Norm of Berezin transform on \(L^p\) space (Q940782)

Let \(D\) be the unit disc in \(\mathbb C\) equipped with the standard Lebesgue measure \(\mu\). For \(K_\alpha(z, \xi)= (\alpha +1)\frac{(1-| \xi| ^2)^\alpha}{^(1-z\overline{\xi})^{\alpha + 2}}\), the operator \(G_\alpha: L^p(D) \to L^p(D)\), \((G_\alpha f)(z):= \frac1\pi \int_D| K_\alpha(z, \xi)| f(\xi)\,d\mu(\xi)\) is introduced. It is shown that for \(1 \leq p < \infty\), \(\alpha > -1\) and \(p(\alpha + 1) >1\), \[ \| G_\alpha\| _p = \frac{\alpha + 1}{\Gamma^2(1 + \alpha/2)}\Gamma(1/p)\Gamma(\alpha + 1 - 1/p). \] This gives an upper estimate for the norm of corresponding Bergman type operator and, in particular (when \(\alpha=0\)), estimates the norm of classical Bergman projection. By a duality argument, it is shown that for \(1 < p \leqslant \infty\), the norm of Berezin transform \(B\) on \(L^p (D)\) equals \( \frac{p + 1}{p^2 }\frac{\pi}{\sin (\pi /p)}\).