Hostname: page-component-65b85459fc-5fwlf Total loading time: 0 Render date: 2025-10-16T11:15:50.810Z Has data issue: false hasContentIssue false
  • English
  • Français

Lorentz boundedness and compactness of Riesz transform commutator on stratified Lie groups

Part of: Abstract harmonic analysis Harmonic analysis in several variables

Published online by Cambridge University Press:  15 September 2025

You-Wei Chen ,
Steven Krantz Open the ORCID record for Steven Krantz [Opens in a new window] ,
Ji Li Open the ORCID record for Ji Li [Opens in a new window] ,
Chun-Yen Shen  and
Daniel Spector
You-Wei Chen
Affiliation:
Department of Mathematics, National Changhua University of Education , Changhua City 500, Taiwan e-mail: bensonchen@cc.ncue.edu.tw
Steven Krantz
Affiliation:
Department of Mathematics, Washington University in St. Louis , St. Louis, MO 63130, United States e-mail: sk@math.wustl.edu
Ji Li
Affiliation:
Department of Mathematics, Macquarie University , Sydney, NSW 2109, Australia e-mail: ji.li@mq.edu.au
Chun-Yen Shen*
Affiliation:
Department of Mathematics, National Taiwan University , Taipei 10617, Taiwan
Daniel Spector
Affiliation:
Department of Mathematics, National Taiwan Normal University , Taipei 106308, Taiwan e-mail: spectda@gapps.ntnu.edu.tw
*
e-mail: cyshen@math.ntu.edu.tw
Get access Rights & Permissions [Opens in a new window]

Abstract

The $L^p$ boundedness of the commutator $[b, T]$ has been intensively studied in recent decades in part because it has important connections and applications to partial differential equations. Inspired by these works, we study the boundedness and compactness of the Riesz transform commutator in a general setting, namely, in the scale of Lorentz spaces and on stratified Lie groups. In this article, we provide a complete characterization between the space of the symbol b and the Lorentz estimates of $[b, R_j]$.

Keywords

Riesz transformcommutatorcompactness

MSC classification

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc.

Information

Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 28
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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.)

Article purchase

Temporarily unavailable

Footnotes

J.L. was supported by ARC DP Grant No. 220100285. C.-Y.S. was supported by NSTC through Grant 111-2115-M-002-010-MY5.

References

Bergh, J. and Löfström, J., Interpolation spaces: An introduction, Grundlehren der Mathematischen Wissenschaften, 223, Springer-Verlag, Berlin, 1976.Google Scholar
Bonfiglioli, A., Lanconelli, E., and Uguzzoni, F., Stratified lie groups and potential theory for their sub-Laplacians, Springer Monographs in Mathematics, Springer, Berlin Heidelberg, New York, 2007.Google Scholar
Calderón, A.-P., Commutators of singular integral operators . Proc. Natl. Acad. Sci. USA 53(1965), 1092–1099.Google Scholar
Castillo, R. E., Fernández, J. C. R., and Trousselot, E., Functions of bounded $\left(\varphi, p\right)$ mean oscillation. Proyecciones 27(2008), 163–177.Google Scholar
Chen, P., Duong, X., Li, J., and Wu, Q., Compactness of Riesz transform commutator on stratified lie groups . J. Funct. Anal. 277(2019), no. 6, 1639–1676.Google Scholar
Christ, M., A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 60–61(1990), no. 2, 601–628. https://doi.org/10.4064/cm-60-61-2-601-628.Google Scholar
Christ, M. and Geller, D., Singular integral characterisations of hardy spaces on homogeneous groups . Duke Math. 51(1984), 547–598.Google Scholar
Coifman, R., Lions, P. L., Meyer, Y., and Semmes, S., Compensated compactness and Hardy spaces . J. Math. Pures Appl. 72(1993), 247–286.Google Scholar
Coifman, R., Rochberg, R., and Weiss, G., Factorization theorems for Hardy spaces in several variables . Ann. Math. 103(1976), 611–635.Google Scholar
Dafni, G., Nonhomogeneous div-curl lemmas and local Hardy spaces . Adv. Differ. Equ. 10(2005), 505–526.Google Scholar
Dao, N. and Krantz, S., Lorentz boundedness and compactness characterization of integral commutators on spaces of homogeneous type . Nonlinear Anal. 203(2021), Article no. 112162, 22 pp.Google Scholar
Duong, X., Li, H.-Q., Li, J., and Wick, B. D., Lower bound of Riesz transform kernels and commutator theorems on stratified nilpotent lie groups . J. Math. Pures Appl. (9) 124(2019), 273–299. (English, with English and French summaries).Google Scholar
Duong, X. T., Li, J., Wick, B. D., and Yang, D. Y., Factorization for Hardy spaces and characterization for BMO spaces via commutators in the Bessel setting . Indiana Univ. Math. J. 66(2017), no. 4, 1081–1106.Google Scholar
Folland, G. B. and Stein, E. M., Hardy spaces on homogeneous groups. Princeton Mathematical Series, 28, Princeton University Press, Princeton, NJ, 1982.Google Scholar
Franchi, B., Tchou, N., and Tesi, M. C., Div-curl type theorem, H-convergence and stokes formula in the Heisenberg group . Commun. Contemp. Math. 8(2006), 67–99.Google Scholar
Górka, P. and Macios, A., The Riesz-Kolmogorov theorem on metric spaces . Miskolc Math. Notes. 15(2014), no. 2, 459–465.Google Scholar
Grafakos, L., Modern Fourier analysis. 2nd ed., Graduate Texts in Mathematics, 250, Springer, New York, NY, 2009.Google Scholar
Grafakos, L., Classical Fourier analysis. 3rd ed., Graduate Texts in Mathematics, 249, Springer, New York, NY, 2014.Google Scholar
Hebisch, W. and Sikora, A., A smooth subadditive homogeneous norm on a homogeneous group . Stud. Math. 96(1990), 231–236.Google Scholar
Holmes, I., Lacey, M., and Wick, B. D., Commutators in the two-weight setting . Math. Ann. 367(2017), 51–80.Google Scholar
Hu, G., Yang, D., and Yang, D., Boundedness of maximal singular integral operators on spaces of homogeneous type and its applications . J. Math. Soc. Japan 59(2007), no. 2, 323–349.Google Scholar
Lacey, M., Lectures on Nehari’s theorem on the polydisk, topics in harmonic analysis and ergodic theory . Contemp. Math. 444(2007), 185–213.Google Scholar
Lacey, M. and Li, J., Compactness of commutator of Riesz transforms in the two weight setting . J. Math. Anal. Appl. 508(2022), no. 1, Article no. 125869, 11 pp.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.Google Scholar
Nehari, Z., On bounded bilinear forms . Ann. Math. 65(1957), 153–162.Google Scholar
Royden, H. L. and Fitzpatrick, P. M.. Real analysis. 4th ed., Pearson Modern Classics for Advanced Mathematics Series, 2010.Google Scholar
Uchiyama, A., The factorization of ${H}^p$ on the space of homogeneous type. Pac. J. Math. 92(1981), 453–468.Google Scholar
Varopoulos, N. T., Saloff-Coste, L., and Coulhon, T., Analysis and geometry on groups, Cambridge Tracts in Mathematics, 100, Cambridge University Press, Cambridge, 1992.Google Scholar
Wheeden, R. L. and Zygmund, A., Measure and integral. 2nd ed., Pure and Applied Mathematics, CRC Press, Boca Raton, FL, 2015. An introduction to real analysis.Google Scholar