(2): 361-372.
[11]	高洁欣, 钱涛. SHANNON SAMPLING AND ESTIMATION OF BAND-LIMITED FUNCTIONS IN THE SEVERAL COMPLEX VARIABLES SETTING[J]. 数学物理学报(英文版), 2005, 25(4): 741-754.
[12]	陆善镇, 孟岩, 吴强. LIPSCHITZ ESTIMATES FOR MULTILINEAR SINGULAR INTEGRALS, II[J]. 数学物理学报(英文版), 2004, 24(2): 291-300.
[13]	XU JINGSHI. A DISCRETE CHARACTERIZATION OF HERZ-TYPE TRIEBEL-LIZORKIN SPACES AND ITS APPLICATIONS[J]. 数学物理学报(英文版), 2004, 24(3): 412-420.
[14]	徐景实, 杨大春. APPLICATIONS OF HERZ-TYPE TRIEBEL-LIZORKI SPACES[J]. 数学物理学报(英文版), 2003, 23(3): 328-.
[15]	江寅生, 唐林, 杨大春. CONTINUITY FOR MAXIMAL COMMUTATORS OF BOCHNER-RIESZ OPERATORS WITH BMO FUNCTIONS[J]. 数学物理学报(英文版), 2001, 21(3): 339-349.