Pavel Shvartsman
We describe a class of Sobolev $W^k_p$-extension domains $Ω\subset R^n$ determined by a certain inner subhyperbolic metric in $Ω$. This enables us to characterize finitely connected Sobolev $W^1_p$-extension domains in $R^2$ for each $p>2$.
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In this paper, we study the Sobolev extension property of Lp-quasidisks which are the generalizations of the classical quasidisks. After that, we also find some applications of their Sobolev extension property.
In this paper we build up a criteria for fractional Orlicz-Sobolev extension and imbedding domains on Ahlfors $n$-regular domains.
We provide an extension operator for weighted Sobolev spaces on bounded polyhedral cones $K$ involving a mixture of weights, which measure the distance to the vertex and the edges of the cone, respectively. Our results are based on Stein's extension operator for Sobolev spaces.
We characterize the restrictions of first order Sobolev functions to regular subsets of a homogeneous metric space and prove the existence of the corresponding linear extension operator.
We show that the extension results by Mazya and Poborchi for polynomial planar cusps can be realized via composition operators generated by reflections.
We deduce an extension theorem for the so-called Sobolev-Grand Lebesgue Spaces defined on the suitable subsets of the whole finite-dimensional Euclidean space, and estimate the norms of correspondent extension operator, which may be choosed as linear.
There are known trace and extension theorems relating functions in a weighted Sobolev space in a domain U to functions in a Besov space on the boundary bU. We extend these theorems to the case where the Sobolev exponent p is less than one by modifying our Sobolev spaces to consider averages of functions in Whitney balls. Averaged Sobolev spaces are also of interest in the applications in the case where p>1, and so we also provide trace and extension results in that case. Finally, we provide some comparable results for Neumann traces and extensions.
Sobolev Versus Homogeneous Sobolev II
本文研究了Sobolev延拓域与齐次Sobolev延拓域的关系，构造了特定指数范围内一个非齐次延拓域的实例。
We study extension operators on Sobolev spaces with decreasing integrability on the base of set functions associated with the operator norms. Sharp necessary conditions in the terms of the generalized density condition and the terms of weak equivalence of Euclidean and intrinsic metrics are given.
We prove that Burenkov's Extension Operator preserves Sobolev spaces built on general Morrey spaces, including classical Morrey spaces. The analysis concerns bounded and unbounded open sets with Lipschitz boundaries in the n-dimensional Euclidean space.
In this paper, we study the traces and the extensions for weighted Sobolev spaces on upper half spaces when the weights reach to the borderline cases. We first give a full characterization of the existence of trace spaces for these weighted Sobolev spaces, and then study the trace parts and the extension parts between the weighted Sobolev spaces and a new kind of Besov-type spaces (on hyperplanes) which are defined by using integral averages over selected layers of dyadic cubes.
We prove that Stein's Extension Operator preserves Sobolev-Morrey spaces, that is spaces of functions with weak derivatives in Morrey spaces. The analysis concerns classical and generalized Morrey spaces on bounded and unbounded domains with Lipschitz boundaries in the n-dimensional Euclidean space.
Trace and extension operators for fractional Sobolev spaces with variable exponent
本文研究变指数分数阶 Sobolev 空间的迹与延拓问题，证明在特定正则性条件下存在线性延拓算子。
Let $d$ be a metric on $R^n$ and let $C^{m,(d)}(R^n)$ be the space of $C^m$-function on $R^n$ whose partial derivatives of order $m$ belong to the space $Lip(R^n;d)$. We show that the homogeneous Sobolev space $L^{m+1}_p(R^n),p>n,$ can be represented as a union of $C^{m,(d)}(R^n)$-spaces where $d$ belongs to a family of metrics on $R^n$ with certain "nice" properties. This enables us in several important cases to give intrinsic characterizations of the restrictions of Sobolev spaces to arbitrary closed subsets of $R^n$. In particular, we generalize the classical Whitney extension theorem for t…
The volume of the boundary of a Sobolev $(p,q)$-extension domain II
本文证明在参数满足 $1\leq q <p< \frac{qn}{(n-q)}$ 的条件下，有界 Sobolev $(p,q)$-延拓域的边界体积为零。
We prove the product of the Sobolev-extension domains is still a Sobolev-extension domain.
Sobolev Versus Homogeneous Sobolev Extension
本文研究了 Sobolev 延拓域与齐次 Sobolev 延拓域之间的关系，并在不同参数条件下给出了两类延拓域相互蕴含或存在反例的严格数学刻画。
We study the existence of an extension operator $Λ\colon W^{1,φ}(Ω)\to W^{1,ψ}(\mathbb{R}^n)$. We assume that $φ\in Φ_\mathrm{w}(Ω)$ has generalized Orlicz growth, $ψ\in Φ_\mathrm{w}(\mathbb{R}^n)$ is an extension of $φ$, and that $Ω\subset\mathbb{R}^n$ is an $(ε,δ)$-domain. Special cases include the classical constant exponent case, the Orlicz case, the variable exponent case, and the double phase case.
For each $p>1$ and each positive integer $m$ we give intrinsic characterizations of the restriction of the homogeneous Sobolev space $L^m_p(R)$ to an arbitrary closed subset $E$ of the real line. We show that the classical one dimensional Whitney extension operator is "universal" for the scale of $L^m_p(R)$ spaces in the following sense: for every $p\in(1,\infty]$ it provides almost optimal $L^m_p$-extensions of functions defined on $E$. The operator norm of this extension operator is bounded by a constant depending only on $m$. This enables us to prove several constructive $L^m_p$-extension c…
Sobolev extensions, interpolation inequalities and consequences
本文引入Lebesgue W^{1,p}扩展域概念，证明任何W^{1,p}扩展域（1<p<∞）都是Lebesgue W^{1,p}扩展域，并推导出类似全空间形式的Sobolev插值不等式，讨论其应用。
We construct whole-space extensions of functions in a fractional Sobolev space of order $s\in (0,1)$ and integrability $p\in (0,\infty)$ on an open set $O$ which vanish in a suitable sense on a portion $D$ of the boundary $\partial O$ of $O$. The set $O$ is supposed to satisfy the so-called interior thickness condition in $\partial O \setminus D$, which is much weaker than the global interior thickness condition. The proof works by means of a reduction to the case $D=\emptyset$ using a geometric construction.
Extension Operators for Fractional Sobolev Spaces on Lipschitz Submanifolds
本文研究了分数Sobolev空间在Lipschitz子流形上的扩展算子，通过局部坐标图表构造并明确了扩展领域大小对估计的影响，应用于几何简化等数值分析领域。
Sobolev extensions over Cantor-cuspidal graphs
本文针对由单位区间经典三分Cantor集距离的α次幂函数定义的Cantor尖点图，研究其上下区域的Sobolev延拓性质。
For each $m\ge 1$ and $p>2$ we characterize bounded simply connected Sobolev $L^m_p$-extension domains $Ω\subset R^2$. Our criterion is expressed in terms of certain intrinsic subhyperbolic metrics in $Ω$. Its proof is based on a series of results related to the existence of special chains of squares joining given points $x$ and $y$ in $Ω$. An important geometrical ingredient for obtaining these results is a new "Square Separation Theorem". It states that under certain natural assumptions on the relative positions of a point $x$ and a square $S\subsetΩ$ there exists a similar square $Q\subsetΩ…
Extension of Sobolev functions on balls in infinite dimensions
提出全新方法证明存在有界Sobolev延拓算子，解决了文献中的开放问题。
Estimates for Kolmogorov and linear widths of some weighted Sobolev classes on a John domain are obtained.
We show that the first order Sobolev spaces on cuspidal symmetric domains can be characterized via pointwise inequalities. In particular, they coincide with the Hajlasz-Sobolev spaces.
We introduce Herz-Sobolev spaces, which unify and generalize the classical Sobolev spaces. We will give a proof of the Sobolev-type embedding for these function spaces. All these results generalize the classical results on Sobolev spaces. Some remarks on Caffarelli--Kohn--Nirenberg inequality are given.
We discuss a few old results concerning embedding theorems for Campanato and Sobolev-Morrey spaces adapting the formulations to the case of domains of class $C^{0,γ}$, and we present more recent results concerning the extension of functions from Sobolev-Morrey spaces defined on those domains. As a corollary of the extension theorem we obtain an embedding theorem for Sobolev-Morrey spaces on arbitrary $C^{0,γ}$ domains.
For each $p>1$ and each positive integer $m$ we give intrinsic characterizations of the restriction of the Sobolev space $W^m_p(R)$ and homogeneous Sobolev space $L^m_p(R)$ to an arbitrary closed subset $E$ of the real line. In particular, we show that the classical one dimensional Whitney extension operator is "universal" for the scale of $L^m_p(R)$ spaces in the following sense: for every $p\in(1,\infty]$ it provides almost optimal $L^m_p$-extensions of functions defined on $E$. The operator norm of this extension operator is bounded by a constant depending only on $m$. This enables us to pr…
We prove an asymptotically sharp dimension upper-bound for the boundary of bounded simply-connected planar Sobolev $W^{1,p}$-extension domains via the weak mean porosity of the boundary. The sharpness of our estimate is shown by examples.
We prove a fractional version of the Hardy--Sobolev--Maz'ya inequality for arbitrary domains and $L^p$ norms with $p\geq 2$. This inequality combines the fractional Sobolev and the fractional Hardy inequality into a single inequality, while keeping the sharp constant in the Hardy inequality.
We prove an estimate on the Hausdorff-dimension of the set of two-sided boundary points of general Sobolev-extension domains on Euclidean spaces. We also present examples showing lower bounds on possible dimension estimates of this type.
Order estimates for Kolmogorov, Gelfand and linear widths of a weighted Sobolev class on a domain with a peak in a weighted Lebesgue space are obtained for some special weights.
Let $n\geq 2$ and $1\leq q<p<\fz$. We prove that if $Ω\subset\mathbb R^n$ is a Sobolev $(p, q)$-extension domain, with additional capacitory restrictions on boundary in the case $q\leq n-1$, $n>2$, then $|\partialΩ|=0$. In the case $1\leq q<n-1$, we give an example of a Sobolev $(p,q)$-extension domain with $|\partialΩ|>0$.
We establish point wise inequalities for Sobolev functions on a wider class of outward cuspidal domains. It is a generalization of an earlier result by the author and his collaborators
Let $L^m_p(R^n)$, $p\in [1,\infty]$, be the homogeneous Sobolev space, and let $E\subset R^n$ be a closed set. For each $p>n$ and each non-negative integer $m$ we give an intrinsic characterization of the restrictions to $E$ of $m$-jets generated by functions $F\in L^{m+1}_p(R^n)$. Our trace criterion is expressed in terms of variations of corresponding Taylor remainders of $m$-jets evaluated on a certain family of "well separated" two point subsets of $E$. For $p=\infty$ this result coincides with the classical Whitney-Glaeser extension theorem for $m$-jets. Our approach is based on a represe…
We provide a necessary condition on the regularity of domains for the optimal embeddings of first order (and higher order) Orlicz-Sobolev spaces into Orlicz spaces in the sense of \cite{Cia96} (and \cite{Cia06}).
We study extensions of sets and functions in general metric measure spaces. We show that an open set has the strong BV extension property if and only if it has the strong extension property for sets of finite perimeter. We also prove several implications between the strong BV extension property and extendability of two different non-equivalent versions of Sobolev $W^{1,1}$-spaces and show via examples that the remaining implications fail.
In this paper we present a method for determining whether there exists a largest Orlicz space $L^A(Ω)$ satisfying the Sobolev embedding $W^mL^A(Ω) \to Y(Ω)$ where $Y(Ω)$ stands for an arbitrary so-called Marcinkiewicz endpoint space. The tool developed in this work enables us to investigate the optimality of Orlicz domain spaces in Sobolev embeddings and also in Sobolev trace embeddings on domains $Ω$ in $\mathbb{R}^n$ with various regularity.
Local Sobolev-Poincare imbedding domains
本文刻画了局部Sobolev-Poincaré嵌入域：在$1\leq p\leq n$时等价于一致域（在弱连通假设下），在$n<p<\infty$时等价于$\alpha$-雪茄域，并由此与全局嵌入域等价。
Composition operators on Sobolev spaces with a metric measure space as a domain. II
本文刻画了在度量测度空间上的Sobolev空间中能够诱导有界复合算子的同胚映射。
In this paper we give an estimate for the Hausdorff dimension of the set of two-sided points of the boundary of bounded simply connected Sobolev $W^{1,p}$-extension domain for $1<p<2$. Sharpness of the estimate is shown by examples. We also prove the equivalence of different definitions of two-sided points.
Widths and entropy numbers of embeddings of Sobolev classes on a Hölder domain
本文研究了Hölder域上Sobolev类的熵数和宽度，改进了Besov的结果并给出了相关宽度的上界估计。
In this paper order estimates for the Kolmogorov widths of weighted Sobolev classes on a multi-dimensional domain are obtained. The classes are defined by conditions on the highest-order derivatives and the derivative of order zero.
Let $Ω$ be a John domain, and let $Γ\subset \partial Ω$ be an $h$-set. For some functions $h$ and some weight functions depending on distance from $Γ$, embedding theorems for a weighted Sobolev class is obtained.
In this note, we prove that the boundary of a $(W^{1, p}, BV)$-extension domain is of volume zero under the assumption that the domain $\boz$ is $1$-fat at almost every $x\in\partial\boz$. Especially, the boundary of any planar $(W^{1, p}, BV)$-extension domain is of volume zero.
We deal with Orlicz-Sobolev embeddings in open subsets of $\mathbb{R}^n$. A necessary and sufficient condition is established for the existence of an optimal, i.e. largest possible, Orlicz-Sobolev space continuously embedded into a given Orlicz space. Moreover, the optimal Orlicz-Sobolev space is exhibited whenever it exists. Parallel questions are addressed for Orlicz-Sobolev embeddings into Orlicz spaces with respect to a Frostman measure, and, in particular, for trace embeddings on the boundary.
The main result of the present paper, combined with earlier results of Hardt and Lin settles the extension problem for $W^{1,p}(\mathcal M, \mathcal N)$, where $\mathcal M$ and $\mathcal N$ are compact riemannian manfolds, $\mathcal M$ having non-empty smooth boundary and assuming moreover that $\mathcal N$ is simply connected. The main question which is studied is the following: Given a map in the trace space $W^{1-\frac{1}{p}, p} (\partial \mathcal M, \mathcal N)$, does it possess an extension in $W^{1,p}(\mathcal M, \mathcal N)$? We show that the answer is negative in the case $\mathfrak p_…
We describe a procedure to introduce Sobolev spaces and the semigroup generated by the fractional Dirichlet Laplacian on an arbitrary domain of $\R^d$. In particular, the well-definedness of the spaces of both non-homogeneous and homogeneous type together with their duality properties, embeddings, and Gagliardo-Nirenberg inequalities will be discussed. We also show the continuity and the smoothing property of the semigroup.
A form of Sobolev inequalities for the symmetric gradient of vector-valued functions is proposed, which allows for arbitrary ground domains in $\mathbb R ^n$. In the relevant inequalities, boundary regularity of domains is replaced with information on boundary traces of trial functions. The inequalities so obtained exhibit the same exponents as in classical inequalities for the full gradient of Sobolev functions, in regular domains. Furthermore, they involve constants independent of the geometry of the domain, and hence yield novel results yet for smooth domains. Our approach relies upon a poi…
Optimal function spaces and Sobolev embeddings
本研究建立特定上确界算子有界性与Sobolev嵌入中函数空间最优性的等价关系，借助高阶Sobolev嵌入与等周不等式的关联给出范数显式计算方法，应用于多类区域嵌入且部分适用于乘积概率空间嵌入。
We prove that for a bounded domain $Ω\subset \mathbb R^n$ which is Gromov hyperbolic with respect to the quasihyperbolic metric, especially when $Ω$ is a finitely connected planar domain, the Sobolev space $W^{1,\,\infty}(Ω)$ is dense in $W^{1,\,p}(Ω)$ for any $1\le p<\infty$. Moreover if $Ω$ is also Jordan or quasiconvex, then $C^{\infty}(\mathbb R^n)$ is dense in $W^{1,\,p}(Ω)$ for $1\le p<\infty$.
Linear extension operators for Sobolev spaces on radially-symmetric binary trees
本文研究径向对称二叉树上叶子函数的最小加权 $W^{1,p}$ 延拓问题，证明存在范数仅依赖 $p$ 的线性算子，即关于由 $p$ 与权重确定的 Markov 核的调和延拓。
In complete metric measure spaces equipped with a doubling measure and supporting a weak Poincaré inequality, we investigate when a given Banach-valued Sobolev function defined on a subset satisfying a measure-density condition is the restriction of a Banach-valued Sobolev function defined on the whole space. We investigate the problem for Hajłasz- and Newton-Sobolev spaces, respectively. First, we show that Hajłasz-Sobolev extendability is independent of the target Banach spaces. We also show that every $c_0$-valued Newton-Sobolev extension set is a Banach-valued Newton-Sobolev extension set …
Fractional Sobolev-Poincare inequalities in irregular domains
本文研究了不规则区域中的分数阶 (q,p)-Sobolev-Poincare 不等式，在 s-John 区域和拟双曲边界条件区域中建立了本质 sharp 不等式，证明了其与通常导数结果的极限联系，并刻画了支撑该不等式的区域几何性质。
We study extensions of Sobolev and BV functions on infinite-dimensional domains. Along with some positive results we present a negative solution of the long-standing problem of existence of Sobolev extensions of functions in Gaussian Sobolev spaces from a convex domain to the whole space.
Hilbertian Hardy--Sobolev Spaces on Tube Domains over Convex Cones
本文引入了在凸锥上管域上的Hilbertian Hardy--Sobolev空间，从傅里叶分析角度发展其结构理论，推导了Paley-Wiener型表示，揭示内在傅里叶结构，并得到边界Sobolev空间的Hardy-Sobolev分解定理。
Zero-extension convergence and Sobolev spaces on changing domains
本文在不假设参考构型的前提下，将弱收敛与强收敛推广到底层区域同时变化的Sobolev函数序列，建立相应理论与紧性定理，并通过实例验证其有效性。
Given $f:\partial (-1,1)^n\to{\mathbb R}$, consider its radial extension $Tf(X):=f(X/\|X\|_{\infty})$, $\forall\, X\in [-1,1]^n\setminus\{0\}$. In "On some questions of topology for $S^1$-valued fractional Sobolev spaces" (RACSAM 2001), the first two authors (HB and PM) stated the following auxiliary result (Lemma D.1). If $0<s<1$, $1< p<\infty$ and $n\ge 2$ are such that $1<sp<n$, then $f\mapsto Tf$ is a bounded linear operator from $W^{s,p}(\partial (-1,1)^n)$ into $W^{s,p}((-1,1)^n)$. The proof of this result contained a flaw detected by the third author (IS). We present a correct proof. We…
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