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 study the extension problem for germs of holomorphic isometries $f:(D;x_0)\to (\Omega ;f\left(x_0\right))$ up to normalizing constants between bounded domains in Euclidean spaces equipped with Bergman metrics $d{s}_D^2$ on $D$ and $d{s}_{\Omega }^2$ on $\Omega$. Our main focus is on boundary extension for pairs of bounded domains $(D,\Omega )$ such that the Bergman kernel $K_D(z,w)$ extends meromorphically in $(z,\overline{w})$ to a neighborhood of $\overline{D}\times D$, and such that the analogous statement holds true for the Bergman kernel $K_{\Omega }(\varsigma ,\xi )$ on $\Omega$. Assuming that $(D;d{s}_D^2)$ and $(\Omega ;d{s}_{\Omega }^2)$ are complete Kähler manifolds, we prove that...

We study the spaces $H_{\mu }\left(\Omega \right)=f:\Omega \to ℂholomorphic:{\int }_0^R{\int }_0^{2\pi }\left|f\left(r{e}^{i\phi }\right)\right|d\phi d\mu \left(r\right)<\infty$ where Ω is a disc with radius R and μ is a given probability measure on [0,R[. We show that, depending on μ, $H_{\mu }\left(\Omega \right)$ is either isomorphic to l₁ or to ${\left(\sum \oplus Aₙ\right)}_{\left(1\right)}$. Here Aₙ is the space of all polynomials of degree ≤ n endowed with the L₁-norm on the unit sphere.

Let $\Omega$ an open set in ${\mathbf{C}}^4$ near $z_0\in \partial \Omega$, $\lambda$ a suitable holomorphic function near $z_0$. If we know that we can solve the following problem (see [M. Derridj, Annali. Sci. Norm. Pisa, Série IV, vol. IX (1981)]) : $\bar{\partial }u=\lambda f$, ($f$ is a $(0,1)$ form, $\bar{\partial }$ closed in $U\left(z_0\right)$ in $U\left(z_0\right)$ with supp$\left(u\right)\subset \bar{\Omega }\cap U(z_0)$, then we deduce an extension result for $C.R.$ functions on $\partial \Omega \cap U\left(z_0\right)$, as holomorphic fonctions in $\Omega \cap V\left(z_0\right)$.

Sia $F$ un endomorfismo olomorfo della palla unitaria aperta $B_n$ di ${𝐂}^{𝐧}$. In questa nota proviamo che certe ipotesi minimali, relative al comportamento di $F$ su un orociclo e vicino ad un punto del bordo, assicurano che $F$ è un automorfismo olomorfo di $B_n$.

Let ${}_{\infty }\left(B_X\right)$ be the Banach space of all bounded and continuous functions on the closed unit ball $B_X$ of a complex Banach space X and holomorphic on the open unit ball, with sup norm, and let ${}_u\left(B_X\right)$ be the subspace of ${}_{\infty }\left(B_X\right)$ of those functions which are uniformly continuous on $B_X$. A subset $B\subset B_X$ is a boundary for ${}_{\infty }\left(B_X\right)$ if $\parallel f\parallel =su{p}_{x\in B}\left|f\left(x\right)\right|$ for every $f{\in }_{\infty }\left(B_X\right)$. We prove that for X = d(w,1) (the Lorentz sequence space) and X = C₁(H), the trace class operators, there is a minimal closed boundary for ${}_{\infty }\left(B_X\right)$. On the other hand, for X = , the Schreier...

We study a class of typical Hartogs domains which is called a generalized Fock-Bargmann-Hartogs domain $D_{n,m}^p\left(\mu \right)$. The generalized Fock-Bargmann-Hartogs domain is defined by inequality ${\mathrm{e}}^{{\mu \parallel z\parallel }^2}{\sum }_{j=1}^m{\left|{\omega }_j\right|}^{2p}<1$, where $(z,\omega )\in {ℂ}^n\times {ℂ}^m$. In this paper, we will establish a rigidity of its holomorphic automorphism group. Our results imply that a holomorphic self-mapping of the generalized Fock-Bargmann-Hartogs domain $D_{n,m}^p\left(\mu \right)$ becomes a holomorphic automorphism if and only if it keeps the function ${\sum }_{j=1}^m{\left|{\omega }_j\right|}^{2p}{\mathrm{e}}^{{\mu \parallel z\parallel }^2}$ invariant.

The purpose of this paper is to prove that proper holomorphic self-mappings of the minimal ball are biholomorphic. The proof uses the scaling technique applied at a singular point and relies on the fact that a proper holomorphic mapping f: D → Ω with branch locus $V_f$ is factored by automorphisms if and only if $f_{*}\left(\pi ₁(D\setminus f^{-1}\left(f\left(V_f\right)\right),x)\right)$ is a normal subgroup of $\pi ₁(\Omega \setminus f\left(V_f\right),b)$ for some $b\in \Omega \setminus f\left(V_f\right)$ and $x\in f^{-1}\left(b\right)$.

The aim of this paper is to establish the equivalence between the non-pluripolarity of a compact set in a complex space and the property $\left(L{B}^{\infty }\right)$ for the dual space of the space of germs of holomorphic functions on that compact set.

We prove that the linearization of a germ of holomorphic map of the type $F_{\lambda }\left(z\right)=\lambda (z+O\left(z^2\right))$ has a ${𝒞}^1$-holomorphic dependence on the multiplier $\lambda$. ${𝒞}^1$-holomorphic functions are ${𝒞}^1$-Whitney smooth functions, defined on compact subsets and which belong to the kernel of the $\overline{\partial }$ operator. The linearization is analytic for $\left|\lambda \right|\ne 1$ and the unit circle ${𝕊}^1$ appears as a natural boundary (because of resonances,roots of unity). However the linearization is still defined at most points of ${𝕊}^1$, namely those points which lie “far enough...

For a given direction $𝐛\in {ℂ}^n\setminus \left\{\mathbf{0}\right\}$ we study non-homogeneous directional linear higher-order equations whose all coefficients belong to a class of joint continuous functions which are holomorphic on intersection of all directional slices with a unit ball. Conditions are established providing boundedness of $L$-index in the direction with a positive continuous function $L$ satisfying some behavior conditions in the unit ball. The provided conditions concern every solution belonging to the same class...

Let H(B) denote the space of all holomorphic functions on the unit ball B of ℂⁿ. Let φ be a holomorphic self-map of B and g ∈ H(B) such that g(0) = 0. We study the integral-type operator $C_{\phi }^{g}f\left(z\right)={\int }_0^1\Re f\left(\phi \left(tz\right)\right)g\left(tz\right)dt/t$, f ∈ H(B). The boundedness and compactness of $C_{\phi }^g$ from Privalov spaces to Bloch-type spaces and little Bloch-type spaces are studied

Sia $E$ uno spazio $DF$ riflessivo e sia $K$ un compatto di $E$. Si dimostra che lo spazio dei germi olomorfi su $K$, con la topologia naturale, è un limite induttivo regolare e quasi completo purché lo spazio dei germi olomorfi all'origine sia un limite induttivo regolare.

Let Ω be either the complex plane or the open unit disc. We completely determine the isomorphism classes of $Hv=f:\Omega \to ℂholomorphic:su{p}_{z\in \Omega }\left|f\left(z\right)\right|v\left(z\right)<\infty$ and investigate some isomorphism classes of $hv=f:\Omega \to ℂharmonic:su{p}_{z\in \Omega }\left|f\left(z\right)\right|v\left(z\right)<\infty$ where v is a given radial weight function. Our main results show that, without any further condition on v, there are only two possibilities for Hv, namely either $Hv\sim l_{\infty }$ or $Hv\sim H_{\infty }$, and at least two possibilities for hv, again $hv\sim l_{\infty }$ and $hv\sim H_{\infty }$. We also discuss many new examples of weights.

Let $X$ denote either ${\mathrm{ℂ\mathbb{P}}}^m$ or ${ℂ}^m$. We study certain analytic properties of the space ${ℰ}^n(X,gp)$ of ordered geometrically generic $n$-point configurations in $X$. This space consists of all $q=(q_1,\cdots ,q_n)\in X^n$ such that no $m+1$ of the points $q_1,\cdots ,q_n$ belong to a hyperplane in $X$. In particular, we show that for a big enough $n$ any holomorphic map $f:{ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)\to {ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)$ commuting with the natural action of the symmetric group $𝐒\left(n\right)$ in ${ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)$ is of the form $f\left(q\right)=\tau \left(q\right)q=(\tau \left(q\right)q_1,\cdots ,\tau \left(q\right)q_n)$, $q\in {ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)$, where $\tau :{ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)\to \mathrm{𝐏𝐒𝐋}(m+1,ℂ)$ is an $𝐒\left(n\right)$-invariant holomorphic map. A similar result holds true for mappings of the configuration...

We give in ${ℝ}^6$ a real analytic almost complex structure $J$, a real analytic hypersurface $M$ and a vector $v$ in the Levi null set at $0$ of $M$, such that there is no germ of $J$-holomorphic disc $\gamma$ included in $M$ with $\gamma \left(0\right)=0$ and $\frac{\partial \gamma }{\partial x}\left(0\right)=v$, although the Levi form of $M$ has constant rank. Then for any hypersurface $M$ and any complex structure $J$, we give sufficient conditions under which there exists such a germ of disc.

Let X be an analytic set defined by polynomials whose coefficients $a₁,...,a_s$ are holomorphic functions. We formulate conditions on sequences $a_{1,\nu },...,a_{s,\nu }$ of holomorphic functions converging locally uniformly to $a₁,...,a_s$, respectively, such that the sequence $X_{\nu }$ of sets obtained by replacing $a_j$’s by $a_{j,\nu }$’s in the polynomials converges to X.