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Singular integrals and commutators on homogeneous groups. (English) Zbl 1026.43007

The authors using the basic idea from the paper of F. Soria and G. Weiss [Indiana Univ. Math. J. 43, 187-204 (1994; Zbl 0803.42004)], prove several general theorems for the boundedness of sublinear operators and commutators generated by linear operators and \(\text{BMO}({\mathbf G})\) functions on the weighted Lebesgue space on the homogeneous group \({\mathbf G}\). The conditions of these theorems are so general that many important operators in analysis satisfy these conditions.
Some of these theorems are best possible even for \(n\)-dimensional Euclidean spaces. They also establish some new applications of these theorems on weighted inequalities for singular integrals (including rough singular integral operators, oscillatory integrals, parabolic singular integrals) and their commutators and the maximal operators associated with them. These methods are more direct than previously used methods.

MSC:

43A85 Harmonic analysis on homogeneous spaces
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory

Citations:

Zbl 0803.42004
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Full Text: DOI

References:

[1] Aguilera, N. E.; Harboure, E. O., Some inequalities for maximal operators, Indiana Univ. Math. J., 29, 559-576 (1980) · Zbl 0512.42019 · doi:10.1512/iumj.1980.29.29042
[2] Bramanti, M.; Cerutti, M. C., W_p^1,2 solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients, Commun. in Partial Diff. Equat., 18, 1735-1763 (1993) · Zbl 0816.35045 · doi:10.1080/03605309308820991
[3] Calderón, A. P., Inequalities for the maximal function relative to a metric, Studia Math., 57, 297-306 (1976) · Zbl 0341.44007
[4] Christ, M., Weak type (1,1) bounds for rough operators, Ann. Math., 128, 19-42 (1988) · Zbl 0666.47027 · doi:10.2307/1971461
[5] Christ, M.; Rubio de Francia, J. L., Weak type (1,1) bounds for rough operators. II, Invent. Math., 93, 225-237 (1988) · Zbl 0695.47052 · doi:10.1007/BF01393693
[6] Coifman, R. R.; Rochberg, R.; Weiss, G., Factorization theorems for Hardy spaces in several variables, Ann. Math., 103, 611-635 (1976) · Zbl 0326.32011 · doi:10.2307/1970954
[7] Coifman, R. R.; Weiss, G., Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83, 569-645 (1977) · Zbl 0358.30023 · doi:10.1090/S0002-9904-1977-14325-5
[8] Duoandikoetxea, J., Weighted norm inequalities for homogeneous singular integrals, Trans. Amer. Math. Soc., 336, 869-880 (1993) · Zbl 0770.42011 · doi:10.2307/2154381
[9] Fan, D.; Pan, Y., Singular integral operators with rough kernels supported by subvarieties, Amer. J. Math., 119, 799-839 (1997) · Zbl 0899.42002 · doi:10.1353/ajm.1997.0024
[10] Fefferman, R., A note on singular integrals, Proc. Amer. Math. Soc., 74, 266-270 (1979) · Zbl 0417.42009 · doi:10.2307/2043145
[11] Folland, G. B.; Stein, E. M., Hardy spaces on homogeneous groups (1982), Princeton, NJ: Princeton Univ. Press and Univ. of Tokyo Press, Princeton, NJ · Zbl 0508.42025
[12] J. García-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Math. Studies 116 (Amsterdam, 1985). · Zbl 0578.46046
[13] Hebisch, W.; Sikora, A., A smooth subadditive homogeneous norm on a homogeneous group, Studia Math., 96, 232-236 (1990) · Zbl 0723.22007
[14] Hofmann, S., Singular integrals with power weight, Proc. Amer. Math. Soc., 110, 343-353 (1990) · Zbl 0717.42018 · doi:10.2307/2048076
[15] Muckenhoupt, B.; Wheeden, R. L., Weighted norm inequalities for singular and fractional integrals, Trans. Amer. Math. Soc., 161, 249-258 (1971) · Zbl 0226.44007 · doi:10.2307/1995940
[16] Ricci, F.; Stein, E. M., Harmonic analysis on nilpotent groups and singular integrals: I. Oscillatory integrals, J. Funct. Anal., 73, 179-194 (1987) · Zbl 0622.42010 · doi:10.1016/0022-1236(87)90064-4
[17] Seeger, A., Singular integral operators with rough convolution kernels, J. Amer. Math. Soc., 9, 95-105 (1996) · Zbl 0858.42008 · doi:10.1090/S0894-0347-96-00185-3
[18] Soria, F.; Weiss, G., A remark on singular integrals and power weights, Indiana Univ. Math. J., 43, 187-204 (1994) · Zbl 0803.42004 · doi:10.1512/iumj.1994.43.43009
[19] Stein, E. M., Note on singular integrals, Proc. Amer. Math. Soc., 8, 250-254 (1957) · Zbl 0077.27301 · doi:10.2307/2033721
[20] Stein, E. M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals (1993), Princeton, NJ: University Press, Princeton, NJ · Zbl 0821.42001
[21] Vargas, A. M., Weighted weak type (1,1) bounds for rough operators, J. London Math. Soc., 54, 297-310 (1996) · Zbl 0884.42011
[22] Watson, D. K., Vector-valued inequalities, factorization, and extrapolation for a family of rough operators, J. Funct. Anal., 121, 389-415 (1994) · Zbl 0811.47050 · doi:10.1006/jfan.1994.1053
[23] D. K. Watson, A_1 weights and weak type (1,1) estimates for rough operators, preprint (1994-1995).
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