Lipschitz estimates for the Berezin transform (Q989233)

Summary: We consider the generalized Fock space \(\mathcal A^2(\mu_m)\), where \(\mu_m\) is the measure with weight \(e^{- |z|^m }\), \(m > 0\), with respect to the Lebesgue measure on \(\mathbb C^n\). Improving upon a recent result of L.\,Coburn and J.-B.\thinspace Xia [cf.\ \textit{L. A. Coburn}, Proc. Am. Math. Soc. 133, No.~1, 127--131 (2005; Zbl 1093.47024); ibid. 135, No.~4, 1163--1168 (2007; Zbl 1137.47018)], we show that for any bounded operator \(X\) on \(\mathcal A^2(\mu_m)\), the Berezin transform of \(X\) satisfies Lipschitz estimates.