Abstract
Let \(\overrightarrow{b}=(b_{1},b_{2},\ldots,b_{m})\) be a collection of locally integrable functions and \(T_{\Sigma\overrightarrow{b}}\) the commutator of multilinear singular integral operator T. Denote by \(\mathbb{L}(\delta)\) and \(\mathbb{L}(\delta(\cdot))\) the Lipschitz spaces and the variable Lipschitz spaces, respectively. The main purpose of this paper is to establish some new characterizations of the (variable) Lipschitz spaces in terms of the boundedness of multilinear commutator \(T_{\Sigma\overrightarrow{b}}\) in the context of the variable exponent Lebesgue spaces, that is, the authors give the necessary and sufficient conditions for bj (j = 1, 2, …, m) to be \(\mathbb{L}(\delta)\) or \(\mathbb{L}(\delta(\cdot))\) via the boundedness of multilinear commutator from products of variable exponent Lebesgue spaces to variable exponent Lebesgue spaces. The authors do so by applying the Fourier series technique and some pointwise estimate for the commutators. The key tool in obtaining such pointwise estimate is a certain generalization of the classical sharp maximal operator.
Similar content being viewed by others
References
Bernardis, A., Dalmasso, E., Pradolini, G.: Generalized maximal functions and related operators on weighted Musielak–Orlicz spaces. Ann. Acad. Sci. Fenn. Math., 39, 23–50 (2014)
Bernardis, A., Hartzstein, S., Pradolini, G., Bernardis, A., Hartzstein, S., Pradolini, G.: Weighted inequalities for commutators of fractional integrals on spaces of homogeneous type. J. Math. Anal. Appl., 322, 825–846 (2006)
Bramanti, M., Cerutti, M.: Commutators of singular integrals and fractional integrals on homogeneous spaces. Contemp. Math., 189, 81–81 (1995)
Cabral, A., Pradolini, G., Ramos, W.: Extrapolation and weighted norm inequalities between Lebesgue and Lipschitz spaces in the variable exponent context. J. Math. Anal. Appl., 436, 620–636 (2016)
Calderón, A., Scott, R.: Sobolev type inequalities for p > 0. Studia Math., 62, 75–92 (1978)
Chaffee, L.: Characterizations of bounded mean oscillation through commutators of bilinear singular integral operators. Proc. Roy. Soc. Edinburgh Sect. A, 146, 1159–1166 (2016)
Chanillo, S.: A note on commutators. Indiana Univ. Math. J., 31, 7–16 (1982)
Coifman, R., Meyer, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Amer. Math. Soc., 212, 315–331 (1975)
Coifman, R., Meyer, Y.: Commutateurs d’intégrales singulières et opérateurs multilineaires. Ann. Inst. Fourier (Grenoble), 28, 177–202 (1978)
Coifman, R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. of Math., 103, 611–635 (1976)
Cruz-Uribe, D., Diening, L., Hästö, P.: The maximal operator on weighted variable Lebesgue spaces. Fract. Calc. Appl. Anal., 14, 361–374 (2011)
Cruz-Uribe, D., Fiorenza, A.: Endpoint estimates and weighted norm inequalities for commutators of fractional integrals. Publ. Mat., 47, 103–131 (2003)
Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Springer Science & Business Media, Heidelberg (2013)
Cruz-Uribe, D., Fiorenza, A., Martell, J., et al.: The boundedness of classical operators on variable Lp spaces. Ann. Acad. Sci. Fenn. Math., 31, 239–264 (2006)
Cruz-Uribe, D., Fiorenza, A., Neugebauer, C.: The maximal function on variable Lp spaces. Ann. Acad. Sci. Fenn. Math., 28, 223–238 (2003)
Cruz-Uribe, D., Fiorenza, A., Neugebauer, C.: Weighted norm inequalities for the maximal operator on variable Lebesgue spaces. J. Math. Anal. Appl., 394, 744–760 (2012)
Cruz-Uribe, D., Naibo, V.: Kato–Ponce inequalities on weighted and variable Lebesgue spaces. Differential Integral Equations, 29, 801–836 (2016)
Cruz-Uribe, D., Wang, L.: Variable Hardy spaces. Indiana Univ. Math. J., 63, 447–493 (2014)
Di Fazio, G., Ragusa, M.: Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients. J. Funct. Anal., 112, 241–256 (1993)
Diening, L.: Maximal function on Musielak–Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math., 129, 657–700 (2005)
Diening, L., Harjulehto, P., Hästö, P., et al.: Lebesgue and Sobolev Spaces with Variable Exponents, vol. 2017 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2011
Diening, L., Růžička, M.: Calderón–Zygmund operators on generalized Lebesgue spaces Lp(·) and problems related to fluid dynamics. J. Reine Angew. Math., 563, 197–220 (2003)
Fan, X., Zhao, D.: On the spaces Lp(x)(Ω) and Wm,p(x)(Ω). J. Math. Anal. Appl., 263, 424–446 (2001)
García-Cuerva, J., De Francia, J.: Weighted Norm Inequalities and Related Topics, vol. 116 of North-Holland Mathematics Studies, Elsevier Science Publishers B.V., Netherlands, 1985
Grafakos, L., He, D.: Multilinear Calderón–Zygmund operators on Hardy spaces, II. J. Math. Anal. Appl., 416, 511–521 (2014)
Grafakos, L., Torres, R.: Multilinear Calderón–Zygmund theory. Adv. Math., 165, 124–164 (2002)
Guo, W., Lian, J., Wu, H.: The unified theory for the necessity of bounded commutators and applications. J. Geom. Anal., 30, 3995–4035 (2020)
Harboure, E., Salinas, O., Viviani, B.: Boundedness of the fractional integral on weighted Lebesgue and Lipschitz spaces. Trans. Amer. Math. Soc., 349, 235–255 (1997)
Harjulehto, P., Haästö, P., Pere, M.: Variable exponent Lebesgue spaces on metric spaces: the Hardy–Littlewood maximal operator. Real Anal. Exchange, 30, 87–104 (2004)
Huang, A., Xu, J.: Multilinear singular integrals and commutators in variable exponent Lebesgue spaces. Appl. Math. J. Chinese Univ. Ser. A, 25, 69–77 (2010)
Hytónen, T.: The Lp-to-Lq boundedness of commutators with applications to the Jacobian operator. J. Math. Pures Appl., 156, 351–391 (2021)
Janson, S.: Mean oscillation and commutators of singular integral operators. Ark. Mat., 16, 263–270 (1978)
John, F., Nirenberg, L.: On functions of bounded mean oscillation. Comm. Pure Appl. Math., 14, 415–426 (1961)
Kováčik, O., Rákosník, J.: On spaces Lp(x) and Wk,p(x). Czechoslovak Math. J., 41, 592–618 (1991)
Lerner, A. K., Ombrosi, S., Pérez, C., et al.: New maximal functions and multiple weights for the multilinear Calderón–Zygmund theory. Adv. Math., 220, 1222–1264 (2009)
Li, K.: Multilinear commutators in the two-weight setting. Bull. Lond. Math. Soc., 54, 568–589 (2022)
Li, W.: John–Nirenberg inequality and self-improving properties. J. Math. Res. Exposition, 25, 42–46 (2005)
Lu, G., Zhang, P.: Multilinear Calderón–Zygmund operators with kernels of Dini’s type and applications. Nonlinear Anal., 107, 92–117 (2014)
Muckenhoupt, B., Wheeden, R.: Weighted bounded mean oscillation and the Hilbert transform. Studia Math., 3, 221–237 (1976)
Nekvinda, A.: Hardy–Littlewood maximal operator on Lp(x)(ℝn). Math. Inequal. Appl., 7, 255–266 (2004)
Paluszyński, M.: Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss. Indiana Univ. Math. J., 44, 1–17 (1995)
Pérez, C.: Endpoint estimates for commutators of singular integral operators. J. Funct. Anal., 128, 163–185 (1995)
Péerez, C., Pradolini, G., Torres, R., Trujillo-Gonzaález, R.: End-point estimates for iterated commutators of multilinear singular integrals. Bull. Lond. Math. Soc., 46, 26–42 (2014)
Péerez, C., Trujillo-Gonzaález, R.: Sharp weighted estimates for multilinear commutators. J. Lond. Math. Soc., 65, 672–692 (2002)
Pradolini, G.: A class of pairs of weights related to the boundedness of the fractional integral operator between Lp and Lipschitz spaces. Comment. Math. Univ. Carolin., 42, 133–152 (2001)
Pradolini, G.: Two-weighted norm inequalities for the fractional integral operator between Lp and Lipschitz spaces. Comment. Math., 41, 147–169 (2001)
Pradolini, G., Ramos, W.: Characterization of Lipschitz functions via the commutators of singular and fractional integral operators in variable Lebesgue spaces. Potential Anal., 46, 499–525 (2017)
Ramseyer, M., Salinas, O., Viviani, B.: Lipschitz type smoothness of the fractional integral on variable exponent spaces. J. Math. Anal. Appl., 403, 95–106 (2013)
Rios, C.: The Lp Dirichlet problem and nondivergence harmonic measure. Trans. Amer. Math. Soc., 355, 665–687 (2003)
Tan, J., Liu, Z., Zhao, J.: On some multilinear commutators in variable Lebesgue spaces. J. Math. Inequal., 11, 715–734 (2017)
Wang, W., Xu, J.: Commutators of multilinear singular integrals with Lipschitz functions on products of variable exponent Lebesgue spaces. Adv. Math. (China), 38, 669–677 (2009)
Wang, W., Xu, J.: Commutators of multilinear singular integrals with Lipschitz functions. Commun. Math. Res., 25, 318–328 (2009)
Xu, J.: Generalized commutators of multilinear singular integrals. Proc. A. Razmadze Math. Inst., 142, 109–119 (2006)
Zhang, J., Liu, Z.: Characterizations of Lipschitz space via commutators of some bilinear integral operators. Ann. Funct. Anal., 8, 291–302 (2017)
Acknowledgements
The authors cordially thank the anonymous referees who gave valuable suggestions and useful comments which have led to the improvement of this paper.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest The authors declare no conflict of interest.
Additional information
Supported by the National Natural Science Foundation of China (Grant No. 11571160), the Research Funds for the Educational Committee of Heilongjiang (Grant No. 2019-KYYWF-0909) and the Reform and Development Foundation for Local Colleges and Universities of the Central Government (Grant No. 2020YQ07)
Rights and permissions
About this article
Cite this article
Wu, J.L., Zhang, P. Characterization of Lipschitz Functions via Commutators of Multilinear Singular Integral Operators in Variable Lebesgue Spaces. Acta. Math. Sin.-English Ser. 39, 2465–2488 (2023). https://doi.org/10.1007/s10114-023-2164-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-023-2164-0