| Title:
|
Commutators of singular integrals on spaces of homogeneous type (English) |
| Author:
|
Pradolini, Gladis |
| Author:
|
Salinas, Oscar |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
57 |
| Issue:
|
1 |
| Year:
|
2007 |
| Pages:
|
75-93 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
commutators |
| Keyword:
|
spaces of homogeneous type |
| Keyword:
|
weights |
| MSC:
|
42B25 |
| idZBL:
|
Zbl 1174.42322 |
| idMR:
|
MR2309950 |
| . |
| Date available:
|
2009-09-24T11:44:01Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128156 |
| . |
| Reference:
|
[1] H. Aimar: Singular integrals and approximate identities on spaces of homogeneous type.Trans. Am. Math. Soc. 292 (1985), 135–153. Zbl 0578.42016, MR 0805957, 10.1090/S0002-9947-1985-0805957-9 |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[5] R. Coifman: Distribution function inequalities for singular integrals.Proc. Natl. Acad. Sci. USA 69 (1972), 2838–2839. Zbl 0243.44006, MR 0303226, 10.1073/pnas.69.10.2838 |
| Reference:
|
[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, 10.1007/BFb0058946 |
| Reference:
|
[7] R. Coifman, R. Rochberg, and G. Weiss: Factorization theorems for Hardy spaces in several variables.Ann. Math. 103 (1976), 611–635. MR 0412721, 10.2307/1970954 |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[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. MR 1213138, 10.1006/jfan.1993.1032 |
| Reference:
|
[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. MR 1265808, 10.1080/03605309408821025 |
| Reference:
|
[12] C. Fefferman, E. M. Stein: Some maximal inequalities.Amer. J. Math. 93 (1971), 107–115. MR 0284802, 10.2307/2373450 |
| Reference:
|
[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 |
| Reference:
|
[14] R. Macías, C. Segovia: Lipschitz functions on spaces of homogeneous type.Adv. Math. 33 (1979), 257–270. MR 0546295, 10.1016/0001-8708(79)90012-4 |
| Reference:
|
[15] R. Macías, C. Segovia: Singular integrals on generalized Lipschitz and Hardy spaces.Studia Math. 65 (1979), 55–75. MR 0554541, 10.4064/sm-65-1-55-75 |
| Reference:
|
[16] R. Macías, C. Segovia: A well behaved quasi-distance for spaces of homogeneous type.Trabajos de Matemática, Serie I 32 (1981). |
| Reference:
|
[17] R. O’Neil: Fractional integration in Orlicz spaces.Trans. Amer. Math. Soc. 115 (1965), 300–328. MR 0194881, 10.1090/S0002-9947-1965-0194881-0 |
| Reference:
|
[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. MR 1481632, 10.1007/BF02648265 |
| Reference:
|
[19] C. Pérez: Endpoint estimates for commutators of singular integral operators.J. Funct. Anal. 128 (1995), 163–185. MR 1317714, 10.1006/jfan.1995.1027 |
| Reference:
|
[20] C. Pérez, R. Wheeden: Uncertainty principle estimates for vector fields.J. Funct. Anal. 181 (2001), 146–188. MR 1818113, 10.1006/jfan.2000.3711 |
| Reference:
|
[21] G. Pradolini, O. Salinas: Maximal operators on spaces of homogeneous type.Proc. Amer. Math. Soc. 132 (2003), 435–441. MR 2022366, 10.1090/S0002-9939-03-07079-5 |
| Reference:
|
[22] C. Ríos: The $L^{p}$ Dirichlet problems and non divergence harmonic measure.Trans. Amer. Math. Soc. 355 (2003), 665–687. MR 1932720, 10.1090/S0002-9947-02-03145-8 |
| Reference:
|
[23] M. Rao, Z. Ren: Theory of Orlicz spaces.Marcel Dekker, New York, 1991. MR 1113700 |
| Reference:
|
[24] R. Rochberg, G. Weiss: Derivatives of analytic families of Banach spaces.Ann. Math. 118 (1983), 315–347. MR 0717826, 10.2307/2007031 |
| Reference:
|
[25] M. Wilson: Weighted norm inequalities for the continuous square function.Trans. Amer. Math. Soc. 314 (1989), 661–692. MR 0972707, 10.1090/S0002-9947-1989-0972707-9 |
| . |
