Norm of the Bergman projection (Q2925745)

The Bloch space \(\mathfrak{B}\) contains all holomorphic functions \(f\) on the unit ball \(\mathbb{B}\) for which the semi-norm \(\displaystyle \|f\|_{\beta} = \sup_{z\in \mathbb{B}} (1-|z|^2)|\nabla f(z)|\) is finite. For \(f\in\mathfrak{B}\), the \(\mathfrak{B}\)-norm is defined as \(\displaystyle \|f\|_{\mathfrak{B}}= |f(0)|+ \|f\|_{\beta}\). The Bloch space \(\mathfrak{B}\), with invariant norm, contains all holomorphic functions \(f\in H(\mathbb{B})\) for which the \(\tilde{\mathfrak{B}}\)-norm \(\displaystyle \|f\|_{\tilde{\mathfrak{B}}}= |f(0)|+ \|f\|_{\tilde{\beta}}\) is finite, where the \(\tilde{\beta}\)-norm is defined as \(\displaystyle \|f\|_{\tilde{\beta}} = \sup_{z\in \mathbb{B}} |\tilde{\nabla} f(z)|\) and the invariant gradient \(\tilde{\nabla}f(z) = \nabla(f\circ \phi_z)(0)\). For \(1\leq p\leq\infty\) and \(\alpha>-1\), the Bergman projection operator \(P_{\alpha}\) is defined by \(\displaystyle P_{\alpha}g(z)=\int_{\mathbb{B}} K_{\alpha}(z, w) g(w) \,dv_{\alpha}(w)\), for \(g\in L^{p}\) and \(z, w \in \mathbb{B}\). The authors consider the \(\beta\)-norm and \(\mathfrak{B}\)-norm of the Bergman projection \(P_{\alpha}: L^{\infty} \rightarrow \mathfrak{B}\) which are defined as \(\displaystyle \|P_{\alpha}\|_{\beta}= \sup_{\|g\|_{\infty}<1}\|P_{\alpha}g\|_{\beta}\) and \(\displaystyle \|P_{\alpha}\|_{\mathfrak{B}}= \sup_{\|g\|_{\infty}<1}\|P_{\alpha}g\|_{\mathfrak{B}}\); similarly, they define the \(\tilde{\beta}\)-norm and \(\tilde{\mathfrak{B}}\)-norm of the Bergman projection.NEWLINENEWLINEIn this paper, the authors present the exact \(\beta\)-norm and an estimate for the \(\tilde{\beta}\)-norm of \(P_{\alpha}\); namely, NEWLINE\[NEWLINE\|P_{\alpha}\|_{\beta} =C_{\alpha},\quad \|P_{\alpha}\|_{\tilde{\beta}} =\max_{0\leq t\leq \pi/2} \ell(t),NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \frac{\pi}{2} C_{\alpha}\leq \|P_{\alpha}\|_{\tilde{\beta}} \leq \frac{\sqrt{\pi^2 +4}}{2} C_{\alpha}.NEWLINE\]NEWLINE Here, \(\displaystyle \theta= n+1+\alpha \), \(\displaystyle C_{\alpha}=\frac{\Gamma(\theta+1)}{\Gamma^2((\theta+1)/2)} \) and \(\displaystyle \ell(t)= \theta \int_{\mathbb{B}} \frac{|(1-w)\cos t + w\sin t|}{|w-1|^{\theta}} \,dv_{\alpha}(w)\). As a corollary, the authors give the following norm estimates of \(P_{\alpha}\): NEWLINE\[NEWLINEC_{\alpha}\leq \|P_{\alpha}\|_{\mathfrak{B}} \leq 1+C_{\alpha}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\displaystyle \frac{\pi}{2} C_{\alpha}\leq \|P_{\alpha}\|_{\tilde{\mathfrak{B}}} \leq 1+\frac{\sqrt{\pi^2 +4}}{2} C_{\alpha}.NEWLINE\]