Derivatives of the Berezin Transform

Abstract

For a rotation invariant domainΩ, we considerA2(Ω,μ)the Bergman space and we investigate some properties of the rank one projectionA(z):=〈⋅,kz〉kz. We prove that the trace of all the strong derivatives ofA(z) is zero. We also focus on the generalized Fock spaceA2(μm), whereμmis the measure with weighte-|z|m,m>0, with respect to the Lebesgue measure onℂnand establish estimations of derivatives of the Berezin transform of a bounded operatorTonA2(μm).