We investigate the Bergman kernel function for the intersection of two complex ellipsoids $\{(z,w_1,w_2)\in {ℂ}^{n+2}:|z_1{|}^2+\cdots +|z_n{|}^2+|w_1{|}^q<1,|z_1{|}^2+\cdots +|z_n{|}^2+|w_2{|}^r<1\}.$ We also compute the kernel function for $\{(z_1,w_1,w_2)\in {ℂ}^3:|z_1{|}^{2/n}+|w_1{|}^q<1,|z_1{|}^{2/n}+|w_2{|}^r<1\}$ and show deflation type identity between these two domains. Moreover in the case that $q=r=2$ we express the Bergman kernel in terms of the Jacobi polynomials. The explicit formulas of the Bergman kernel function for these domains enables us to investigate whether the Bergman kernel has zeros or not. This kind of problem is called a Lu Qi-Keng problem.

We compute the Bergman kernel functions of the unbounded domains ${\Omega }_p=(z^{\text{'}},z)\in ℂ²:z>p\left(z^{\text{'}}\right)$, where $p\left(z^{\text{'}}\right)={\left|z^{\text{'}}\right|}^{\alpha }/\alpha$. It is also shown that these kernel functions have no zeros in ${\Omega }_p$. We use a method from harmonic analysis to reduce the computation of the 2-dimensional case to the problem of finding the kernel function of a weighted space of entire functions in one complex variable.

In this note we compute the Bergman kernel of the unit ball with respect to the smallest norm in ${ℂ}^n$ that extends the euclidean norm in ${ℝ}^n$ and give some applications.

We consider a certain class of unbounded nonhyperbolic Reinhardt domains which is called the twisted Fock-Bargmann-Hartogs domains. By showing Cartan's linearity theorem for our unbounded nonhyperbolic domains, we give a complete description of the automorphism groups of twisted Fock-Bargmann-Hartogs domains.

Let $\Omega$ be a complex analytic manifold of dimension $n$ with a hermitian metric and $C^{\infty }$ boundary, and let $\square =\overline{\partial }{\overline{\partial }}^{*}+{\overline{\partial }}^{*}\overline{\partial }$ be the self-adjoint $\overline{\partial }$-Neumann operator on the space $L_{0,q}^2\left(\Omega \right)$ of forms of type $(0,q)$. If the Levi form of $\partial \Omega$ has everywhere at least $n-q$ positive or at least $q+1$ negative eigenvalues, it is well known that $\mathrm{Ker}$$\square$ has finite dimension and that the range of $\square$ is the orthogonal complement. In...

It is shown that the weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with $C^{1+\epsilon }$-smooth boundary. On the other hand, it is proved that the weak converse to the Suita conjecture holds for any finitely connected planar domain.

We give some explicit values of the constants $C_1$ and $C_2$ in the inequality $C_1/sin\left(\frac{\pi }{p}\right)\le {\left|P\right|}_p\le C_2/sin\left(\frac{\pi }{p}\right)$ where ${\left|P\right|}_p$ denotes the norm of the Bergman projection on the $L^p$ space.