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 $r\in [1, \infty)$.
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
.
DOI: 10.1007/s10473-023-0409-8
[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
