BOUNDEDNESS AND COMPACTNESS FOR THE COMMUTATOR OF THE ω-TYPE CALDERÓN-ZYGMUND OPERATOR ON LORENTZ SPACE∗

doi:10.1007/s10473-023-0409-8

Acta mathematica scientia,Series B ›› 2023, Vol. 43 ›› Issue (4): 1587-1602.doi: 10.1007/s10473-023-0409-8

Previous Articles Next Articles

BOUNDEDNESS AND COMPACTNESS FOR THE COMMUTATOR OF THE ω-TYPE CALDERÓN-ZYGMUND OPERATOR ON LORENTZ SPACE^∗

Xiangxing TAO¹, Yuan ZENG^1,†, Xiao YU²

1. Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou 310023, China;
2. Department of Mathematics, Shangrao Normal University, Shangrao 334001, China

Received:2022-01-24 Published:2023-08-08
Contact: †Xiangxing TAO, E-mail: zy1347256531@163.com
About author:Xiangxing TAO, E-mail: xxtau@163.com; Xiao YU, E-mail: yx2000s@163.com
Supported by:
*This research was supported by the NNSF of China (12271483, 11961056) and the NSF of Jiangxi Province (20192BAB201004). Zeng's research supported by the “Xin-Miao” Program of Zhejiang Province (2021R415027) and the Innovation Fund of ZUST (2020yjskc06).

Abstract

Abstract: In this paper, the authors consider the $\omega$-type Calderón-Zygmund operator $T_{\omega}$ and the commutator $[b,T_{\omega}]$ generated by a symbol function $b$ on the Lorentz space $L^{p,r}(X)$ over the homogeneous space $(X,d,\mu)$. The boundedness and the compactness of the commutator $[b,T_{\omega}]$ on Lorentz space $L^{p,r}(X)$ are founded for any $p\in (1, \infty)$ and $r\in [1, \infty)$.

Key words: $\omega$-type Calderón-Zygmund operator, commutators, Lorentz space, homogeneous space

Xiangxing TAO, Yuan ZENG, Xiao YU. BOUNDEDNESS AND COMPACTNESS FOR THE COMMUTATOR OF THE ω-TYPE CALDERÓN-ZYGMUND OPERATOR ON LORENTZ SPACE^∗[J].Acta mathematica scientia,Series B, 2023, 43(4): 1587-1602.

Trendmd

References

[1] Calderón A P. Commutators of singular integral operators. Proc Natl Acad Sci USA,1965, 53: 1092-1099
[2] Pérez C. Endpoint estimates for commutators of singular integral operators. J Funct Anal,1995, 128(1): 163-185
[3] Yabuta K. Generalization of Calderón-Zygmund operators. Studia Math,1985, 82(1): 17-31
[4] Stein E M.Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton: Princeton Univ Press, 1993
[5] García-Cuerva J, De Francia J L. Weighted Norm Inequalities and Related Topics. Amsterdam: North Holland, 1985
[6] Journé J L. Calderón-Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral of Calderón. Berlin: Springer, 1983
[7] Bramanti M, Cerutti M C. Commutators of singular integrals on homogeneous spaces. Bollettino della Unione Matematica Italiana-B, 1996, 10(4): 843-884
[8] Macías R, Segovia C. Lipschitz functions on spaces of homogeneous type. Adv Math,1979, 33(3): 257-270
[9] Dao N A, Krantz S G. Lorentz boundedness and compactness characterization of integral commutators on spaces of homogeneous type. Nonlinear Anal, 2021, 203: 112162
[10] Stempak K, Tao X X. Local Morrey and Campanato spaces on quasimetric measure spaces. J Funct Spaces, 2014, Art: 172486
[11] Duoandikoetxea J. Fourier Analysis.Providence, RI: Amer Math Soc, 2001
[12] Coifman R, Rochberg R, Weiss G. Factorization theorems for Hardy spaces in several variables. Ann of Math, 1976, 103: 611-635
[13] Uchiyama A. On the compactness of operators of Hankel type. Tohoku Math J, 1978, 30(1): 163-171
[14] Janson S. Mean oscillation and commutators of singular integral operators. Ark Mat, 1978, 16(2): 263-270
[15] Krantz S, Li S Y. Boundedness and compactness of integral operators on spaces of homogeneous type and applications, II. J Math Anal Appl, 2001, 258(2): 642-657
[16] Grafakos L. Classical Fourier Analysis. New York: Springer, 2014
[17] John F, Nirenberg L. On functions of bounded mean oscillation. Comm Pure Appl Math, 1961, 14: 415-426
[18] Quek T S, Yang D C. Calderón-Zgymund-type operators on weighted weak Hardy spaces over $R^n$. Acta Math Sin,2000, 16(1): 141-160
[19] Cwikel M. The dual of weak $L^p$. Ann Inst Fourier (Grenoble), 1975, 25(2): 81-126
[20] Torchinsky A.Real-Variable Methods in Harmonic Analysis. San Diego: Academic Press, 1986
[21] Brudnyi Y. Compactness criteria for spaces of measurable functions. St Petersburg Math J, 2015, 26(1): 49-68
[22] Clop A, Cruz V. Weighted estimates for Beltrami equations. Ann Acad Sci Fenn Math, 2013, 38(1): 91-113

BOUNDEDNESS AND COMPACTNESS FOR THE COMMUTATOR OF THE ω-TYPE CALDERÓN-ZYGMUND OPERATOR ON LORENTZ SPACE^∗

Abstract

Cite this article

share this article

References

Related Articles 12

Metrics

Comments

Recommended 10

[1]	Qianjun HE, Xinfeng WU, Dunyan YAN. BOUNDS FOR MULTILINEAR OPERATORS UNDER AN INTEGRAL TYPE CONDITION ON MORREY SPACES [J]. Acta mathematica scientia,Series B, 2022, 42(3): 1191-1208.
[2]	Yunan CUI, Paweƚ FORALEWSKI, Joanna KOŃCZAK. ORLICZ-LORENTZ SEQUENCE SPACES EQUIPPED WITH THE ORLICZ NORM [J]. Acta mathematica scientia,Series B, 2022, 42(2): 623-652.
[3]	Kwok-Pun HO. A GENERALIZATION OF BOYD'S INTERPOLATION THEOREM [J]. Acta mathematica scientia,Series B, 2021, 41(4): 1263-1274.
[4]	Jun LIU, Dachun YANG, Wen YUAN. LITTLEWOOD-PALEY CHARACTERIZATIONS OF ANISOTROPIC HARDY-LORENTZ SPACES [J]. Acta mathematica scientia,Series B, 2018, 38(1): 1-33.
[5]	SHAO Jing-Jing, HAN Ya-Zhou. SZEGÖ\|TYPE FACTORIZATION THEOREM FOR NONCOMMUTATIVE HARDY-LORENTZ SPACES [J]. Acta mathematica scientia,Series B, 2013, 33(6): 1675-1684.
[6]	REN Yan-Bo, GUO Tie-Xin. WEAK ATOMIC DECOMPOSITIONS OF MARTINGALE HARDY-LORENTZ SPACES AND APPLICATIONS [J]. Acta mathematica scientia,Series B, 2012, 32(3): 1157-1166.
[7]	HAN Ya-Zhou, Turdebek N. Bekjan. THE DUAL OF NONCOMMUTATIVE LORENTZ SPACES [J]. Acta mathematica scientia,Series B, 2011, 31(5): 2067-2080.
[8]	Ayse Sandikci A. Turan G¨urkanli. GABOR ANALYSIS OF THE SPACES {\boldmath M( p, q, w) ({R^d) AND S( p, q, r, w, ω) (R^d) [J]. Acta mathematica scientia,Series B, 2011, 31(1): 141-158.
[9]	FAN Li-Ping, JIAO Yong, LIU Pei-De. LORENTZ MARTINGALE SPACES AND INTERPOLATION [J]. Acta mathematica scientia,Series B, 2010, 30(4): 1143-1153.
[10]	CHEN Li-Gong, LIU Pei-De. THE DYADIC DERIVATIVE AND CESÀRO MEAN OF BANACH-VALUED MARTINGALES [J]. Acta mathematica scientia,Series B, 2009, 29(2): 265-275.
[11]	Hakan Avci; A. Turan Gurkanli. MULTIPLIERS AND TENSOR PRODUCTS OF L(p, q) LORENTZ SPACES [J]. Acta mathematica scientia,Series B, 2007, 27(1): 107-116.
[12]	Cenap Duyar A. Turan G¨urkanli. MULTIPLIERS AND RELATIVE COMPLETION IN WEIGHTED LORENTZ SPACES [J]. Acta mathematica scientia,Series B, 2003, 23(4): 467-.