Sharp Forelli-Rudin estimates and the norm of the Bergman projection
Let \(\mathbb B_n\) denote the unit ball in \(\mathbb C^n\), \(n\geq 1\), let \(\nu\) be the normalized volume Lebesgue measure on \(\mathbb B_n\) and let \(\sigma\) be the normalized surface measure on the unit sphere \(\mathbb S_n\). The Bergman space \(A^p(B_n)\), \(p>0\), consists of all holomorphic functions from \(L^p(\mathbb B_n)\) -- the space of all \(p\)-integrable measurable functions on \(\mathbb B_n\). If \(1<p<\infty\), then the Bergman projection \[ Pf(z)=\int_{\mathbb B_n}\frac{f(w)d\nu(w)}{(1-\langle z,w\rangle)^{n+1}}, \] where \(\langle z,w\rangle=z_1\overline{w}_1+z_2\overline{w}_2+\ldots +z_n\overline{w}_n\), maps \(L^p(\mathbb B_n)\) boundedly onto \(A^p(\mathbb B_n)\). The author obtains a new estimate for the \(L^p\)-norm of the Bergman projection: {Theorem 1.1.} For \(1<p<\infty\), we have \[ \frac{\Gamma(\frac{n+1}{p})\Gamma(\frac{n+1}{q})}{\Gamma^2(\frac{n+1}{2})}\leq \|P\|_p\leq \frac{\pi n!}{\Gamma^2(\frac{n+1}{2})}\csc\frac{\pi}{p}, \] where \(q:=\frac{p}{p-1}\) is the conjugate exponent of \(p\), and \(\Gamma(x)\) is the Euler gamma function. The author also gives new sharp estimates for the integrals \[ I_c(z):=\int_{\mathbb S_n}\frac{d\sigma(\zeta)}{|1-\langle z,\zeta\rangle|^{n+c}} \quad\text{and}\quad J_{c,t}(z):=\int_{\mathbb B_n}\frac{(1-|w|^2)^td\nu(w)}{|1-\langle z,w\rangle|^{n+1+t+c}}, \] where \(z\in \mathbb B_n\), \(c\) is real and \(t>-1\).
scientific article; zbMATH DE number 6376101
scientific article