Estimations optimales pour le projecteur de Bergman dans le disque unité

Abstract

This work focuses on the sharp estimates of the Bergman projection in the unit disk. We denote by Aα (D) the weighted Bergman space on the unit disk of C : it is the space of analytic function in L (D, dA ). Here, α > −1 and dA (z ) = (α + 1)(1 − |z | ) dA(z ) where dA(z ) is the normalized Lebesgue area measure on D. The Bergman projection is defined on L (D, dA ) by :2P f (z ) := Zf (w)D (1 − z w)2+α dAα (w) ,z ∈ D.In this thesis, from the work of David Békollé and Aline Bonami in 1978, we prove that given a weighted ω on D, the Bergman projection is of type (p, p) on L (D, ω dA ) (1 < p < +∞) (respectivel of weak type (1, 1) on L (D, ω dA )) if and only if ω belong to (B ) class (respectively (B1,α ) class)p,αof Békollé-Bonami. Later, thanks to a work of Sandra Pott and Maria Carmen Reguera, we give sharp estimates for the Bergman projection on L (D, ω dA ) (1 < p < +∞) in terms of Békollé-Bonami constant Bp,α (ω ) by the means of a dyadic model approach and an adaptation of a method of Cruz-Uribe, Martell and Pérez