Boundedness of Commutators with Lipschitz Functions in Non-homogeneous Spaces*

Abstract

Under the assumption that the underlying measure is a non-negative Radon measure which only satisfies some growth condition, the authors prove that for a class of commutators with Lipschitz functions which include commutators generated by Calderón-Zygmund operators and Lipschitz functions as examples, their boundedness in Lebesgue spaces or the Hardy space H ¹(μ) is equivalent to some endpoint estimates satisfied by them. This result is new even when the underlying measure μ is the d-dimensional Lebesgue measure.

Keywords

Commutator / Lipschitz function / Lebesgue space / Hardy space / RBMO space / Non-doubling measure / 47B47 / 42B20

References

[1] García-Cuerva Publ. Mat., 2005, 49: 285

[2] García-Cuerva Indiana Univ. Math. J., 2001, 50: 1241

[3] García-Cuerva, J. and Rubio de Francia, J. L., Weighted Norm Inequalities and Related Topics, North-Holland Math. Studies, Vol. 16, North-Holland, Amesterdan, 1985.

[4] Hu, G., Meng, Y. and Yang, D. C., Boundedness of Riesz potentials in non-homogeneous spaces, Acta Math. Sci. Ser. B Engl. Ed., to appear, 2008.

[5] Meng Taiwanese J. Math., 2006, 10: 1443

[6] Nazarov Int. Math. Res. Not., 1998, 9: 463

[7] Nazarov Duke Math. J., 2002, 113: 259

[8] Nazarov Acta Math., 2003, 190: 151

[9] Orobitg Trans. Amer. Math. Soc., 2002, 354: 2013

[10] Tolsa Proc. London Math. Soc., 2001, 82: 195

[11] Tolsa Adv. Math., 2001, 164: 57

[12] Tolsa Math. Ann., 2001, 319: 89

[13] Tolsa Publ. Mat., 2001, 45: 163

[14] Tolsa Acta Math., 2003, 190: 105

[15] Tolsa Trans. Amer. Math. Soc., 2003, 355: 315

[16] Verdera, J., The fall of the doubling condition in Calderón-Zygmund theory, Publ. Mat., Vol. Extra, 2002, 275–292.