Norm estimate of composition operators on the weighted Hardy spaces of the ball and of the polydisk

By applying the reproducing kernel approach of Jury Jury MT. Reproducing kernels, de-Branges-Rovnyak spaces, and norm of weighted composition operators. Proc Am Math Soc. 2007;135:3669–3675, we give norm estimates for a class of composition operators on Hilbert spaces of holomorphic functions on the unit ball or polydisk. The results roughly assert that if a certain multiplier norm of a symbol φ is bounded by 1 or a kernel related to φ is positive semi-definite, then the squared norm of the composition operator Cφ is bounded by (1+|φ(0)|)/(1−|φ(0)|). This generalizes the classical result on the Hardy and weighted Bergman spaces of the disk and Jury's result on Drury-Arveson space, Hardy and weighted Bergman spaces of the unit ball to a class of reproducing kernel spaces of holomorphic functions including Dirichlet space of the unit ball and the Hardy space of the polydisk.