Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-05-23T22:04:38.708Z Has data issue: false hasContentIssue false
  • English
  • Français

A revisit on the compactness of commutators

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  20 August 2020

Weichao Guo ,
Huoxiong Wu  and
Dongyong Yang
Weichao Guo*
Affiliation:
School of Science, Jimei University, Xiamen, 361021, P.R.China
Huoxiong Wu
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen, 361005, P.R. China e-mail: huoxwu@xmu.edu.cndyyang@xmu.edu.cn
Dongyong Yang
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen, 361005, P.R. China e-mail: huoxwu@xmu.edu.cndyyang@xmu.edu.cn
*
e-mail: weichaoguomath@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

A new characterization of $\text {CMO}(\mathbb R^n)$ is established replying upon local mean oscillations. Some characterizations of iterated compact commutators on weighted Lebesgue spaces are given, which are new even in the unweighted setting for the first order commutators.

Keywords

Compactnesscommutatorsingular integralfractional integralCMO ℝn

MSC classification

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 42B25: Maximal functions, Littlewood-Paley theory
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 6 , December 2021 , pp. 1667 - 1697
Check for updates
Copyright
© Canadian Mathematical Society 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Supported by the NSF of China (Nos.11771358, 11871101, 11701112, 11671414, 11971402, and 11871254).

References

Bényi, A., Martell, J. M., Moen, K., Stachura, E., and Torres, R. H., Boundedness results for commutators with BMO functions via weighted estimates: a comprehensive approach . Math. Ann. 376(2020), nos. 1–2, 61–102.CrossRefGoogle Scholar
Chaffee, L. and Torres, R. H., Characterization of compactness of the commutators of bilinear fractional integral operators . Potential Anal. 43(2015), no. 3, 481–494.CrossRefGoogle Scholar
Chen, J. and Hu, G., Compact commutators of rough singular integral operators . Canad. Math. Bull. 58(2015), no. 1, 19–29.CrossRefGoogle Scholar
Chen, Y., Ding, Y., and Wang, X., Compactness of commutators of Riesz potential on Morrey spaces . Potential Anal. 30(2009), no. 4, 301–313.CrossRefGoogle Scholar
Clop, A. and Cruz, V., Weighted estimates for Beltrami equations . Ann. Acad. Sci. Fenn. Math. 38(2013), no. 1, 91–113.CrossRefGoogle Scholar
Coifman, R. R., Rochberg, R., and Weiss, G., Factorization theorems for Hardy spaces in several variables . Ann. Math. 103(1976), no. 3, 611–635.CrossRefGoogle Scholar
Evans, L. C. and Gariepy, R. F., Measure theory and fine properties of functions . Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.Google Scholar
Grafakos, L., Classical Fourier analysis . 2nd ed., Graduate Texts in Mathematics, 249, Springer, New York, NY, 2008.Google Scholar
Holmes, I., Rahm, R., and Spencer, S., Commutators with fractional integral operators . Stud. Math. 233(2016), no. 3, 279–291.Google Scholar
Hytönen, T. P. and Pérez, C., Sharp weighted bounds involving $ A_\infty $ . Anal. PDE 6(2013), no. 4, 777–818.CrossRefGoogle Scholar
Hytönen, T. P., Roncal, L., and Tapiola, O., Quantitative weighted estimates for rough homogeneous singular integrals . Israel J. Math. 218(2017), no. 1, 133–164.CrossRefGoogle Scholar
Iwaniec, T., LP-theory of quasiregular mappings . In: Vuorinen, M. (ed.), Quasiconformal space mappings, Lecture Notes in Mathematics, 1508, Springer, Berlin, Germany, 1992, pp. 39–64.Google Scholar
John, F., Quasi-isometric mappings. In: Seminari 1962/63 Analist Algebra Geometria. e Topologia. Vol. 2, Ist. Naz. Alta Mat Ediz, Cremonese, Rome, Italy, 1965, pp. 462–473.Google Scholar
Krantz, S. G. and Li, S.-Y., Boundedness and compactness of integral operators on spaces of homogeneous type and applications. II . J. Math. Anal. Appl. 258(2001), no. 2, 642–657.CrossRefGoogle Scholar
Kunwar, I. and Ou, Y., Two-weight inequalities for multilinear commutators . N.Y. J. Undergrad. Math. 24(2018), 980–1003.Google Scholar
Lacey, M. T., An elementary proof of the A2 bound . Israel J. Math. 217(2017), no. 1, 181–195.CrossRefGoogle Scholar
Lerner, A. K., A “local mean oscillation” decomposition and some of its applications . In: M. Lacey and A. K. Lerner (eds.), Function spaces, approximation, inequalities and lineability, Lectures of the Spring School in Analysis, Prague, Czech Republic, 2011, pp. 71–106.Google Scholar
Lerner, A. K., Ombrosi, S., and Rivera-Ríos, I. P., Commutators of singular integrals revisited . Bull. Lond. Math. Soc. 51(2019), no. 1, 107–119.CrossRefGoogle Scholar
Muckenhoupt, B., Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165(1972), 207–226.CrossRefGoogle Scholar
Muckenhoupt, B. and Wheeden, R., Weighted norm inequalities for fractional integrals . Trans. Am. Math. Soc. 192(1974), 261–274.CrossRefGoogle Scholar
Strömberg, J.-O., Bounded mean oscillation with Orlicz norms and duality of Hardy spaces . Indiana Univ. Math. J. 28(1979), no. 3, 511–544.CrossRefGoogle Scholar
Tao, J., Yang, D., and Yang, D., Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces . Math. Methods Appl. Sci. 42(2019), no. 5, 1631–1651.CrossRefGoogle Scholar
Taylor, M. E., Partial differential equations II . In: S. S. Antman, J. E. Marsden, and L. Sirovich (eds.), Qualitative studies of linear equations, 2nd ed., Applied Mathematical Sciences, 116. Springer, New York, NY, 2011.Google Scholar
Uchiyama, A., On the compactness of operators of Hankel type . Tôhoku. Math. J. 30(1978), no. 1, 163–171.CrossRefGoogle Scholar