Pavel Shvartsman
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.
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For each $p>1$ and each positive integer $m$ we use divided differences to give intrinsic characterizations of the restriction of the Sobolev space $W^m_p(R)$ to an arbitrary closed subset of the real line.
For each $p>n$ we use local oscillations and doubling measures to give intrinsic characterizations of the restriction of the Sobolev space $W_p^1(R^n)$ to an arbitrary closed subset of $R^n$.
We study totally bounded subsets in weighted variable exponent amalgam and Sobolev spaces. Moreover, this paper includes several detailed generalized results of some compactness criterions in these spaces.
For each $p>2$ we give intrinsic characterizations of the restriction of the homogeneous Sobolev space $L^1_p(R^2)$ to an arbitrary finite subset $E$ of $R^2$. The trace criterion is expressed in terms of certain weighted oscillations of the second order with respect to a measure generated by the Menger curvature of triangles with vertices in $E$.
Given $p \in (1,\infty)$, let $(\operatorname{X},\operatorname{d},μ)$ be a metric measure space with uniformly locally doubling measure $μ$ supporting a weak local $(1,p)$-Poincaré inequality. For each $θ\in [0,p)$, we characterize the trace space of the Sobolev $W^{1}_{p}(\operatorname{X})$-space to lower codimension-$θ$ content regular closed subsets $S \subset \operatorname{X}$. In particular, if the space $(\operatorname{X},\operatorname{d},μ)$ is Ahlfors $Q$-regular for some $Q \geq 1$ and $p \in (Q,\infty)$, then we get an intrinsic description of the trace-space of the Sobolev $W^{1}_{p…
In this paper we build up a criteria for fractional Orlicz-Sobolev extension and imbedding domains on Ahlfors $n$-regular domains.
We prove that functions in the Hajłasz-Sobolev space $M^{1,s}$ on an $s$-Ahlfors regular metric space are uniformly continuous when $s\le 1$.
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 study properties of the classical fractional Sobolev spaces (or Bessel potential spaces) on non-Lipschitz subsets of $\mathbb{R}^n$. We investigate the extent to which the properties of these spaces, and the relations between them, that hold in the well-studied case of a Lipschitz open set, generalise to non-Lipschitz cases. Our motivation is to develop the functional analytic framework in which to formulate and analyse integral equations on non-Lipschitz sets. In particular we consider an application to boundary integral equations for wave scattering by planar screens that are non-Lipschit…
We study Sobolev type spaces defined in terms of sharp maximal functions on Ahlfors regular subsets of the Euclidean space and the relation between these spaces and traces of classical Sobolev spaces.
Let $S \subset \mathbb{R}^{n}$ be an arbitrary nonempty compact set such that the $d$-Hausdorff content $\mathcal{H}^{d}_{\infty}(S) > 0$ for some $d \in (0,n]$. For each $p \in (\max\{1,n-d\},n]$, an almost sharp intrinsic description of the trace space $W_{p}^{1}(\mathbb{R}^{n})|_{S}$ of the Sobolev space $W_{p}^{1}(\mathbb{R}^{n})$ to the set $S$ is obtained. Furthermore, for each $p \in (\max\{1,n-d\},n]$ and $\varepsilon \in (0, \min\{p-(n-d),p-1\})$, new bounded linear extension operators from the trace space $W_{p}^{1}(\mathbb{R}^{n})|_{S}$ into the space $W_{p-\varepsilon}^{1}(\mathbb{…
We are concerned here with Sobolev-type spaces of vector-valued functions. For an open subset $Ω\subset\mathbb{R}^N$ and a Banach space $V$, we compare the classical Sobolev space $W^{1,p}(Ω, V)$ with the so-called Sobolev-Reshetnyak space $R^{1,p}(Ω, V)$. We see that, in general, $W^{1,p}(Ω, V)$ is a closed subspace of $R^{1,p}(Ω, V)$. As a main result, we obtain that $W^{1,p}(Ω, V)=R^{1,p}(Ω, V)$ if, and only if, the Banach space $V$ has the Radon-Nikodým property
Optimal stability of regularized spectral differentiation in Sobolev spaces
该研究针对带噪数据下Sobolev空间函数的稳定谱微分问题，引入容许傅里叶乘子，证明正则化微分算子达极小极大最优稳定性速率，扩展了相关结果并验证了多种乘子的表现。
Sobolev regularity for a class of local fractional new maximal operators
本文研究局部新极大算子及其分数版本的正则性，给出逐点梯度估计，证明其在Sobolev空间上的有界性，并推广到零边值Sobolev空间。
This paper concerns the following question: given a subset $E$ of $\mathbb{R}^n$ with empty interior and an integrability parameter $1<p<\infty$, what is the maximal regularity $s\in\mathbb{R}$ for which there exists a non-zero distribution in the Bessel potential Sobolev space $H^{s,p}(\mathbb{R}^n)$ that is supported in $E$? For sets of zero Lebesgue measure we apply well-known results on set capacities from potential theory to characterise the maximal regularity in terms of the Hausdorff dimension of $E$, sharpening previous results. Furthermore, we provide a full classification of all poss…
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…
Let $S \subset \mathbb{R}^{n}$ be a~closed set such that for some $d \in [0,n]$ and $\varepsilon > 0$ the~$d$-Hausdorff content $\mathcal{H}^{d}_{\infty}(S \cap Q(x,r)) \geq \varepsilon r^{d}$ for all cubes~$Q(x,r)$ centered in~$x \in S$ with side length $2r \in (0,2]$. For every $p \in (1,\infty)$, denote by $W_{p}^{1}(\mathbb{R}^{n})$ the classical Sobolev space on $\mathbb{R}^{n}$. We give an~intrinsic characterization of the restriction $W_{p}^{1}(\mathbb{R}^{n})|_{S}$ of the space $W_{p}^{1}(\mathbb{R}^{n})$ to~the set $S$ provided that $p > \max\{1,n-d\}$. Furthermore, we prove the exist…
We prove that if the Sobolev embedding $M^{1,p}(X)\hookrightarrow L^q(X)$ holds for some $q>p\geq 1$ in a metric measure space $(X,d,μ),$ then a constant $C$ exists such that $μ(B(x,r))\geq Cr^n$ for all $x\in X$ and all $0<r\leq 1,$ where $\frac{1}{p}-\frac{1}{q}=\frac{1}{n}.$ This was proved in \cite{Gor17} assuming a doubling condition on the measure $μ.$
We give a simple characterization of chaos for weighted composition $C_0$-semigroups on $L^p_ρ(Ω)$ for an open interval $Ω\subseteq\mathbb{R}$. Moreover, we characterize chaos for these classes of $C_0$-semigroups on the closed subspace $W^{1,p}_*(Ω)$ of the Sobolev space $W^{1,p}(Ω)$ for a bounded interval $Ω\subset\mathbb{R}$. These characterizations simplify previously obtained characterization of chaos for these classes of $C_0$-semigroups.
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 $p>2$.
For each $p>n$ we use local oscillations to give intrinsic characterizations of the trace of the Sobolev space $W^1_p(Ω)$ to the boundary of an arbitrary domain $Ω\subset R^n$.
We investigate two density questions for Sobolev, Besov and Triebel--Lizorkin spaces on rough sets. Our main results, stated in the simplest Sobolev space setting, are that: (i) for an open set $Ω\subset\mathbb R^n$, $\mathcal{D}(Ω)$ is dense in $\{u\in H^s(\mathbb R^n):{\rm supp}\, u\subset \overlineΩ\}$ whenever $\partialΩ$ has zero Lebesgue measure and $Ω$ is "thick" (in the sense of Triebel); and (ii) for a $d$-set $Γ\subset\mathbb R^n$ ($0<d<n$), $\{u\in H^{s_1}(\mathbb R^n):{\rm supp}\, u\subset Γ\}$ is dense in $\{u\in H^{s_2}(\mathbb R^n):{\rm supp}\, u\subset Γ\}$ whenever $-\frac{n-d…
Existence of extremal functions in higher-order affine Sobolev inequalities
本文证明了高阶仿射Sobolev不等式中极值函数的存在性，采用集中紧致性方法，在整数或分数正则性空间中进行证明，所用工具在任意正则性空间中具有独立价值。
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.
Given the standard Gaussian measure $γ$ on the countable product of lines $\mathbb{R}^{\infty}$ and a probability measure $g \cdot γ$ absolutely continuous with respect to $γ$, we consider the optimal transportation $T(x) = x + \nabla φ(x)$ of $g \cdot γ$ to $γ$. Assume that the function $|\nabla g|^2/g$ is $γ$-integrable. We prove that the function $φ$ is regular in a certain Sobolev-type sense and satisfies the classical change of variables formula $g = {\det}_2(I + D^2 φ) \exp \bigl(\mathcal{L} φ- 1/2 |\nabla φ|^2 \bigr)$. We also establish sufficient conditions for the existence of third o…
On the equivalence of Sobolev norms in infinite dimensions
该研究针对无穷维希尔伯特空间的真开子集，利用在配备高斯测度的有限维开凸集上证明的与维度无关的高阶索伯列夫范数估计，得到非退化高斯测度下的范数等价性，还解决了马利亚万-索伯列夫范数等价性的遗留情形，是相关领域的首个真开子集等价定理。
We show that given a homeomorphism $f:G\rightarrowΩ$ where $G$ is a open subset of $\mathbb{R}^2$ and $Ω$ is a open subset of a $2$-Ahlfors regular metric measure space supporting a weak $(1,1)$-Poincaré inequality, it holds $f\in BV_{\operatorname{loc}}(G,Ω)$ if and only $f^{-1}\in BV_{\operatorname{loc}}(Ω,G)$. Further if $f$ satisfies the Luzin N and N$^{-1}$ conditions then $f\in W^{1,1}_{\operatorname{loc}}(G,Ω)$ if and only if $f^{-1}\in W^{1,1}_{\operatorname{loc}}(Ω,G)$.
We give some necessary conditions and sufficient conditions for the compactness of the embedding of Sobolev spaces $W^{1,p}(Ω,w) \to L^p(Ω,w),$ where $w$ is some weight on a domain $Ω\subset \Real^n$.
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.
It is common that a Sobolev space defined on $\mathbb{R}^m$ has a non-compact embedding into an $L^p$-space, but it has subspaces for which this embedding becomes compact. There are three well known cases of such subspaces, the Rellich compactness, for a subspace of functions on a bounded domain (or an unbounded domain, sufficiently thin at infinity), the Strauss compactness, for a subspace of radially symmetric functions in $\mathbb{R}^m$, and the weighted Sobolev spaces. Known generalizations of Strauss compactness include subspaces of functions with block-radial symmetry, subspaces of funct…
We prove non-autonomous maximal $L^p$-regularity results on UMD spaces replacing the common Hölder assumption by a weaker fractional Sobolev regularity in time. This generalizes recent Hilbert space results by Dier and Zacher. In particular, on $L^q(Ω)$ we obtain maximal $L^p$-regularity for $p \ge 2$ and elliptic operators in divergence form with uniform $VMO$-modulus in space and $W^{α,p}$-regularity for $α> \frac{1}{2}$ in time.
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.
On Sobolev and Besov Spaces of Hybrid Regularity
本文研究单位 n-立方体上 Sobolev 空间 H^q 中函数关于张量积或双曲小波基的非线性逼近，推导 Jackson 和 Bernstein 不等式，证明逼近类包含混合正则性 Besov 空间，并表明张量积小波逼近能力不弱于经典小波。
Density properties for fractional Musielak-Sobolev spaces
本文定义了一个新的分数阶 Musielak-Sobolev 空间，并证明了其中光滑紧支撑函数的密度性质。
We prove a result on the fractional Sobolev regularity of composition of paths of low fractional Sobolev regularity with functions of bounded variation. The result relies on the notion of variability, proposed by us in the previous article [43, arXiv:2003.11698]. Here we work under relaxed hypotheses, formulated in terms of Sobolev norms, and we can allow discontinuous paths, which is new. The result applies to typical realizations of certain Gaussian or Lévy processes, and we use it to show the existence of Stieltjes type integrals involving compositions.
In this article, we fully characterize the measurable Gaussian processes $(U(x))_{x\in\mathcal{D}}$ whose sample paths lie in the Sobolev space of integer order $W^{m,p}(\mathcal{D}),\ m\in\mathbb{N}_0,\ 1 <p<+\infty$, where $\mathcal{D}$ is an arbitrary open set. The result is phrased in terms of a form of Sobolev regularity of the covariance function on the diagonal. This is then linked to the existence of suitable Mercer or otherwise nuclear decompositions of the integral operators associated to the covariance function and its cross-derivatives. In the Hilbert case $p=2$, additional links a…
We establish trace theorems for function spaces defined on general Ahlfors regular metric spaces $Z$. The results cover the Triebel-Lizorkin spaces and the Besov spaces for smoothness indices $s<1,$ as well as the first order Hajłasz-Sobolev space $M^{1,p}(Z)$. They generalize the classical results from the Euclidean setting, since the traces of these function spaces onto any closed Ahlfors regular subset $F \subset Z$ are Besov spaces defined intrinsically on $F$. Our method employs the definitions of the function spaces via hyperbolic fillings of the underlying metric space.
We give a quantitative characterization of traces on the boundary of Sobolev maps in $\dot{W}^{1,p}(\mathcal M, \mathcal N)$, where $\mathcal{M}$ and $\mathcal{N}$ are compact Riemannian manifolds, $\partial \mathcal{M} \neq \emptyset$: the Borel-measurable maps $u\colon \partial \mathcal M \to \mathcal{N}$ that are the trace of a map $U\in \dot{W}^{1,p}(\mathcal M, \mathcal{N})$ are characterized as the maps for which there exists an extension energy density $w \colon \partial \mathcal{M} \to [0,\infty]$ that controls the Sobolev energy of extensions from $\lfloor p - 1 \rfloor$-dimensional s…
A theory of Sobolev inequalities in arbitrary open sets of Euclidean space is established. Boundary regularity of domains is replaced with information on boundary traces of trial functions and of their derivatives up to some explicit minimal order. The relevant Sobolev inequalities involve constants independent of the geometry of the domain, and exhibit the same critical exponents as in the classical inequalities on regular domains. Our approach relies upon new representation formulas for Sobolev functions, and on ensuing pointwise estimates which hold in any open set.
Let $X_N$ be a symmetric $N\times N$ random matrix whose $\sqrt{N}$-scaled centered entries are uniformly square integrable. We prove that if the entries of $X_N$ can be partitioned into independent subsets each of size $o(\log N)$, then the empirical eigenvalue distribution of $X_N$ converges weakly to its mean in probability. This significantly extends the best previously known results on convergence of eigenvalues for matrices with correlated entries (where the partition subsets are blocks and of size $O(1)$.) we prove this result be developing a new log-Sobolev inequality, generalizing the…
This paper studies the inclusions between different Sobolev-Lorentz spaces $W^{1,(p,q)}(Ω)$ defined on open sets $Ω\subset {\mathbf{R}^n},$ where $n \ge 1$ is an integer, $1<p<\infty$ and $1 \le q \le \infty.$ We prove that if $1 \le q<r \le \infty,$ then $W^{1,(p,q)}(Ω)$ is strictly included in $W^{1,(p,r)}(Ω).$ We show that although $H^{1,(p,\infty)}(Ω) \subsetneq W^{1,(p,\infty)}(Ω)$ where $Ω\subset {\mathbf{R}}^n$ is open and $n \ge 1,$ there exists a partial converse. Namely, we show that if a function $u$ in $W^{1,(p,\infty)}(Ω), n \ge 1$ is such that $u$ and its distributional gradient …
We show that independent and uniformly distributed sampling points are as good as optimal sampling points for the approximation of functions from the Sobolev space $W_p^s(Ω)$ on bounded convex domains $Ω\subset \mathbb{R}^d$ in the $L_q$-norm if $q<p$. More generally, we characterize the quality of arbitrary sampling points $P\subset Ω$ via the $L_γ(Ω)$-norm of the distance function $\rm{dist}(\cdot,P)$, where $γ=s(1/q-1/p)^{-1}$ if $q<p$ and $γ=\infty$ if $q\ge p$. This improves upon previous characterizations based on the covering radius of $P$.
In this paper we consider an abstract Wiener space $(X,γ,H)$ and an open subset $O\subseteq X$ which satisfies suitable assumptions. For every $p\in(1,+\infty)$ we define the Sobolev space $W_{0}^{1,p}(O,γ)$ as the closure of Lipschitz continuous functions which support with positive distance from $\partial O$ with respect to the natural Sobolev norm, and we show that under the assumptions on $O$ the space $W_{0}^{1,p}(O,γ)$ can be characterized as the space of functions in $W^{1,p}(O,γ)$ which have null trace at the boundary $\partial O$, or, equivalently, as the space of functions defined on…
We prove a Hardy-Sobolev-Maz'ya inequality for arbitrary domains Ω\subset\R^N with a constant depending only on the dimension N\geq 3. In particular, for convex domains this settles a conjecture by Filippas, Maz'ya and Tertikas. As an application we derive Hardy-Lieb-Thirring inequalities for eigenvalues of Schrödinger operators on domains.
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