Operators on weighted Bergman spaces (0<p≤1) and applications

Abstract

We describe the boundedness of a linear operator from Bp(ρ) = {f : D → C analytic : (∫ D ρ(1 − |z|) (1 − |z|) |f(z)| dA(z) )1/p < ∞} , for 0 < p ≤ 1 under some conditions on the weight function ρ, into a general Banach space X by means of the growth conditions at the boundary of certain fractional derivatives of a single X-valued analytic function. This, in particular, allows us to characterize the dual of Bp(ρ) for 0 < p < 1 and to give a formulation of generalized Carleson measures in terms of the inclusion B1(ρ) ⊂ L(D,μ). We then apply the result to the study of multipliers, Hankel operators and composition operators acting on Bp(ρ) spaces. 1991 Math. Subject Class. : Primary 47B38, 47B35 Secondary 42A45