For a domain D ⊂ ℂ the Kobayashi-Royden ϰ and Hahn h pseudometrics are equal iff D is simply connected. Overholt showed that for $D\subset {ℂ}^n$, n ≥ 3, we have $h_D\equiv {\varkappa }_D$. Let D₁, D₂ ⊂ ℂ. The aim of this paper is to show that $h_{D₁\times D₂}$ iff at least one of D₁, D₂ is simply connected or biholomorphic to ℂ 0. In particular, there are domains D ⊂ ℂ² for which $h_D\not\equiv {\varkappa }_D$.

We present new sharp embedding theorems for mixed-norm analytic spaces in pseudoconvex domains with smooth boundary. New related sharp results in minimal bounded homogeneous domains in higher dimension are also provided. Last domains we consider are domains which are direct generalizations of the well-studied so-called bounded symmetric domains in ${ℂ}^n.$ Our results were known before only in the very particular case of domains of such type in the unit ball. As in the unit ball case, all our...

For a domain $\Omega \subset {ℂ}^n$ let $H\left(\Omega \right)$ be the holomorphic functions on $\Omega$ and for any $k\in \mathbb{N}$ let $A^k\left(\Omega \right)=H\left(\Omega \right)\cap C^k\left(\overline{\Omega }\right)$. Denote by ${𝒜}_D^k\left(\Omega \right)$ the set of functions $f\Omega \to [0,\infty )$ with the property that there exists a sequence of functions $f_j\in A^k\left(\Omega \right)$ such that $\left\{\right|f_j\left|\right\}$ is a nonincreasing sequence and such that $f\left(z\right)={lim}_{j\to \infty }\left|f_j\left(z\right)\right|$. By ${𝒜}_I^k\left(\Omega \right)$ denote the set of functions $f\Omega \to (0,\infty )$ with the property that there exists a sequence of functions $f_j\in A^k\left(\Omega \right)$ such that $\left\{\right|f_j\left|\right\}$ is a nondecreasing sequence and such that $f\left(z\right)={lim}_{j\to \infty }\left|f_j\left(z\right)\right|$. Let $k\in \mathbb{N}$ and let ${\Omega }_1$ and ${\Omega }_2$ be bounded $A^k$-domains of holomorphy in ${ℂ}^{m_1}$ and ${ℂ}^{m_2}$ respectively. Let $g_1\in {𝒜}_D^k\left({\Omega }_1\right)$, $g_2\in {𝒜}_I^k\left({\Omega }_1\right)$ and $h\in {𝒜}_D^k\left({\Omega }_2\right)\cap {𝒜}_I^k\left({\Omega }_2\right)$. We prove...

Let $D$ be a domain in $C^2$. Given $w\in C$, set $D_w=\left\{z\in C\mid \left(z,w\right)\in D\right\}$. If $f$ is a holomorphic and square-integrable function in $D$, then the set $E\left(D,f\right)$ of all $w$ such that $f(\cdot ,w)$ is not square-integrable in $D_w$ has measure zero. We call this set the exceptional set for $f$. In this Note we prove that whenever $0<r<1$ there exists a holomorphic square-integrable function $f$ in the unit ball $B$ in $C^2$ such that $E\left(B,f\right)$ is the circle $C\left(0,r\right)=\left\{z\in C\mid \left|z\right|=r\right\}$.

Let $B$ be the open unit ball of a Banach space $E$, and let $f:B\to B$ be a holomorphic map with $f\left(0\right)=0$. In this paper, we discuss a condition whereby $f$ is a linear isometry on $E$.