arXiv subjects

Andrei V. Vasin

Publications and source records attributed to Andrei V. Vasin.

9 recordsLinked to original sources

arXiv2026

Composition operators between de Branges-Rovnyak and Hardy spaces

Let $d\ge 1$ and $φ: B_d\to\mathbb{D}$ be a holomorphic function, where $B_d$ denotes the open unit ball of $\mathbb{C}^d$ and $\mathbb{D} = B_1$. Let $b: \mathbb{D} \to \mathbb{D}$ be a holomorphic function and $\mathcal {H}(b)$ denote the corresponding de Branges-Rovnyak space. We show that compactness of the composition operator $C_φ$ from $\mathcal{H}(b)$ to the Hardy space $H^2(B_d)$ is related to natural restrictions on the Nevanlinna counting functions of the slice-functions $φ_ζ$, $ζ\in \partial B_d$.

math.CV↗

arXiv2026

Carleson families of cubes related to porous sets

Given a porous set $E\in \mathbb{R}^d$ and a dyadic lattice $\mathcal{D}$, we refine the Carleson packing condition and the sparseness property for the dyadic cover $\mathcal{D}_E=\{Q \in \mathcal{D}: \: Q \cap E \neq \varnothing\}$. We study the inverse problem, when a Carleson family $\mathcal{S} \subset \mathcal{D}$ generates the porous set $E$ such that $\mathcal{S} \subset \mathcal{D}_E$.

math.FA↗

arXiv2025

On harmonic approximation of Lipschitz functions on compacts in $\mathbb{R}^d$

Given a porous compact $K \subset \mathbb{R}^d$ and a continuity modulus $ω$, we prove a quantitative Jackson-Bernstein type theorem on harmonic approximation. That is, a function $f$ belongs to the class $\mathrm{Lip}_ω(K)$ if and only if $f$ can be approximated uniformly on $K$ with a rate of $ω(δ)$ by a function that is harmonic in the $δ$-neighborhood of $K$, provided the uniform estimate $ω(δ)/δ$ on the gradient holds.

math.FA↗

arXiv2024

A characterization of Calderón-Zygmund operators on RBMO

Let $\mathrm{RBMO}(μ) = \mathrm{RBMO}(\mathbb{R}^m, μ)$ denote the regular BMO space introduced by X. Tolsa for an $n$-dimensional finite positive measure on $\mathbb{R}^m$, $0<n \le m$. We characterize the bounded Calderón-Zygmund operators $T:\mathrm{RBMO}(μ) \to \mathrm{RBMO}(μ)$ in terms of the function $T1$.

math.FA↗

arXiv2023

Calderón-Zygmund operators on Zygmund spaces on domains

Given a bounded Lipschitz domain $D\subset \mathbb{R}^d$ and a Calderón-Zygmund operator $T$, we study the relations between smoothness properties of $\partial D$ and the boundedness of $T$ on the Zydmund space $\mathcal{C}_ω(D)$ defined for a general growth function $ω$. In the proof we obtain a T(P) theorem for the Zygmund spaces, when one checks boundedness not only of the characteristic function, but a finite collection of polynomials restricted to the domain. Also, a new form of extra cancellation property of the even Calderón-Zygmund operators in polynomial domains is stated.

math.FA↗

arXiv2022

A T(P) theorem for Zygmund spaces on domains

Let $D\subset \mathbb{R}^d$ be a bounded Lipschitz domain, $ω$ be a high order modulus of continuity and let $T$ be a convolution Calderón-Zygmund operator. We characterize the bounded restricted operators $T_D$ on the Zygmund space $\mathcal{C}_ω(D)$. The characterization is based on properties of $T_D P$ for appropriate polynomials $P$ restricted to $D$.

math.FA↗

arXiv2021

Calderón-Zygmund operators on RBMO

Let $μ$ be an $n$-dimensional finite positive measure on $\mathbb{R}^m$. We obtain a $T1$ condition sufficient for the boundedness of Calderón-Zygmund operators on $\textrm{RBMO}(μ)$, the regular BMO space of Tolsa.

math.CA↗

arXiv2017

T1 theorem for Campanato spaces on domains

Given a Lipschitz domain $D\subset \mathbb{R}^d,$ a Calderón-Zygmund operator $T$ and a modulus of continuity $ω(x),$ we solve a problem when the restricted operator $T_Df=T(fχ_D)χ_D$ sends the Campanato space $\mathcal{C}_ω(D)$ into itself. The solution is a T1 type sufficient and necessary condition for the characteristic function $χ_D$ of $D$: $$(Tχ_D)χ_D \in \mathcal{C}_{\tildeω}(D),$$ assumed $\tildeω(x)= ω(x)/\int_x^1 ω(t)dt/t.$ To check the hypotheses of T1 theorem we need extra restrictions on both the boundary of $D$ and the operator $T.$ It is proved that the restricted Calderón-Zygmund operator $T_D$ with the even kernel is bounded on $\mathcal{C}_ω(D),$ provided $D$ be $C^{1,\tildeω}-$smooth domain. This result is sharp.

math.FA↗

arXiv2016

Restricted Beurling transforms on Campanato spaces

Let $Ω\subset \mathbb{C}$ be a bounded domain with $\mathcal{C}^{1,ω}$-smooth boundary, where $ω$ is a Dini-smooth modulus of continuity. We prove that the restricted Beurling transform is bounded on the Campanato space $\mathrm{BMO}_ω(Ω)$.

math.CV↗