Sharp bounds for general commutators on weighted Lebesgue spaces
We show that if a linear operator T is bounded on weighted Lebesgue space L2(w) and obeys a linear bound with respect to the A2 constant of the weight, then its commutator [b, T ] with a function b in BMO will obey a quadratic bound with respect to the A2 constant of the weight. We also prove that t...
| Autores: | , , |
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| Formato: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2012 |
| País: | España |
| Recursos: | Universidad de Sevilla (US) |
| Repositorio: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/42392 |
| Acesso em linha: | http://hdl.handle.net/11441/42392 https://doi.org/10.1090/S0002-9947-2011-05534-0 |
| Access Level: | acceso abierto |
| Palavra-chave: | commutators singular integrals BMO A2 Ap |
| Resumo: | We show that if a linear operator T is bounded on weighted Lebesgue space L2(w) and obeys a linear bound with respect to the A2 constant of the weight, then its commutator [b, T ] with a function b in BMO will obey a quadratic bound with respect to the A2 constant of the weight. We also prove that the kth-order commutator T k b = [b, T k−1 b ] will obey a bound that is a power (k + 1) of the A2 constant of the weight. Sharp extrapolation provides corresponding Lp(w) estimates. In particular these estimates hold for T any Calder´on-Zygmund singular integral operator. The results are sharp in terms of the growth of the operator norm with respect to the Ap constant of the weight for all 1 < p < ∞, all k, and all dimensions, as examples involving the Riesz transforms, power functions and power weights show. |
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