Related papers: Homogeneous Subsets of a Lipschitz Graph and the C…

We prove the corona theorem for domains whose boundary lies in certain smooth quasicircles. These curves, which are not necessarily Dini-smooth, are defined by quasiconformal mappings whose complex dilatation verifies certain conditions.…

Let $D\subset\Co$ be a bounded domain, whose boundary $B$ consists of $k$ simple closed continuous curves and $H^{\infty}(D)$ be the algebra of bounded analytic functions on $D$. We prove the matrix-valued corona theorem for matrices with…

For a wide class of domains $G\subset\mathbb C^d$ including balls and polydisks we prove the density of their canonical image in the spectrum of $H^\infty(G)$. This Corona Theorem is proved first in its abstract version for certain uniform…

We study classes of domains in $\mathbb{R}^{d+1},\ d \geq 2$ with sufficiently flat boundaries that admit a decomposition or covering of bounded overlap by Lipschitz graph domains with controlled total surface area. This study is motivated…

Using a description of the spectrum of bidual algebra $A^{**}$ of a uniform algebra $A$ we obtain abstract corona theorem for certain uniform algebras. It asserts the density of a specific Gleason part in the spectrum of an $H^\infty$ --…

This note concerns low-dimensional intrinsic Lipschitz graphs, in the sense of Franchi, Serapioni, and Serra Cassano, in the Heisenberg group $\mathbb{H}^n$, $n\in \mathbb{N}$. For $1\leq k\leq n$, we show that every intrinsic $L$-Lipschitz…

In this paper continuing our work started in our earlier papers we prove the corona theorem for the algebra of bounded holomorphic functions defined on an unbranched covering of a Caratheodory hyperbolic Riemann surface of finite type.

A graph $H$ is said to be positive if the homomorphism density $t_H(G)$ is non-negative for all weighted graphs $G$. The positive graph conjecture proposes a characterisation of such graphs, saying that a graph is positive if and only if it…

We obtain estimates in the corona theorem for the algebra of analytic functions in the unit disc whose nth derivative is bounded, and its subalgebras defined by the boundary continuity of the nth derivative. The corona theorem for such…

Let $f$ be a Lipschitz map from a subset $A$ of a stratified group to a Banach homogeneous group. We show that directional derivatives of $f$ act as homogeneous homomorphisms at density points of $A$ outside a $\sigma$-porous set. At…

In this paper, we introduce a generalization of corona of graphs. This construction generalizes the generalized corona of graphs (consequently, the corona of graphs), the cluster of graphs, the corona-vertex subdivision graph of graphs and…

The main result of the present paper is a Rademacher-type theorem for intrinsic Lipschitz graphs of codimension $k\leq n$ in sub-Riemannian Heisenberg groups $\mathbb H^n$. For the purpose of proving such a result we settle several related…

We prove that a homogeneous bounded domain admits a Berezin quantization.

We call a subset $K$ of $\mathbb C$ \emph{biholomorphically homogeneous} if for any two points $p,q\in K$ there exists a neighborhood $U$ of $p$ and a biholomorphism $\psi:U\to \psi(U)\subset \mathbb C$ such that $\psi(p)=q$ and $\psi(K\cap…

Let $K$ be a square Cantor set, i.e. the Cartesian product $K=E\times E$ of two linear Cantor sets. Let $\delta_n$ denote the proportion of the intervals removed in the $n$th stage of the construction of $E$. It is shown that if…

We provide a novel approach to approximate bounded Lipschitz domains via a sequence of smooth, bounded domains. The flexibility of our method allows either inner or outer approximations of Lipschitz domains which also possess weakly defined…

Assume that $A$ is a closed linear operator defined on all of a Hilbert space $H$. Then $A$ is bounded. A new short proof of this classical theorem is given on the basis of the uniform boundedness principle. The proof can be easily extended…

In this paper, for a graph G and a family of partitions P of vertex neighborhoods of G, we define the general corona G \circ P of G. Among several properties of this new operation, we focus on application general coronas to a new kind of…

On a non-compact, smooth, connected, boundaryless, complete Riemannian manifold $(M,g)$, one can define its ideal boundary by rays (or equivalently, Busemann functions). From the viewpoint of Mather theory, boundary elements could be…

We call a bipartite graph {\it homogeneous} if every finite partial automorphism which respects left and right can be extended to a total automorphism. A $(\kappa,{\lambda} )$ bipartite graph is a bipartite graph with left side of size…