Previous |  Up |  Next


commutators; spaces of homogeneous type; weights
In this work we prove some sharp weighted inequalities on spaces of homogeneous type for the higher order commutators of singular integrals introduced by R. Coifman, R. Rochberg and G. Weiss in Factorization theorems for Hardy spaces in several variables, Ann. Math. 103 (1976), 611–635. As a corollary, we obtain that these operators are bounded on $L^{p}(w)$ when $w$ belongs to the Muckenhoupt’s class $A_{p}$, $p>1$. In addition, as an important tool in order to get our main result, we prove a weighted Fefferman-Stein type inequality on spaces of homogeneous type, which we have not found previously in the literature.
[1] H.  Aimar: Singular integrals and approximate identities on spaces of homogeneous type. Trans. Am. Math. Soc. 292 (1985), 135–153. DOI 10.1090/S0002-9947-1985-0805957-9 | MR 0805957 | Zbl 0578.42016
[2] H.  Aimar: Rearrangement and continuity properties of  ${\mathrm BMO}(\phi )$ functions on spaces of homogeneous type. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV.  Ser. 18 (1991), 353–362. MR 1145315
[3] M.  Bramanti, M. C.  Cerutti: Commutators of singular integrals and fractional integrals on homogeneous spaces. In: Harmonic Analysis and Operator Theory. Proceedings of the conference in honor of Mischa Cotlar, January 3–8, 1994, Caracas, Venezuela, S. A. M. Marcantognini et al. (eds.), Am. Math. Soc., Providence. MR 1347007
[4] M.  Bramanti, M. C.  Cerutti: Commutators of singular integrals on homogeneous spaces. Boll. Unione Mat. Ital., VII.  Ser. B 10 (1996), 843–883. MR 1430157
[5] R.  Coifman: Distribution function inequalities for singular integrals. Proc. Natl. Acad. Sci. USA 69 (1972), 2838–2839. DOI 10.1073/pnas.69.10.2838 | MR 0303226 | Zbl 0243.44006
[6] R.  Coifman, G.  Weiss: Analyse harmonique non-commutative sur certains espaces homogènes. Lecture Notes in Mathematics, Vol.  242. Springer-Verlag, Berlin-New York, 1971. MR 0499948
[7] R.  Coifman, R.  Rochberg, and G.  Weiss: Factorization theorems for Hardy spaces in several variables. Ann. Math. 103 (1976), 611–635. DOI 10.2307/1970954 | MR 0412721
[8] F.  Chiarenza, M.  Frasca, and P.  Longo: Interior $W^{2,p}$  estimates for non divergence elliptic equations with discontinuous coefficients. Ric. Mat. 40 (1991), 149–168. MR 1191890
[9] F.  Chiarenza, M.  Frasca, and P. Longo: $W^{2,p}$-solvability of the Dirichlet problem for non divergence elliptic equations with VMO coefficients. Trans. Am. Math. Soc. 336 (1993), 841–853. MR 1088476
[10] G.  Di Fazio, M. A.  Ragusa: Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients. J.  Funct. Anal. 112 (1993), 241–256. DOI 10.1006/jfan.1993.1032 | MR 1213138
[11] B.  Franchi, C.  E.  Gutiérrez, and R.  Wheeden: Weighted Sobolev-Poincaré inequalities for Grushin type operators. Comm. Partial Differential Equations 19 (1994), 523–604. DOI 10.1080/03605309408821025 | MR 1265808
[12] C.  Fefferman, E. M.  Stein: Some maximal inequalities. Amer. J.  Math. 93 (1971), 107–115. DOI 10.2307/2373450 | MR 0284802
[13] J. L.  Journé: Calderón Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral of Calderón. Lecture Notes in Mathematics Vol.  994. Springer-Verlag, Berlin-New York, 1983. MR 0706075
[14] R.  Macías, C. Segovia: Lipschitz functions on spaces of homogeneous type. Adv. Math. 33 (1979), 257–270. DOI 10.1016/0001-8708(79)90012-4 | MR 0546295
[15] R.  Macías, C.  Segovia: Singular integrals on generalized Lipschitz and Hardy spaces. Studia Math. 65 (1979), 55–75. MR 0554541
[16] R.  Macías, C.  Segovia: A well behaved quasi-distance for spaces of homogeneous type. Trabajos de Matemática, Serie  I 32 (1981).
[17] R.  O’Neil: Fractional integration in Orlicz spaces. Trans. Amer. Math. Soc. 115 (1965), 300–328. DOI 10.1090/S0002-9947-1965-0194881-0 | MR 0194881
[18] C.  Pérez: Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function. J.  Fourier Anal. Appl. 3 (1997), 743–756. DOI 10.1007/BF02648265 | MR 1481632
[19] C.  Pérez: Endpoint estimates for commutators of singular integral operators. J.  Funct. Anal. 128 (1995), 163–185. DOI 10.1006/jfan.1995.1027 | MR 1317714
[20] C.  Pérez, R.  Wheeden: Uncertainty principle estimates for vector fields. J.  Funct. Anal. 181 (2001), 146–188. DOI 10.1006/jfan.2000.3711 | MR 1818113
[21] G.  Pradolini, O.  Salinas: Maximal operators on spaces of homogeneous type. Proc. Amer. Math. Soc. 132 (2003), 435–441. DOI 10.1090/S0002-9939-03-07079-5 | MR 2022366
[22] C. Ríos: The $L^{p}$ Dirichlet problems and non divergence harmonic measure. Trans. Amer. Math. Soc. 355 (2003), 665–687. DOI 10.1090/S0002-9947-02-03145-8
[23] M.  Rao, Z.  Ren: Theory of Orlicz spaces. Marcel Dekker, New York, 1991. MR 1113700
[24] R.  Rochberg, G.  Weiss: Derivatives of analytic families of Banach spaces. Ann. Math. 118 (1983), 315–347. DOI 10.2307/2007031 | MR 0717826
[25] M.  Wilson: Weighted norm inequalities for the continuous square function. Trans. Amer. Math. Soc. 314 (1989), 661–692. DOI 10.1090/S0002-9947-1989-0972707-9 | MR 0972707
Partner of
EuDML logo