Restricted Weak Type Extrapolation of Multi-Variable Operators and Related Topics
A remarkable result in Harmonic Analysis is the so-called Rubio de Francia’s extrapolation theorem. Roughly speaking, it says that if one has an operator T that is bounded on Lp(v), for some p 1 and every weight v in Ap, then T is bounded in Lq(w), for every q > 1 and every weight w in Aq. Rubio...
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| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2019 |
| País: | España |
| Institución: | CBUC, CESCA |
| Repositorio: | TDR. Tesis Doctorales en Red |
| OAI Identifier: | oai:www.tdx.cat:10803/668407 |
| Acceso en línea: | http://hdl.handle.net/10803/668407 |
| Access Level: | acceso abierto |
| Palabra clave: | Espais de Lorentz Espacios de Lorentz Lorentz spaces Ciències Experimentals i Matemàtiques 51 |
| Sumario: | A remarkable result in Harmonic Analysis is the so-called Rubio de Francia’s extrapolation theorem. Roughly speaking, it says that if one has an operator T that is bounded on Lp(v), for some p 1 and every weight v in Ap, then T is bounded in Lq(w), for every q > 1 and every weight w in Aq. Rubio de Francia’s extrapolation theory is very useful in practice, but there is an issue: it does not allow to produce estimates for q = 1. The works of M. J. Carro, L. Grafakos, and J. Soria [9], and M. J. Carro and J. Soria [14] give a solution to this problem, allowing to extrapolate down to the endpoint q = 1. In this project, we started building upon these works to produce multi- variable extensions of the extrapolation results that they presented. We have succeeded in this endeavor, and now we possess extrapolation schemes in the setting of weighted Lorentz spaces that are of great use when trying to bound multi-variable operators for which no sparse domination is known, and also when working with Lorentz spaces outside the Banach-range. As a particular case, we have studied product-type operators, two-variable commutators, averaging operators, and bi-linear multipliers. Sawyer-type inequalities play a fundamental role in the proof of our multi-variable extrapolation schemes and are essential to complete the charac- terization of the weighted restricted weak type bounds for the point-wise product of Hardy-Littlewood maximal operators. In this work, we have ex- tended the classical weak (1, 1) Sawyer-type inequalities proved in [27] to the general restricted weak type case, even in the multi-variable setting. In 2017, at the University of Alabama, we started a collaboration with David V. Cruz-Uribe to produce restricted weak type bounds for fractional operators, Calderón-Zygmund operators, and commutators of these operators. We managed to obtain satisfactory results on this matter, even two-weight norm inequalities, applying a wide variety of techniques on sparse domination, function spaces, and weighted theory. |
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