Related papers: Sobolev homeomorphisms and Brennan's conjecture

Let $n \geq 4$ and let $\Omega$ be a bounded hyperconvex domain in $\mathbb{C}^{n}$. Let $\varphi$ be a negative exhaustive smooth plurisubharmonic function on $\Omega$. We show that any holomorphic function defined on a connected open…

In this paper, we use Takeuchi's Theorem to show that for every Lipschitz pseudoconvex domain $\Omega$ in $\mathbb{CP}^n$ there exists a Lipschitz defining function $\rho$ and an exponent $0<\eta<1$ such that $-(-\rho)^\eta$ is strictly…

If $\Omega$ is a simply connected domain in $\overline{{\mathbb C}}$ then, according to the Ahlfors-Gehring theorem, $\Omega$ is a quasidisk if and only if there exists a sufficient condition for the univalence of holomorphic functions in…

We prove that any metric surface (that is, metric space homeomorphic to a 2-manifold with boundary) with locally finite Hausdorff 2-measure is the Gromov-Hausdorff limit of polyhedral surfaces with controlled geometry. We use this result,…

We study the density of polynomials in $H^2(\Omega,e^{-\varphi})$, the space of square integrable holomorphic functions in a bounded domain $\Omega$ in $\mathbb{C}$, where $\varphi$ is a subharmonic function. In particular, we prove that…

We first prove that all the functions in L 2 whose directional derivative is in L 2 have a directional trace on the boundary of any open bounded domain, without assumptions on its regularity. This enables us to define the omnidirectional…

In this paper, we study the holomorphicity of totally geodesic Kobayashi isometric embeddings between bounded symmetric domains. First we show that for a $C^1$-smooth totally geodesic Kobayashi isometric embedding $f\colon \Omega\to\Omega'$…

Let $\Omega \subset \mathbb{R}^{n+1}$, $n \geq 1$, be an open and connected set. Set $\mathcal{T}_n$ to be the set of points $\xi \in \partial \Omega$ so that there exists an approximate tangent $n$-plane for $\partial\Omega$ at $\xi$ and…

The Fefferman--Szeg\H{o} metric \(g_{\operatorname{FS}}^\Omega\) on a \(C^\infty\)-smooth bounded strongly pseudoconvex domain \(\Omega\subset\mathbb C^n\) is an invariant metric defined via the Fefferman surface measure. For this metric,…

We consider a smooth boundary b\Omega which is q-convex in the sense that its Levi-form has positive trace on every complex q-plane. We prove that b\Omega is tangent of infinite order to the complexification of each of its submanifolds…

Using quasiconformal mappings, we prove that any Riemann surface of finite connectivity and finite genus is conformally equivalent to an intrinsic circle domain U in a compact Riemann surface S. This means that each connected component B of…

We introduce a general $L_p$-solvability result for the Poisson equation in non-smooth domains $\Omega\subset \mathbb{R}^d$, with the zero Dirichlet boundary condition. Our sole assumption on the domain $\Omega$ is the Hardy inequality:…

We construct a bounded $C^{1}$ domain $\Omega$ in $R^{n}$ for which the $H^{3/2}$ regularity for the Dirichlet and Neumann problems for the Laplacian cannot be improved, that is, there exists $f$ in $C^{\infty}(\overline\Omega)$ such that…

Let $\Omega$ be a domain of $\mathbb R^n$ with $n\ge 2$ and $p(\cdot)$ be a local Lipschitz funcion in $\Omega$ with $1<p(x)<\infty$ in $\Omega$. We build up an interior quantitative second order Sobolev regularity for the normalized…

For a measure space $\Omega$ we extend the theory of Orlicz spaces generated by an even convex integrand $\varphi \colon \Omega \times X \to \left[ 0, \infty \right]$ to the case when the range Banach space $X$ is arbitrary. Besides…

The current article stems from our study on the asymptotic behavior of holomorphic isometric embeddings of the Poincar\'e disk into bounded symmetric domains. As a first result we prove that any holomorphic curve exiting the boundary of a…

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^2$ with Lipschitz boundary or a bounded convex domain in $\mathbb{C}^n$ and $\phi\in C(\overline{\Omega})$ such that $H_{\phi}$ is compact on $A^2(\Omega)$. Then $\phi\circ f$ is…

Let $\Omega\subset R^n$ be a bounded convex domain with $n\ge2$. Suppose that $A$ is uniformly elliptic and belongs to $W^{1,n}$ when $n\ge 3$ or $W^{1,q}$ for some $q>2$ when $n=2$. For $1<p<\infty$, we build up a global second order…

In this paper we study integral estimates of derivatives of conformal mappings $\varphi:\mathbb D\to\Omega$ of the unit disc $\mathbb D\subset\mathbb C$ onto bounded domains $\Omega$ that satisfy the Ahlfors condition. These integral…

We prove precise growth and cancellation estimates for the Szeg\H{o} kernel of an unbounded model domain $\Omega\subset\mathbb{C}^2$ under the assumption that ${\rm b}\Omega$ satisfies a uniform finite-type hypothesis. Such domains have…