Optimal exponents in weighted estimates without examples
We present a general approach for proving the optimality of the exponents on weighted estimates. We show that if an operator T satisfies a bound like ∥ T ∥ Lp(w) ≤ c [w]βAp w ε Ap, then the optimal lower bound for β is closely related to the asymptotic behaviour of the unweighted Lp norm ∥ T ∥ Lp(Rn...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2015 |
| País: | Argentina |
| Institución: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/160873 |
| Acceso en línea: | http://hdl.handle.net/11336/160873 |
| Access Level: | acceso abierto |
| Palabra clave: | Muckenhoupt weights Calderon-Zygmund operators Maximal functions https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Sumario: | We present a general approach for proving the optimality of the exponents on weighted estimates. We show that if an operator T satisfies a bound like ∥ T ∥ Lp(w) ≤ c [w]βAp w ε Ap, then the optimal lower bound for β is closely related to the asymptotic behaviour of the unweighted Lp norm ∥ T ∥ Lp(Rn) as p goes to 1 and +∞. By combining these results with the known weighted inequalities, we derive the sharpness of the exponents, without building any specific example, for a wide class of operators including maximaltype, Caldeŕon-Zygmund and fractional operators. In particular, we obtain a lower bound for the best possible exponent for Bochner- Riesz multipliers.We also present a new result concerning a continuum family of maximal operators on the scale of logarithmic Orlicz functions. Further, our method allows to consider in a unified way maximal operators defined over very general Muckenhoupt bases. |
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