Abstract
Let X be a ball Banach function space on \({\mathbb R}^{n}\). Let Ω be a Lipschitz function on the unit sphere of \({\mathbb R}^{n}\), which is homogeneous of degree zero and has mean value zero, and let TΩ be the convolutional singular integral operator with kernel Ω(⋅)/|⋅|n. In this article, under the assumption that the Hardy–Littlewood maximal operator \({\mathscr{M}}\) is bounded on both X and its associated space, the authors prove that the commutator [b, TΩ] is compact on X if and only if \(b\in \text {CMO }({\mathbb R}^{n})\). To achieve this, the authors mainly employ three key tools: some elaborate estimates, given in this article, on the norm of X about the commutators and the characteristic functions of some measurable subsets, which are implied by the assumed boundedness of \({\mathcal M}\) on X and its associated space as well as the geometry of \(\mathbb R^{n}\); the complete John–Nirenberg inequality in X obtained by Y. Sawano et al.; the generalized Fréchet–Kolmogorov theorem on X also established in this article. All these results have a wide range of applications. Particularly, even when \(X:=L^{p(\cdot )}({\mathbb R}^{n})\) (the variable Lebesgue space), \(X:=L^{\vec {p}}({\mathbb R}^{n})\) (the mixed-norm Lebesgue space), \(X:=L^{\Phi }({\mathbb R}^{n})\) (the Orlicz space), and \(X:=(E_{\Phi }^{q})_{t}({\mathbb R}^{n})\) (the Orlicz-slice space or the generalized amalgam space), all these results are new.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adams, D.R.: Morrey Spaces. Birkhäuser/Springer, Cham (2015)
Adams, D.R., Xiao, J.: Nonlinear potential analysis on Morrey spaces and their capacities. Ind. Univ. Math. J. 53, 1629–1663 (2004)
Andersen, K.F., John, R.T.: Weighted inequalities for vecter-valued maximal functions and singular integrals. Studia Math. 69, 19–31 (1980)
Arai, R., Nakai, E.: Commutators of Calderón–Zygmund and generalized fractional integral operators on generalized Morrey spaces. Rev. Mat. Complut. 31, 287–331 (2018)
Arai, R., Nakai, E.: An extension of the characterization of CMO and its application to compact commutators on Morrey spaces. J. Math. Soc. Japan 72, 507–539 (2020)
Astala, K., Iwaniec, T., Koskela, P., Martin, G.: Mappings of BMO-bounded distortion. Math. Ann. 317, 703–726 (2000)
Auscher, P., Mourgoglou, M.: Representation and uniqueness for boundary value elliptic problems via first order systems. Rev. Mat. Iberoam. 35, 241–315 (2019)
Auscher, P., Prisuelos-Arribas, C.: Tent space boundedness via extrapolation. Math. Z. 286, 1575–1604 (2017)
Benedek, A., Panzone, R.: The space Lp, with mixed norm. Duke Math. J. 28, 301–324 (1961)
Bennett, C., Sharpley, R.: Interpolation of Operators. Pure Appl. Math., vol. 129. Academic Press, Boston, MA (1988)
Birnbaum, Z., Orlicz, W.: ÜBer die verallgemeinerung des begriffes der zueinander konjugierten potenzen. Studia Math. 3, 1–67 (1931)
Bokayev, N.A., Burenkov, V.I., Matin, D.T.: On precompactness of a set in general local and global Morrey-type spaces. Eurasian. Math. J. 8, 109–115 (2017)
Chaffee, L., Cruz-Uribe, D.: Necessary conditions for the boundedness of linear and bilinear commutators on Banach function spaces. Math. Inequal. Appl. 21, 1–16 (2018)
Chang, D.-C., Wang, S., Yang, D., Zhang, Y.: Littlewood–Paley characterizations of Hardy-type spaces associated with ball quasi-Banach function spaces. Complex Anal. Oper. Theory 14, Paper No. 40, 1–33 (2020)
Chen, J., Hu, G.: Compact commutators of rough singular integral operators. Canad. Math. Bull. 58, 19–29 (2015)
Chen, Y., Deng, Q., Ding, Y.: Commutators with fractional differentiation for second-order elliptic operators on \(\mathbb {R}^{n}\). Commun. Contemp. Math. 22(1950010), 1–29 (2020)
Chen, Y., Ding, Y.: Lp bounds for the commutators of singular integrals and maximal singular integrals with rough kernels. Trans. Amer. Math. Soc. 367, 1585–1608 (2015)
Chen, Y., Ding, Y., Hong, G.: Commutators with fractional differentiation and new characterizations of BMO-Sobolev spaces. Anal. PDE 9, 1497–1522 (2016)
Chen, Y., Ding, Y., Wang, X.: Compactness of commutators for singular integrals on Morrey spaces. Canad. J. Math. 64, 257–281 (2012)
Cheung, K., Ho, K.-P.: Boundedness of Hardy–Littlewood maximal operator on block spaces with variable exponent. Czechoslovak Math. J. 64(139), 159–171 (2014)
Chiarenza, F., Frasca, M.: Morrey spaces and Hardy–Littlewood maximal function. Rend. Mat. Appl. (7) 7, 273–279 (1987)
Cleanthous, G., Georgiadis, A.G., Nielsen, M.: Discrete decomposition of homogeneous mixed-norm Besov spaces. In: Functional Analysis, Harmonic Analysis, and Image Processing: a Collection of Papers in Honor of Björn Jawerth, vol. 693, pp 167–184. Contemp. Math. Amer. Math. Soc., Providence (2017)
Clop, A., Cruz, V.: Weighted estimates for Beltrami equations. Ann. Acad. Sci. Fenn. Math. 38, 91–113 (2013)
Coifman, R.R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. of Math. (2) 103, 611–635 (1976)
Cruz-Uribe, D.V.: Extrapolation and factorization. arXiv:1706.02620
Cruz-Uribe, D.V., Fiorenza, A.: Variable Lebesgue Space. Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, Heidelberg (2013)
Cruz-Uribe, D.V., Wang, L.A.D.: Variable Hardy spaces. Ind. Univ. Math. J. 63, 447–493 (2014)
Diening, L., Hästö, P., Roudenko, S.: Function spaces of variable smoothness and integrability. J. Funct. Anal. 256, 1731–1768 (2009)
Diening, L., Harjulehto, P., Hästö, P., Ruz̆ic̆ka, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017. Springer, Heidelberg (2011)
Di Fazio, G., Ragusa, M.A.: Commutators and Morrey spaces. Boll. Un. Mat. Ital. A (7)5, 323–332 (1991)
Duoandikoetxea, J.: Fourier Analysis. Graduate Studies in Mathematics, vol. 29. American Mathematical Society, Providence (2001)
Georgiadis, A.G., Johnsen, J., Nielsen, M.: Wavelet transforms for homogeneous mixed-norm Triebel–Lizorkin spaces. Monatsh Math. 183, 587–624 (2017)
Grafakos, L.: Classical Fourier Analysis. Third edition. Graduate Texts in Mathematics, vol. 249. Springer, New York (2014)
Guliyev, V., Omarova, M., Sawano, Y.: Boundedness of intrinsic square functions and their commutators on generalized weighted Orlicz–Morrey spaces. Banach J. Math. Anal. 9, 44–62 (2015)
Guo, X., Hu, G.: On the commutators of singular integral operators with rough convolution kernels. Canad. J. Math. 68, 816–840 (2016)
Guo, W., Lian, J., Wu, H.: The unified theory for the necessity of bounded commutators and applications. J. Geom. Anal. 30, 3995–4035 (2020)
Guo, W., Wu, H., Yang, D.: A revised on the compactness of commutators. Canad. J. Math. https://doi.org/10.4153/S0008414X20000644 (2020)
Guo, W., Zhao, G.: On relatively compact sets in quasi-Banach function spaces. Proc. Amer. Math. Soc. 148, 3359–3373 (2020)
Ho, K.-P.: Atomic decomposition of Hardy–Morrey spaces with variable exponents. Ann. Acad. Sci. Fenn. Math. 40, 31–62 (2015)
Ho, K.-P.: Dilation operators and integral operators on amalgam space (Lp, lq). Ric. Mat. 68, 661–677 (2019)
Holland, F.: Harmonic analysis on amalgams of Lp and lq. J. London Math. Soc. (2) 10, 295–305 (1975)
Hörmander, L.: Estimates for translation invariant operators in Lp spaces. Acta Math. 104, 93–140 (1960)
Huang, L., Chang, D.-C., Yang, D.: Fourier transform of Hardy spaces associated with ball quasi-Banach function spaces. Appl Anal. https://doi.org/10.1142/S0219530521500135 (2021)
Huang, L., Liu, J., Yang, D., Yuan, W.: Atomic and Littlewood–Paley characterizations of anisotropic mixed-norm Hardy spaces and their applications. J. Geom. Anal. 29, 1991–2067 (2019)
Huang, L., Liu, J., Yang, D., Yuan, W.: Dual spaces of anisotropic mixed-norm Hardy spaces. Proc. Amer. Math. Soc. 147, 1201–1215 (2019)
Huang, L., Yang, D.: On function spaces with mixed norms — a survey. J. Math. Study 54, 262–336 (2021)
Iwaniec, T.: Lp-theory of quasiregular mappings. In: Quasiconformal Space Mappings. Lecture Notes in Math. vol. 1508, pp. 39–64. Springer, Berlin (1992)
Izuki, M., Noi, T., Sawano, Y.: The John–Nirenberg inequality in ball Banach function spaces and application to characterization of BMO. J. Inequal. Appl. Paper No. 268, pp. 11 (2019)
Izuki, M., Sawano, Y.: Characterization of BMO via ball Banach function spaces. Vestn. St.-Peterbg. Univ. Mat. Mekh. Astron. 4(62), 78–86 (2017)
Jia, H., Wang, H.: Decomposition of Hardy–Morrey spaces. J. Math. Anal. Appl. 354, 99–110 (2009)
John, F., Nirenberg, L.: On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14, 415–426 (1961)
Karlovich, A., Lerner, A.: Commutators of singular integrals on generalized Lp spaces with variable exponent. Publ. Mat. 49, 111–125 (2005)
Kikuchi, N., Nakai, E., Tomita, N., Yabuta, K., Yoneda, T.: Calderón–Zygmund operators on amalgam spaces and in the discrete case. J. Math. Anal. Appl. 335, 198–212 (2007)
Kováčik, O., Rákosník, J.: On spaces Lp(x) and Wk, p(x). Czechoslovak. Math. J. 41(116), 592–618 (1991)
Krantz, S.G., Li, S.Y.: Boundedness and compactness of integral operators on spaces of homogeneous type and applications. II. J. Math. Anal. Appl. 258, 642–657 (2001)
Lerner, A.K., Ombrosi, S., Rivera-Ríos, I.P.: Commutators of singular integrals revisited. Bull. Lond. Math. Soc. 51, 107–119 (2019)
Lizorkin, P.I.: Multipliers of Fourier integrals and estimates of convolutions in spaces with mixed norm. applications. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 34, 218–247 (1970)
Lu, S., Ding, Y., Yan, D.: Singular Integrals and Related Topics. World Scientific Publishing Co. Pte. Ltd., Hackensack (2007)
Martínez, S., Wolanski, N.: A minimum problem with free boundary in Orlicz spaces. Adv. Math. 218, 1914–1971 (2008)
Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math. Soc. 43, 126–166 (1938)
Nakano, H.: Modulared Semi-Ordered Linear Spaces. Maruzen, Tokyo (1950)
Nakano, H.: Topology of Linear Topological Spaces. Maruzen, Tokyo (1951)
Nakamura, S., Sawano, Y.: The singular integral operator and its commutator on weighted Morrey spaces. Collect. Math. 68, 145–174 (2017)
Nogayama, T.: Mixed Morrey spaces. Positivity 23, 961–1000 (2019)
Nogayama, T., Ono, T., Salim, D., Sawano, Y.: Atomic decomposition for mixed Morrey spaces. J. Geom. Anal. 31, 9338–9365 (2021)
Orlicz, W.: ÜBer eine gewisse Klasse von räumen vom typus B. Bull. Inst. Acad. Pol. Ser. A 8, 207–220 (1932)
Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Monographs and Textbooks in Pure and Applied Mathematics, vol. 146. Marcel Dekker, Inc., New York (1991)
Sawano, Y.: Theory of Besov Spaces. Developments in Mathematics, vol. 56. Springer, Singapore (2018)
Sawano, Y., Shirai, S.: Compact commutators on Morrey spaces with non-doubling measures. Georgian Math. J. 15, 353–376 (2008)
Sawano, Y., Tanaka, H.: The Fatou property of block spaces. J. Math. Sci. Univ. Tokyo 22, 663–683 (2015)
Sawano, Y., Ho, K.-P., Yang, D., Yang, S.: Hardy spaces for ball quasi-Banach function spaces. Dissertationes. Math. (Rozprawy Mat.) 525, 1–102 (2017)
Sawano, Y., Di Fazio, G., Hakim, D.: Morrey Spaces. Introduction and Applications to Integral Operators and PDE’s, vol. I. Chapman and Hall/CRC, New York (2020)
Sawano, Y., Di Fazio, G., Hakim, D.: Morrey Spaces: Introduction and Applications to Integral Operators and PDE’s, vol. II. Chapman and Hall/CRC, New York (2020)
Tao, J., Yang, D.: Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces. Math. Methods Appl. Sci. 42, 1631–1651 (2019)
Tao, J., Yang, Da, Yang, Do: Beurling–Ahlfors commutators on weighted Morrey spaces and applications to Beltrami equations. Potential Anal. 53, 1467–1491 (2020)
Uchiyama, A.: On the compactness of operators of Hankel type. Tôhoku Math. J. (2) 30, 163–171 (1978)
Wang, F., Yang, D., Yang, S.: Applications of Hardy spaces associated with ball quasi-Banach function spaces. Results Math. 75, Paper No. 26, 1–58 (2020)
Wang, S., Yang, D., Yuan, W., Zhang, Y.: Weak Hardy-type spaces associated with ball quasi-Banach function spaces II: Littlewood–Paley characterizations and real interpolation. J. Geom. Anal. 31, 631–696 (2021)
Yan, X., Yang, D., Yuan, W.: Intrinsic square function characterizations of several Hardy-type spaces — a survey. Anal. Theory Appl. (to appear)
Yan, X., Yang, D., Yuan, W.: Intrinsic square function characterizations of Hardy spaces associated with ball quasi-Banach function spaces. Front. Math. China 15, 769–806 (2020)
Yosida, K.: Functional Analysis. Classics in Mathematics. Springer, Berlin (1995)
Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Springer, Berlin (2010)
Zhang, Y., Yang, D., Yuan, W., Wang, S.: Real-variable characterizations of Orlicz-slice Hardy spaces. Anal. Appl. (Singap.) 17, 597–664 (2019)
Zhang, Y., Huang, L., Yang, D., Yuan, W.: New ball Campanato-type function spaces and their applications. J. Geom. Anal. (to appear)
Zhang, Y., Wang, S., Yang, D., Yuan, W.: Weak Hardy-type spaces associated with ball quasi-Banach function spaces I: Decompositions with applications to boundedness of Calderón–Zygmund operators. Sci. China Math. 64, 2007–2064 (2021)
Acknowledgements
The authors would like to thank Professors Huoxiong Wu and Qingying Xue for some useful discussions on the subject of this article. The authors would also like to thank both two referees and the associate editor for their carefully reading and several motivating remarks which indeed improve the quality of this article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This project is partially supported by the National Natural Science Foundation of China (Grant Nos. 11971058, 12071197, 12122102 and 11871100) and the National Key Research and Development Program of China (Grant No. 2020YFA0712900).
Rights and permissions
About this article
Cite this article
Tao, J., Yang, D., Yuan, W. et al. Compactness Characterizations of Commutators on Ball Banach Function Spaces. Potential Anal 58, 645–679 (2023). https://doi.org/10.1007/s11118-021-09953-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-021-09953-w
Keywords
- Ball Banach function space
- Commutator
- Convolutional singular integral operator
- BMO
- CMO
- Extrapolation
- Fréchet–Kolmogorov theorem