Final Report Summary - ANPROB (Analytic-probabilistic methods for borderline singular integrals)
The aim of AnProb was to combine analytic and probabilistic methods in order to advance the theory of singular integrals of Harmonic Analysis in various situations at the borderline of the existing theory. This aim was successfully accomplished through a broad range of contributions. Among the highlights are solutions of two well-known problems previously posed and tried by some leading experts: a long-standing question of Rubio de Francia on the pointwise converge of vector-valued Fourier series dating back to 1986, achieved in collaboration with M. Lacey, and a problem of Hofmann from 2008 on the minimal integrability conditions in a characterisation of singular integrals by so-called local Tb conditions, solved in collaboration with F. Nazarov. We also generalised the first solution, by Lacey et al. in 2014, to a famous two-weight problem of Muckenhoupt from the 1970’s. In addition to the key achievements, several complementary results and extensions of the theory were obtained by the project, published in over 20 refereed articles, with over 10 further works still waiting for publication at the end of the project. Moreover, two PhD theses and a research monograph - joint work with J. van Neerven, M. Veraar and L. Weis - were completed. The results have gained high international recognition, manifested by the PI’s invitation as a section speaker in the International Congress of Mathematicians in 2014.