Following the line of Ouyang et al. (1998) to study the ${}_p$ spaces of holomorphic functions in the unit ball of ℂⁿ, we present in this paper several results and relations among ${}_p\left(ₙ\right)$, the α-Bloch, the Dirichlet ${}_p$ and the little ${}_{p,0}$ spaces.

We show that the holomorphic functions on polysectors whose derivatives remain bounded on proper subpolysectors are precisely those strongly asymptotically developable in the sense of Majima. This fact allows us to solve two Borel-Ritt type interpolation problems from a functional-analytic viewpoint.

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 that the...

Let $Z$ be a germ of a reduced analytic space of pure dimension. We provide an analytic proof of the uniform Briançon-Skoda theorem for the local ring ${𝒪}_Z$; a result which was previously proved by Huneke by algebraic methods. For ideals with few generators we also get much sharper results.

We study the integral representation of solutions to the Cauchy problem for a differential equation with constant coefficients. The Cauchy data and the right-hand of the equation are given by entire functions on a complex hyperplane of ${ℂ}^{n+1}$. The Borel transformation of power series and residue theory are used as the main methods of investigation.

We apply the Rudin idea to represent the Bergman kernel of the Hartogs domain as the sum of a series of weighted Bergman functions in the study of the dependence of this kernel on deformations of the domain. We prove that the Bergman function depends smoothly on the function defining the Hartogs domain.

We establish an inversion formula for the M. M. Djrbashian A. H. Karapetyan integral transform (cf. [6]) on the Siegel domain ${\Omega }_n=\zeta \in {ℂ}^n:\varrho \left(\zeta \right)>0$, $\varrho \left(\zeta \right)=Im\left({\zeta }_1\right)-{\left|{\zeta }^{\text{'}}\right|}^2$. We build a family of Kähler metrics of constant holomorphic curvature whose potentials are the ${\varrho }^{\alpha }$-Bergman kernels, α > -1, (in the sense of Z. Pasternak-Winiarski [20] of ${\Omega }_n$. We build an anti-holomorphic embedding of ${\Omega }_n$ in the complex projective Hilbert space $ℂ\mathbb{P}\left(H_{\alpha }^2\left({\Omega }_n\right)\right)$ and study (in connection with work by A. Odzijewicz [18] the corresponding transition probability amplitudes....