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...

Let f be an analytic function on the unit disk . We define a generalized Hilbert-type operator ${}_{a,b}$ by ${}_{a,b}\left(f\right)\left(z\right)=\Gamma (a+1)/\Gamma (b+1){\int }_0^1\left(f\left(t\right){(1-t)}^b\right)/\left({(1-tz)}^{a+1}\right)dt$, where a and b are non-negative real numbers. In particular, for a = b = β, ${}_{a,b}$ becomes the generalized Hilbert operator ${}_{\beta }$, and β = 0 gives the classical Hilbert operator . In this article, we find conditions on a and b such that ${}_{a,b}$ is bounded on Dirichlet-type spaces $S^p$, 0 < p < 2, and on Bergman spaces $A^p$, 2 < p < ∞. Also we find an upper bound for the norm of the operator ${}_{a,b}$....

Let $\mu$ be a finite positive measure on the unit disk and let $j\ge 1$ be an integer. D. Suárez (2015) gave some conditions for a generalized Toeplitz operator $T_{\mu }^{\left(j\right)}$ to be bounded or compact. We first give a necessary and sufficient condition for $T_{\mu }^{\left(j\right)}$ to be in the Schatten $p$-class for $1\le p<\infty$ on the Bergman space $A^2$, and then give a sufficient condition for $T_{\mu }^{\left(j\right)}$ to be in the Schatten $p$-class $(0<p<1)$ on $A^2$. We also discuss the generalized Toeplitz operators with general bounded symbols. If $\varphi \in L^{\infty }(D,\mathrm{d}A)$ and $1<p<\infty$, we define the generalized...

Let $D_j$ be a bounded hyperconvex domain in ${ℂ}^{n_j}$ and set $D=D₁\times \cdots \times D_s$, j=1,...,s, s ≥ 3. Also let ${\Omega }_{\pi }$ be the image of D under the proper holomorphic map π. We characterize those continuous functions $f:\partial {\Omega }_{\pi }\to ℝ$ that can be extended to a real-valued pluriharmonic function in ${\Omega }_{\pi }$.

Motivated by the relationship between the area of the image of the unit disk under a holomorphic mapping $h$ and that of $zh$, we study various $L^2$ norms for $T_{\varphi }\left(h\right)$, where $T_{\varphi }$ is the Toeplitz operator with symbol $\varphi$. In Theorem , given polynomials $p$ and $q$ we find a symbol $\varphi$ such that $T_{\varphi }\left(p\right)=q$. We extend some of our results to the polydisc.

Let $\mu$ be a positive Borel measure on the complex plane ${ℂ}^n$ and let $j=(j_1,\cdots ,j_n)$ with $j_i\in \mathbb{N}$. We study the generalized Toeplitz operators $T_{\mu }^{\left(j\right)}$ on the Fock space $F_{\alpha }^2$. We prove that $T_{\mu }^{\left(j\right)}$ is bounded (or compact) on $F_{\alpha }^2$ if and only if $\mu$ is a Fock-Carleson measure (or vanishing Fock-Carleson measure). Furthermore, we give a necessary and sufficient condition for $T_{\mu }^{\left(j\right)}$ to be in the Schatten $p$-class for $1\le p<\infty$.

We give a concrete description of complex symmetric monomial Toeplitz operators $T_{z^p{\overline{z}}^q}$ on the weighted Bergman space $A^2\left(\Omega \right)$, where $\Omega$ denotes the unit ball or the unit polydisk. We provide a necessary condition for $T_{z^p{\overline{z}}^q}$ to be complex symmetric. When $p,q\in {\mathbb{N}}^2$, we prove that $T_{z^p{\overline{z}}^q}$ is complex symmetric on $A^2\left(\Omega \right)$ if and only if $p_1=q_2$ and $p_2=q_1$. Moreover, we completely characterize when monomial Toeplitz operators $T_{z^p{\overline{z}}^q}$ on $A^2\left({𝔻}_n\right)$ are $J_U$-symmetric with the $n\times n$ symmetric unitary matrix $U$.

The objective of this paper is to obtain sharp upper bound for the function $f$ for the second Hankel determinant $|a_2{a}_4-a_3^2|$, when it belongs to the class of functions whose derivative has a positive real part of order $\alpha$ $(0\le \alpha <1)$, denoted by $RT\left(\alpha \right)$. Further, an upper bound for the inverse function of $f$ for the nonlinear functional (also called the second Hankel functional), denoted by $|t_2{t}_4-t_3^2|$, was determined when it belongs to the same class of functions, using Toeplitz determinants.

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...

Let $E$ denote a holomorphic bundle with fiber $D$ and with basis $B$. Both $D$ and $B$ are assumed to be Stein. For $D$ a Reinhardt bounded domain of dimension $d=2$ or $3$, we give a necessary and sufficient condition on $D$ for the existence of a non-Stein such $E$ (Theorem $1$); for $d=2$, we give necessary and sufficient criteria for $E$ to be Stein (Theorem $2$). For $D$ a Reinhardt bounded domain of any dimension not intersecting any coordinate hyperplane, we give a sufficient criterion for $E$ to be Stein (Theorem...

We consider a convexity notion for complex spaces $X$ with respect to a holomorphic line bundle $L$ over $X$. This definition has been introduced by Grauert and, when $L$ is analytically trivial, we recover the standard holomorphic convexity. In this circle of ideas, we prove the counterpart of the classical Remmert’s reduction result for holomorphically convex spaces. In the same vein, we show that if $H^0(X,L)$ separates each point of $X$, then $X$ can be realized as a Riemann domain over the complex projective...

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.