Pavel Shvartsman
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 $R^n$.
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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.
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.
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…
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
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 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 $p>2$.
Sobolev subspaces of nowhere bounded functions
本文证明在次临界Sobolev空间中存在闭无限维子空间，其非零元素无处有界或无处$L^q$（$q$大于Sobolev指数）。
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…
Improved Fractional Sobolev Embeddings on Closed Riemannian Manifolds under Isometric Group Actions
本文研究在等距群作用下闭合黎曼流形上的对称改进分数Sobolev嵌入，证明了G不变分数Sobolev空间嵌入到更高L^p空间，并探讨了有限轨道对称下的最优常数。
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.
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 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.
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 construct explicit examples of Frostman-type measures concentrated on arbitrary planar rectifiable curves of positive length. Based on such constructions we obtain for each $p \in (1,\infty)$ an exact description of the trace space of the first-order Sobolev space $W^{1}_{p}(\mathbb{R}^{2})$ to an arbitrary planar rectifiable curve $Γ\subset \mathbb{R}^{2}$ of positive length.
Traces of Sobolev spaces to piecewise Ahlfors--David regular sets
本文在支持弱局部 Poincaré 不等式的加倍度量测度空间中，刻画 Sobolev 空间到不同余维数的有限个 Ahlfors–David 正则子集并集上的迹，并给出迹范数的定量估计。
In our companion paper (S.N. Chandler Wilde, D.P. Hewett, A. Moiola, Sobolev spaces on non-Lipschitz subsets of $\mathbb{R}^n$ with application to boundary integral equations on fractal screens, 2016) we studied a number of different Sobolev spaces on a general (non-Lipschitz) open subset $Ω$ of $\mathbb{R}^n$, defined as closed subspaces of the classical Bessel potential spaces $H^s(\mathbb{R}^n)$ for $s\in\mathbb{R}$. These spaces are mapped by the restriction operator to certain spaces of distributions on $Ω$. In this note we make some observations about the relation between these spaces of…
Higher-order Sobolev embeddings into spaces of Campanato and Morrey type
该研究给出了重排不变空间上Sobolev型空间嵌入广义Campanato与Morrey空间的充要条件，确定了最优目标与定义域空间，并应用准则得到Orlicz-Sobolev空间的精确嵌入结果。
We give an intrinsic characterization of the restrictions of Sobolev, Triebel-Lizorkin and Besov spaces to regular subsets of $R^n$ via sharp maximal functions and local approximations.
Liftings of Sobolev maps into closed Riemannian manifolds via double coverings and minimal connections relative to planar sets, with an application to ferronematics
本文通过双覆盖和最小连接方法研究Sobolev映射的提升，并应用于铁磁流体模型的最小化问题。
Let $Ω\subset \mathbb{R}^n$ be a domain that supports the $p$-Poincaré inequality. Given a homeomorphism $φ\in L^1_p(Ω)$, for $p>n$ we show the domain $φ(Ω)$ has finite geodesic diameter. This result has a direct application to Brennan's conjecture and quasiconformal homeomorphisms. {\bf The Inverse Brennan's conjecture} states that for any simply connected plane domain $Ω' \subset\mathbb C$ with nonempty boundary and for any conformal homeomorphism $φ$ from the unit disc $\mathbb{D}$ onto $Ω'$ the complex derivative $φ'$ is integrable in the degree $s$, $-2<s<2/3$. If $Ω'$ is bounded than $-2…
The Riemann Mapping Theorem states existence of a conformal homeomorphism $φ$ of a simply connected plane domain $Ω\subset\mathbb C$ with non-empty boundary onto the unit disc $\mathbb D\subset \mathbb C$. In the first part of the paper we study embeddings of Sobolev spaces $\overset{\circ}{W_{p}^{1}}(Ω)$ into weighted Lebesgue spaces $L_{q}(Ω,h)$ with an {}"universal" weight that is Jacobian of $φ$ i.e. $h(z):=J(z,φ)=| φ'(z)|^2$. Weighted Lebesgue spaces with such weights depend only on a conformal structure of $Ω$. By this reason we call the weights $h(z)$ conformal weights. In the second pa…
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 $1\le p<\infty$.
We study a question of density of Lipschitz mappings in the Sobolev class of mappings from a closed manifold into a singular space. The main result of the paper shows that if we change the metric in the target space to a bi-Lipschitz equivalent one, than the property of the density of Lipschitz mappings may be lost.
In this paper, we investigate the relation between Sobolev-type embeddings of Hajłasz-Besov spaces (and also Hajłasz-Triebel-Lizorkin spaces) defined on a metric measure space $(X,d,μ)$ and lower bound for the measure $μ.$ We prove that if the measure $μ$ satisfies $μ(B(x,r))\geq cr^Q$ for some $Q>0$ and for any ball $B(x,r)\subset X,$ then the Sobolev-type embeddings hold on balls for both these spaces. On the other hand, if the Sobolev-type embeddings hold in a domain $Ω\subset X,$ then we prove that the domain $Ω$ satisfies the so-called measure density condition, i.e., $μ(B(x,r)\capΩ)\geq …
An apparently new concept of maximal mean difference quotient is defined for functions in the Lebesgue space $L_{loc}(R^n)$. Our definitions are meaningful for vector valued functions on general measure metric spaces as well and seem to lead to the most natural class of metric Sobolev spaces. The discussion of higher order Sobolev spaces and higher order mean difference quotients on regular subsets of Euclidean spaces is also possible in the context of the generalized Taylor-Whitney jets. This paper is a direct sequel to the papers: B. Bojarski, Taylor expansions and Sobolev spaces, Bull. Geor…
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…
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.
We study higher-order compact Sobolev embeddings on a domain $Ω\subseteq \mathbb R^n$ endowed with a probability measure $ν$ and satisfying certain isoperimetric inequality. Given $m\in \mathbb N$, we present a condition on a pair of rearrangement-invariant spaces $X(Ω,ν)$ and $Y(Ω,ν)$ which suffices to guarantee a compact embedding of the Sobolev space $V^mX(Ω,ν)$ into $Y(Ω,ν)$. The condition is given in terms of compactness of certain one-dimensional operator depending on the isoperimetric function of $(Ω,ν)$. We then apply this result to the characterization of higher-order compact Sobolev …
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.
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 study conditions on closed sets $C,F \subset \mathbb{R}$ making the product $C \times F$ removable or non-removable for $W^{1,p}$. The main results show that the Hausdorff-dimension of the smaller dimensional component $C$ determines a critical exponent above which the product is removable for some positive measure sets $F$, but below which the product is not removable for another collection of positive measure totally disconnected sets $F$. Moreover, if the set $C$ is Ahlfors-regular, the above removability holds for any totally disconnected $F$.
We give precise conditions under which the real interpolation space [Y_0,X_1]_{s,p} coincides with a closed subspace of the corresponding interpolation space [X_0,X_1]_{s,p} when Y_0 is a closed subspace of X_0 of codimension one. This result is applied to study the basis properties of nonharmonic Fourier series in Sobolev spaces H^s on an interval when 0<s<1. The main result: let E be a family of exponentials exp(i λ_n t) and E forms an unconditional basis in L^2 on an interval. Then there exist two number s_0, s_1 such that E forms an unconditional basis in H^s for s<s_0, E forms an uncondit…
Brezis and Mironescu have announced several years ago that for a compact manifold $N^n \subset \mathbb{R}^ν$ and for real numbers $0 < s < 1$ and $1 \le p < \infty$ the class $C^\infty(\overline{Q}^m; N^n)$ of smooth maps on the cube with values into $N^n$ is dense with respect to the strong topology in the Sobolev space $W^{s, p}(Q^m; N^n)$ when the homotopy group $π_{\lfloor sp \rfloor}(N^n)$ of order $\lfloor sp \rfloor$ is trivial. The proof of this beautiful result is long and rather involved. Under the additional assumption that $N^n$ is $\lfloor sp \rfloor$ simply connected, we give a s…
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…
Newtonian spaces generalize first-order Sobolev spaces to abstract metric measure spaces. In this paper, we study regularity of Newtonian functions based on quasi-Banach function lattices. Their (weak) quasi-continuity is established, assuming density of continuous functions. The corresponding Sobolev capacity is shown to be an outer capacity. Assuming sufficiently high integrability of upper gradients, Newtonian functions are shown to be (essentially) bounded and (Hölder) continuous. Particular focus is put on the borderline case when the degree of integrability equals the "dimension of the m…
In this paper, we study a newly developed shearlet system on bounded domains which yields frames for $H^s(Ω)$ for some $s\in \mathbb{N}$, $Ω\subset \mathbb{R}^2$. We will derive approximation rates with respect to $H^s(Ω)$ norms for functions whose derivatives admit smooth jumps along curves and demonstrate superior rates to those provided by pure wavelet systems. These improved approximation rates demonstrate the potential of the novel shearlet system for the discretization of partial differential equations. Therefore, we implement an adaptive shearlet-based algorithm for the solution of an e…
An optimal time-singularity of the estimate for the heat semigroup related to the critical Sobolev embedding
本文研究了热半群与临界Sobolev嵌入相关的估计的最优时间奇异性问题。
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…
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 …
Given a compact manifold $N^n \subset \mathbb{R}^ν$, $s \ge 1$ and $1 \le p < \infty$, we prove that the class of smooth maps on the cube with values into $N^n$ is strongly dense in the fractional Sobolev space $W^{s, p}(Q^m; N^n)$ when $N^n$ is $\lfloor sp \rfloor$ simply connected. For $sp$ integer, we prove weak density of smooth maps with values into $N^n$ when $N^n$ is $sp - 1$ simply connected. The proofs are based on the existence of a retraction of $\mathbb{R}^ν$ onto $N^n$ except for a small subset of $N^n$ and on a pointwise estimate of fractional derivatives of composition of maps i…
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.
Parameter--elliptic pseudodifferential operators given on a closed smooth manifold are investigated on the extended Sobolev scale. This scale consists of all Hilbert spaces that are interpolation spaces with respect to the Hilbert Sobolev scale. We prove that these operators set isomorphisms between appropriate spaces of the scale provided the parameter is modulo large enough. For solutions to the corresponding parameter--elliptic equations, we establish two-sided a priori estimates, in which the constants are independent of the parameter.
We prove that for a bounded simply connected domain $Ω\subset \mathbb R^2$, the Sobolev space $W^{1,\,\infty}(Ω)$ is dense in $W^{1,\,p}(Ω)$ for any $1\le p<\infty$. Moreover, we show that if $Ω$ is Jordan, then $C^{\infty}(\mathbb R^2)$ is dense in $W^{1,\,p}(Ω)$ for $1\le p<\infty$.
Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $μ$. The associated Cameron--Martin space is denoted by $H$. Consider two sufficiently regular convex functions $U:X\rightarrow\mathbb{R}$ and $G:X\rightarrow \mathbb{R}$. We let $ν=e^{-U}μ$ and $Ω=G^{-1}(-\infty,0]$. In this paper we are interested in the $W^{2,2}$ regularity of the weak solutions of elliptic equations of the type \begin{align}\label{Probelma in abstract} λu-L_{ν,Ω} u=f, \end{align} where $λ>0$, $f\in L^2(Ω,ν)$ and $L_{ν,Ω}$ is the self-adjoint operator associated with the quadratic fo…
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