Weighted restricted weak-type extrapolation on classical Lorentz spaces
[eng] An important result in Harmonic Analysis is the extrapolation theorem of Rubio de Francia. In its original version says that if T is a sublinear operator that is bounded in Lp0 pvq, for some p0 ě 1 and every v P Ap , then T is bounded in Lppvq for any p ą 1 and v P Ap. Although the Rubio de Fr...
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| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/190606 |
| Acceso en línea: | https://hdl.handle.net/2445/190606 http://hdl.handle.net/10803/675928 |
| Access Level: | acceso abierto |
| Palabra clave: | Anàlisi matemàtica Espais de Lorentz Teoria d'operadors Mathematical analysis Lorentz spaces Operator theory |
| Sumario: | [eng] An important result in Harmonic Analysis is the extrapolation theorem of Rubio de Francia. In its original version says that if T is a sublinear operator that is bounded in Lp0 pvq, for some p0 ě 1 and every v P Ap , then T is bounded in Lppvq for any p ą 1 and v P Ap. Although the Rubio de Francia theorem has proven to be very useful in practice, it does not allow to get estimates at p “ 1. That is why in [61] it was developed a new extrapolation theory in order to give a solution to this issue, showing that weighted restricted weak-type pp, pq estimates for p ą 1 and for an slightly bigger class than Ap (denoted by Ap) yield estimates at p “ 1. Indeed, in this thesis we start by seeing that the converse of the previous result is also true; that is, we study boundedness properties for operators T that are of restricted weak- type p1, 1q for weights in A1 and we prove that this condition is a “norm” condition since it is equivalent to weighted restricted weak-type pp, pq for Aˆp weights. As a consequence, we can obtain, for instance, boundedness for operators which are given as an average of operators of the above type. As well, we present new weighted restricted estimates on classical Lorentz spaces for operators that satisfy weighted restricted weak-type pp, pq estimates, p ě 1, extending then these results to the limited setting and, as well, to the multi-variable setting. As a conse- quence, we obtain new weighted estimates on classical Lorentz spaces for important operators in Harmonic Analysis such as operators that satisfy a Fefferman-Stein’s inequality, Fourier multipliers of Hörmander type, rough operators, sparse operators, the Bochner-Riesz oper- ator, among others. Further, from the previous estimates we prove pointwise estimates for the decreasing rearrangement of such operators. Finally, we also study strong-type estimates on weighted classical Lorentz spaces. |
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