Abstract
In this paper, the weak type LlogL estimate for the multilinear fractional commutator is obtained by introducing a new kind of maximal operator of the multilinear fractional order associated with the mean Luxumburg norm and using the technique of sharp function.
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Muckenhoupt, B., Wheeden, R. L.: Weighted norm inequalities for fractional integrals. Tran. Amer. Math. Soc., 192, 261–274 (1974)
Elias, M. S.: Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1971
Taibleson, M. H., Weiss, G.: The molecular of certain Hardy spaces. Asterisque, 77 67–149 (1980)
Ding, Y.: Weak type bounds for a class of rough operators with power weights. Proc. Amer. Math. Soc., 125, 2939–2942 (1997)
Ding, Y., Lu, S. Z.: Weighted norm inequalities for fractional integral operarors with rough kernel. Canad. J. Math., 50, 29–39 (1998)
Ding, Y., Lu, S. Z.: Higher order commutators for a class of rough operators. Ark. Math., 37, 33–44 (1999)
Fan, D. S., Le, H. V.: Singular integrals and fractional integrals in Triebel-Lizorkin spaces and in weighted L p spaces. Math. Inequal. Appl., 11, 127–147 (2008)
Fan, D. S., Sato, S.: Singular and fractional integrals along variable surfaces. Hokkaido Math. J., 1, 61–85 (2006)
Chanillo, S.: A note on commutators. Indina. Univ. Math. J., 31, 7–16 (1982)
Pérez, C.: Endpoint estimates for commutators of singular integral operators. J. Func. Anal., 128, 163–185 (1995)
Ding, Y., Lu, S. Z., Zhang, P.: Weak estimates for commutators of fractional integral operators. Sci. China, Ser. A, 44(7), 877–888 (2001)
Ding, Y., Lu, S. Z., Yabuta, K.: Multilinear singular and fractional integrals. Acta Mathematica Sinica, English Series, 22(2), 347–356 (2006)
Grafakos, L., Torres, R. H.: Multilinear Calder’on-Zygmund theory. Adv. Math., 165, 124–164 (2002)
Grafakos, L., Torres, R. H.: Maximal operator and weighted norm inequalities for multilinear singular integrals. Indina, Univ. Math. J., 51(5), 1261–1276 (2002)
Kenig, C. E., Stein, E. M.: Multilinear estimates and fractional integration. Math. Res. Lett., 6, 1–15 (1999)
Pérez, C., Torres, R. H.: Sharp maximal function estimates for multilinear singular operators. Contemp. Math., 320, 323–331 (2003)
Lerner, A. K., Ombrosi, S., Pérez, C., et al.: New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory. Advances in Math., 220(4), 1222–1264 (2009)
Fefferman, C., Stein, E. M.: H p spaces of several variables. Acta Math., 129, 137–193 (1972)
Rao, M. M., Ren, Z. D.: Theory of Orlicz Spaces, Marcel Dekker, New York, 1991
Adams, R.: A note on Riesz potential. Duke. Math. J., 42, 765–778 (1975)
Garcia-Cuerva, J., Rubio de Francia, J. L.: Weighted Norm Inequalities and Related Topics, Amsterdam, North-Holland, 1985
Chen, X., Xue, Q. Y.: Weighted estimates for a class of multilinear fractional type operators. J. Math. Anal. Appl., 362(2), 355–373 (2010)
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Supported by National Natural Science Foundation of China (Grant Nos. 10571156 and 10871173)
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Yu, X., Chen, J.C. Endpoint estimates for commutators of multilinear fractional integral operators. Acta. Math. Sin.-English Ser. 26, 433–444 (2010). https://doi.org/10.1007/s10114-010-8421-z
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DOI: https://doi.org/10.1007/s10114-010-8421-z