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Harmonic Analysis

In this line of research we focus in different aspects of the so called Calderón-Zygmund theory. We study qualitative and quantitative properties of some of the most central operators in Harmonic Analysis such as Singular Integrals (Calderón-Zygmund operators), commutators of these operators, square or maximal functions. In particular, we study the special relationship between the operator norm of these operators and the growth of the Ap constants or their variants in different natural spaces involving weights. We mainly focus in the most classical linear theory where we address some striking questions and open problems but we also consider multilinear aspects of the theory due to its potential applications.

We are also interested in other aspects of the theory which in principle may be thought as instrumental tools but they are deep part of the general theory. Some are the following: extrapolation theory being Rubio de Francia's extrapolation theorem the classical model which can be used as a very useful tool in many situations, the Reverse Hölder and open properties of the Muckenhoupt Ap class of weights where better new proofs have recently been found which may lead to potential applications in connection with the theory of degenerate Poincaré-Sobolev inequalities.

A good portion of this research has its counterpart within the so called multiparameter Harmonic Analysis. This is a large subarea of Harmonic Analysis dealing with classes of operators commuting with multiparameter families of dilations as is the case of the strong maximal function. We are mainly interested in the interplay with the theory of weights. Indeed, some fundamental results from the one-parameter setting are not available in the multiparameter context and we try to address them.

Eusko Jaurlaritza - Gobierno Vasco ikerbasque - Basque Foundation for Science Bizkaia xede. Bizkaiko Foru Aldundia innobasque - Agencia vasca de la innovación Universidad del País Vasco (UPV/EHU) Ayuntamiento de Bilbao - Bilboko Udala