Mapping properties of the Bergman projections on elementary Reinhardt domains

The elementary Reinhardt domain associated to multi-index k = (k 1, …, k n ) ∈ ℤ n is defined by ℋ ( k ) : = { z ∈ D n : z k is defined and | z k | < 1 } . $$\mathcal{H}(\mathbf{k}):=\{z\in\mathbb{D}^n: z^{\mathbf{k}}\ \text{is defined and}\ |z^{\mathbf{k}}|<1\}.$$ In this paper, we study the mapping properties of the associated Bergman projection on L p spaces and L p Sobolev spaces of order ≥ 1.

The convergence classes for analytic functions in the Reinhardt domains

Let $L^0$ be the class of positive increasing on $[1,+\infty)$ functions $l$ such that $l((1+o(1))x)=(1+o(1))l(x)$ $(x\to +\infty)$. We assume that $\alpha$ is a concave function such that $\alpha(e^x)\in L^0$ and function $\beta\in L^0$ such that $\displaystyle\int_1^{+\infty}\frac{\alpha(x)}{\beta(x)}dx<+\infty$. In the article it is proved the following theorem: if $\displaystyle f(z)=\sum_{\|n\|=0}^{+\infty}a_nz_n$, $z\in \mathbb{C}^p$, is analytic function in the bounded Reinhard domain $G\subset \mathbb{C}^p$, then the condition $\displaystyle \int\limits_{R_0}^{1} \frac{\alpha(\ln^{+} M_{G}(R,f))} {(1-R)^2\beta(1/(1-R))}d\,R<+\infty,$ $M_{G}(R,f)=\sup\{|F(Rz)|\colon z\in G\},$ yields that $$\sum_{k=0}^{+\infty}(\alpha(k)-\alpha(k-1)) \beta_1\left({k}/{\ln^{+}|A_k|}\right)<+\infty,$$ $$\beta_1(x)= \int\limits_{x}^{+\infty} \frac{dt}{\beta(t)},\quad A_k=\max\{|a_n|\colon\|n\|=k\}. $$

Remarks on quasi-Reinhardt domains

We present some fundamental properties of quasi-Reinhardt domains, in connection with Kobayashi hyperbolicity, minimal domains and representative domains. We also study proper holomorphic correspondences between quasi-Reinhardt domains.

Norm convergence of partial sums of H1 functions

A classical observation of Riesz says that truncations of a general [Formula: see text] in the Hardy space [Formula: see text] do not converge in [Formula: see text]. A substitute positive result is proved: these partial sums always converge in the Bergman norm [Formula: see text]. The result is extended to complete Reinhardt domains in [Formula: see text]. A new proof of the failure of [Formula: see text] convergence is also given.