Related papers: Real Variable Things in Bergman Theory

The $p-$Bergman kernel $K_p(\cdot)$ is shown to be of $C^{1,1/2}$ for $1<p<\infty$. An unexpected relation between the off-diagonal $p-$Bergman kernel $K_p(\cdot,z)$ and certain weighted $L^2$ Bergman kernel is given for $1\le p\le 2$. As…

The Bergman theory of domains $\{ |{z_{1} |^{\gamma}} < |{z_{2}} | < 1 \}$ in $\mathbb{C}^2$ is studied for certain values of $\gamma$, including all positive integers. For such $\gamma$, we obtain a closed form expression for the Bergman…

For $d\geq 2$ and $\frac{2d+2}{d+2} < p < \infty $, we prove a strict Faber-Krahn type inequality for the first eigenvalue $\lambda _1(\Omega )$ of the $p$-Laplace operator on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ (with…

For any open hyperbolic Riemann surface $X$, the Bergman kernel $K$, the logarithmic capacity $c_{\beta}$, and the analytic capacity $c_{B}$ satisfy the inequality chain $\pi K \geq c^2_{\beta} \geq c^2_B$; moreover, equality holds at a…

Let $\Omega\subset {\mathbb C}^n$ be a bounded domain with the hyperconvexity index $\alpha(\Omega)>0$. Let $\varrho$ be the relative extremal function of a fixed closed ball in $\Omega$ and set $\mu:=|\varrho|(1+|\log|\varrho||)^{-1}$,…

We use the Suita conjecture (now a theorem) to prove that for any domain $\Omega \subset \mathbb{C}$ its Bergman kernel $K(\cdot, \cdot)$ satisfies $K(z_0, z_0) = \hbox{Volume}(\Omega)^{-1}$ for some $z_0 \in \Omega$ if and only if $\Omega$…

In this paper, we give a complete real-variable theory of local variable Hardy spaces. First, we present various real-variable characterizations in terms of several local maximal functions. Next, the new atomic and the finite atomic…

We show that the $p-$Bergman kernel $K_p(z)$ on a bounded domain $\Omega$ is of locally $C^{1,1}$ for $p\geq1$.The proof is based on the locally Lipschitz continuity of the off-diagonal $p-$Bergman kernel $K_p(\zeta,z)$ for fixed $\zeta\in…

On a pseudoconvex Reinhardt domain $\Omega\subset\mathbb{C}^n$ the $p$-Bergman space $A^p(\Omega)$ admits a canonical basis of monomials indexed by a subset $S_p(\Omega)\subset\mathbb{Z}^n$. The corresponding $p$-Monomial Basis Kernel (or…

In this paper we attempt to develop a general $p-$Bergman theory on bounded domains in $\mathbb C^n$. To indicate the basic difference between $L^p$ and $L^2$ cases, we show that the $p-$Bergman kernel $K_p(z)$ is not real-analytic on some…

Let L be an ample holomorphic line bundle over a compact complex Hermitian manifold X. Any fixed smooth Hermitian metric on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k:th tensor power…

We prove a rigidity theorem for the Bergman metric on Hartogs domains over bounded homogeneous domains. Let $\Omega\subset \mathbb C^n$ be a bounded homogeneous domain, let $K_\Omega$ denote its Bergman kernel, and consider $$…

We prove a lower bound on the eigenvalues $\lambda_k$, $k\in\mathbb{N}$, of the Dirichlet Laplacian of a bounded domain $\Omega\subset\mathbb{R}^n$ of volume $V$: $$ \lambda_k \geq C_n\bigg( \delta\frac{k}{V}\bigg)^{2/n} $$ where $\delta$…

Let $\Omega$ be a convex domain in $\mathbb{C}^n$ and $\varphi$ a convex function on $\Omega$. We prove that $\log{K_{\Omega,\varphi}(z)}$ is a convex function (might be identically $-\infty$) on $\Omega$, where $K_{\Omega,\varphi}$ is the…

In the setting of tube domains over symmetric cones, $T_\Omega$, we study the characterization of the positive Borel measures $\mu$ for which the Hardy space $H^p$ is continuously embedded into the Lebesgue space $L^q (T_\Omega, d\mu)$,…

We prove that in variable exponent spaces $L^{p(\cdot)}(\Omega)$, where $p(\cdot)$ satisfies the log-condition and $\Omega$ is a bounded domain in $\mathbf R^n$ with the property that $\mathbf R^n \backslash \bar{\Omega}$ has the cone…

For any bounded convex domain $\Omega$ with $C^{2}$ boundary in $\mathbb{C}^{n}$, we show that there exist positive constants $C_{1}$ and $C_{2}$ such that \[ C_{1}\sqrt{\dfrac{K\left(w,w\right)}{\delta\left(w\right)}}\leq\left\Vert…

Let $(X,\omega)$ be a compact K\"{a}hler manifold. Let $(L,h)$ be a hermitian holomorphic line bundle over $X$, such that $\Theta_{L,h}\geq -\varepsilon\omega$ for a small $\varepsilon>0$, $E$ be a holomorphic line bundle over $X$. For…

Let $M$ be a closed complex submanifold in ${\mathbb C}^N$ with the complete K\"ahler metric induced by the Euclidean metric. Several finiteness theorems on the $L^p$ Bergman space of holomorphic sections of a given Hermitian line bundle…

We give a H\"ormander type $L^2-$estimate for the $\bar{\partial}-$equation with respect to the measure $\delta_\Omega^{-\alpha}dV$, $\alpha<1$, on any bounded pseudoconvex domain with $C^2-$boundary. Several applications to the function…