Related papers: A note on the squeezing function

We consider weighted $L^p$-Hardy inequalities involving the distance to the boundary of a domain in the $n$-dimensional Euclidean space with nonempty boundary. Using criticality theory, we give an alternative proof of the following result…

We present a new application of the squeezing function $s_D$, using which one may detect when a given bounded pseudoconvex domain $D\varsubsetneq \mathbb{C}^n$, $n\geq 2$, is not biholomorphic to any product domain. One of the ingredients…

For an analytic family $\{f_t\}_{t\in\mathbb{D}^*}$ on the unit punctured disk that meromorphically degenerates at the origin, we show that its limiting measure on an snc model is given by the push forward of the canonical measure attached…

Let $f:\mathbb{D}\to\mathbb{C}$ be a bounded analytic function. A set $K\subset\mathbb{D}$ which contains the point $1$ in its boundary is called a convergence set for $f$ at $1$ if $f(z)$ converges to some value $\zeta$ as $z\to1$ with…

For a domain $D \subset \mathbb C^n$, the relationship between the squeezing function and the Fridman invariants is clarified. Furthermore, localization properties of these functions are obtained. As applications, some known results…

Let U be the open unit disc in C and let B be the open unit ball in C^2. We prove that every discrete subset of B is contained in the range f(U) of a complete, proper holomorphic embedding f:U-->B. Here the completeness of f means that for…

We show uniqueness for overdetermined elliptic problems defined on topological disks $\Omega$ with $C^2$ boundary, i.e., positive solutions $u$ to $\Delta u + f(u)=0$ in $\Omega \subset (M^2,g)$ so that $u = 0$ and $\frac{\partial…

Let $\phi$ be a real-valued smooth function on $\mathbf{C}$ satisfying $0 \le \Delta \phi \le M$ for some $M \ge 0$. We consider the space of all holomorphic functions which are square-integrable with respect to the measure $e^{-\phi(z)}…

Consider an open, bounded set $\Omega\subset \mathbb{C}$, a positive integer $k$ and a compact $\mathcal{K}\subset \Omega$ of cardinality strictly greater than $k$. We prove that, for any function $f$ which is holomorphic in $\overline…

Let $\Omega$ be a pseudoconvex domain in $\mathbb C^n$ satisfying an $f$-property for some function $f$. We show that the Bergman metric associated to $\Omega$ has the lower bound $\tilde g(\delta_\Omega(z)^{-1})$ where $\delta_\Omega(z)$…

A holomorphic function f on a simply connected domain {\Omega} is said to possess a universal Taylor series about a point in {\Omega} if the partial sums of that series approximate arbitrary polynomials on arbitrary compacta K outside…

We consider a bounded open subset $\Omega$ of ${\mathbb{R}}^n$ of class $C^{1,\alpha}$ for some $\alpha\in]0,1[$ and we solve the Neumann problem for the Helmholtz equation both in $\Omega$ and in the exterior of $\Omega$. We look for…

For a domain $\Omega\subset\mathbb R^n$, we introduce the concept of a uniformly $C^m$ defining function. We characterize uniformly $C^m$ defining functions in terms of the signed distance function for the boundary and provide a large class…

Given a control region $\Omega$ on a compact Riemannian manifold $M$, we consider the heat equation with a source term $g$ localized in $Omega$. It is known that any initial data in $L^2(M)$ can be stirred to 0 in an arbitrarily small time…

Let $(M,g)$ be a Riemannian manifold with a distinguished point $O$ and assume that the geodesic distance $d$ from $O$ is an isoparametric function. Let $\Omega\subset M$ be a bounded domain, with $O \in \Omega$, and consider the problem…

For \Omega a C^{2}-smooth domain, and a positive bounded continuous map a \in C(\Omega), we prove existence of a minimizer of the functional u \mapsto $\int_{\Omega} a|Du| over the space BV(\Omega) of functions of bounded variation with…

Let U be the closed unit disc in C and let p be a point on the unit circle. Let f be a continuous function on U which extends holomorphically from each circle contained in U and centered at the origin, and from each circle contained in U…

We define the notion of a thick open set $\Omega$ in a Euclidean space and show that a local Hardy-Littlewood inequality holds in $L^p(\Omega)$, $p \in (1, \infty]$. We then establish pointwise and $L^p(\Omega)$ convergence for families of…

Let $\Omega$ be a domain in $\Ri^d$ with boundary $\Gamma$ and let $d_\Gamma$ denote the Euclidean distance to $\Gamma$. Further let $H=-\divv(C\nabla)$ where $C=(\,c_{kl}\,)>0$ with $c_{kl}=c_{lk}$ are real, bounded, Lipschitz continuous…

Let $\Omega$ be homogeneous of degree zero, have vanishing moment of order one on the unit sphere $\mathbb {S}^{d-1}$($d\ge 2$). In this paper, our object of investigation is the following rough non-standard singular integral operator…