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.

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....