On complete pseudoconvex Reinhardt domains in ${ℂ}^2$, we show that there is no nonzero Hankel operator with anti-holomorphic symbol that is Hilbert-Schmidt. In the proof, we explicitly use the pseudoconvexity property of the domain. We also present two examples of unbounded non-pseudoconvex domains in ${ℂ}^2$ that admit nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols. In the first example the Bergman space is finite dimensional. However, in the second example the Bergman space is infinite...

We consider a class of unbounded nonhyperbolic complete Reinhardt domains $$D_{n,m,k}^{\mu ,p,s}:=\left\{(z,w_1,\cdots ,w_m)\in {ℂ}^n\times {ℂ}^{k_1}\times \cdots \times {ℂ}^{k_m}:\frac{\parallel w_1{\parallel }^{2p_1}}{{\mathrm{e}}^{-{\mu }_1{\parallel z\parallel }^s}}+\cdots +\frac{\parallel w_m{\parallel }^{2p_m}}{{\mathrm{e}}^{-{\mu }_m{\parallel z\parallel }^s}}<1\right\},$$ where $s$, $p_1,\cdots ,p_m$, ${\mu }_1,\cdots ,{\mu }_m$ are positive real numbers and $n$, $k_1,\cdots ,k_m$ are positive integers. We show that if a Hankel operator with anti-holomorphic symbol is Hilbert-Schmidt on the Bergman space $A^2\left(D_{n,m,k}^{\mu ,p,s}\right)$, then it must be zero. This gives an example of high dimensional unbounded complete Reinhardt domain that does not admit nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols.

We introduce an alternative proof of the existence of certain Ck barrier maps, with polynomial explosion of the derivatives, on weakly pseudoconvex domains in Cn. Barriers of this sort have been constructed very recently by J. Michel and M.-C. Shaw, and have various applications. In our paper, the adaptation of Hörmander's L2 techniques to suitable vector-valued functions allows us to give a very simple approach of the problem and to improve some aspects of the result of Michel and Shaw, regarding...

It seems impossible to extend the boundary value theory of Hardy spaces to Bergman spaces since there is no boundary value for a function in a Bergman space in general. In this article we provide a new idea to show what is the correct version of Bergman spaces by demonstrating the extension to Bergman spaces of a result of Hardy-Littlewood in Hardy spaces, which characterizes the Hölder class of boundary values for a function from Hardy spaces in the unit disc in terms of the growth of its derivative....

This work is an introduction to anisotropic spaces of holomorphic functions, which have $\omega$-weight and are generalizations of Bloch spaces on a unit ball. We describe the holomorphic Bloch space in terms of the corresponding $L_{\omega }^{\infty }$ space. We establish a description of ${\left(A^p\left(\omega \right)\right)}^{*}$ via the Bloch classes for all $0<p\le 1$.

We introduce a decomposition of holomorphic functions on Fréchet spaces which reduces to the Taylor series expansion in the case of Banach spaces and to the monomial expansion in the case of Fréchet nuclear spaces with basis. We apply this decomposition to obtain examples of Fréchet spaces E for which the τ_{ω} and τ_{δ} topologies on H(E) coincide. Our result includes, with simplified proofs, the main known results-Banach spaces with an unconditional basis and Fréchet nuclear spaces with DN [2,...