Lipschitz estimates for the Berezin transform

Abstract

We consider the generalized Fock space A²(μ_m), where μ_m is the measure with weight e^−|z|m, m > 0, with respect to the Lebesgue measure on Cⁿ. Improving upon a recent result of L. Coburn and J. Xia, we show that for any bounded operator X on A²(μ_m) , the Berezin transform of X satisfies Lipschitz estimates.