Inspired by the work of Engliš, we study the asymptotic behavior of the weighted Bergman kernel together with an application to the Lu Qi-Keng conjecture. Some comparison results between the weighted and the classical Bergman kernel are also obtained.

Let Ω be a bounded strictly pseudoconvex domain in ${ℂ}^n$. In this paper we find sufficient conditions on a function f defined on Ω in order that the weighted Bergman projection $P_{s}f$ belong to the Hardy-Sobolev space $H_k^p\left(\partial \Omega \right)$. The conditions on f we consider are formulated in terms of tent spaces and complex tangential vector fields. If f is holomorphic then these conditions are necessary and sufficient in order that f belong to the Hardy-Sobolev space $H_k^p\left(\partial \Omega \right)$.

This paper characterizes the boundedness and compactness of weighted composition operators between a weighted-type space and the Hardy space on the unit ball of ℂⁿ.

Let H() denote the space of all holomorphic functions on the unit ball ⊂ ℂⁿ. Let φ be a holomorphic self-map of and u∈ H(). The weighted composition operator $u{C}_{\phi }$ on H() is defined by $u{C}_{\phi }f\left(z\right)=u\left(z\right)f\left(\phi \left(z\right)\right)$. We investigate the boundedness and compactness of $u{C}_{\phi }$ induced by u and φ acting from Zygmund spaces to Bloch (or little Bloch) spaces in the unit ball.

We study the limit behavior of weighted Bergman kernels on a sequence of domains in a complex space ${ℂ}^N$, and show that under some conditions on domains and weights, weighed Bergman kernels converge uniformly on compact sets. Then we give a weighted generalization of the theorem given by M. Skwarczyński (1980), highlighting some special property of the domains, on which the weighted Bergman kernels converge uniformly. Moreover, we show that convergence of weighted Bergman kernels implies this property,...

The integral representation for the multiplicity of an isolated zero of a holomorphic mapping $f:({ℂ}^n,0)\to ({ℂ}^n,0)$ by means of Weil’s formulae is obtained.

For complete Reinhardt pairs “compact set - domain” K ⊂ D in ℂⁿ, we prove Zahariuta’s conjecture about the exact asymptotics $ln{d}_s\left(A_K^D\right)-{((n!s)/\tau (K,D))}^{1/n}$, s → ∞, for the Kolmogorov widths $d_s\left(A_K^D\right)$ of the compact set in C(K) consisting of all analytic functions in D with moduli not exceeding 1 in D, τ(K,D) being the condenser pluricapacity of K with respect to D.

The classical Wold decomposition theorem applied to the multiplication by an inner function leads to a special decomposition of the Hardy space. In this paper we obtain norm estimates for componentwise projections associated with this decomposition. An application to operators similar to a contraction is given.