Douadi Drihem
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.
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Multiplication properties and the Rellich-Kondrachov theorem in Herz-type Sobolev spaces
证明赫茨型索伯列夫空间构成巴拿赫代数并建立其雷利奇 - 康德拉乔夫紧致性定理，扩展经典理论，表明该空间是经典索伯列夫空间的自然推广。
Trace Principle for Riesz Potentials on Herz-Type Spaces and Applications
该研究建立了Herz型空间上Riesz位势的迹不等式，论证了参数条件的最优性，给出了相关Sobolev型不等式应用，并得到Herz型Sobolev空间的嵌入定理。
In this paper we prove the Sobolev embeddings for Herz-type Triebel-Lizorkin spaces, \begin{equation*} \dot{K}_{q}^{α_{2},r}F_{θ}^{s_{2}}\hookrightarrow \dot{K}%_{s}^{α_{1},p}F_{β}^{s_{1}} \end{equation*} where the parameters $α_{1},α_{2},s_{1},s_{2},s,q,r,p,β$ and $θ$ satisfy some suitable conditions. An application we obtain new embeddings between Herz and Triebel-Lizorkin spaces. Moreover, we present the Sobolev embeddings for Triebel-Lizorkin spaces equipped with power weights. All these results cover the results on classical Triebel-Lizorkin spaces.
Mixed-norm Herz-type Besov-Triebel-Lizorkin spaces
基于混合范数Herz空间，引入混合范数Herz型Besov和Triebel-Lizorkin空间，建立其φ-变换刻画并证明Sobolev、Franke和Jawerth嵌入定理，推广了已知结果。
Lorentz Herz-type Besov-Triebel-Lizorkin spaces
本文引入了一类新的Lorentz Herz型Besov和Triebel-Lizorkin函数空间，给出了其$φ$-变换刻画、多种嵌入定理、原子与小波分解及差分球平均刻画，并探讨了相关实例。
Let $G:\mathbb{R\rightarrow R}$ be a continuous function. In the first part of this paper, we investigate sufficient conditions on $G$ such that \begin{equation*} \{G(f):f\in \dot{K}_{p,q}^{α}F_{β}^{s}\}\subset \dot{K}_{p,q}^{α}F_{β}^{s} \end{equation*} holds. Here $\dot{K}_{p,q}^{α}F_{β}^{s}$ are Herz-type Triebel-Lizorkin spaces. These spaces unify and generalize many classical function spaces such as Lebesgue spaces of power weights, Sobolev and Triebel-Lizorkin spaces of power weights. In the second part of this paper we will study local and global Cauchy problems for the semilinear parabo…
In this paper we build up a criteria for fractional Orlicz-Sobolev extension and imbedding domains on Ahlfors $n$-regular domains.
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 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 $p>2$.
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.
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.
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 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.
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
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}).
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。
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.
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.
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.
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.
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。
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 $1\le p<\infty$.
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 prove the product of the Sobolev-extension domains is still a Sobolev-extension domain.
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。
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 compute the precise value of the measure of noncompactness of Sobolev embeddings $W_0^{1,p}(D)\hookrightarrow L^p(D)$, $p\in(1,\infty)$, on strip-like domains $D$ of the form $\mathbb{R}^k\times\prod\limits_{i=1}^{n-k}(a_i,b_i)$. We show that such embeddings are always maximally noncompact, that is, their measure of noncompactness coincides with their norms. Furthermore, we show that not only the measure of noncompactness but also all strict $s$-numbers of the embeddings in question coincide with their norms. We also prove that the maximal noncompactness of Sobolev embeddings on strip-like …
The question is addressed of when a Sobolev type space, built upon a general rearrangement-invariant norm, on an $n$-dimensional domain, is a Banach algebra under pointwise multiplication of functions. A sharp balance condition among the order of the Sobolev space, the strength of the norm, and the (ir)regularity of the domain is provided for the relevant Sobolev space to be a Banach algebra. The regularity of the domain is described in terms of its isoperimetric function. Related results on the boundedness of the multiplication operator into lower-order Sobolev type spaces are also establishe…
This paper is concerned with an explicit value of the embedding constant from $W^{1,q}(Ω)$ to $L^{p}(Ω)$ for a bounded domain $Ω\subset\mathbb{R}^N~(N\in\mathbb{N})$, where $1\leq q\leq p\leq \infty$. To obtain this value, we previously proposed a formula for estimating the embedding constant on bounded and unbounded Lipschitz domains by estimating the norm of Stein's extension operator, in the article (K. Tanaka, K. Sekine, M. Mizuguchi, and S. Oishi, Estimation of Sobolev-type embedding constant on domains with minimally smooth boundary using extension operator, Journal of Inequalities and A…
On optimization of cubature formulae for Sobolev classes of functions defined on star domains
本文将凸域上Sobolev类函数求积公式的渐近最优恢复方法推广到有界星形域，基于有限点值和分布梯度的$L_p$有界性给出积分算子的最优恢复。
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 …
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…
We survey a few trace theorems for Sobolev spaces on $N$-dimensional Euclidean domains. We include known results on linear subspaces, in particular hyperspaces, and smooth boundaries, as well as less known results for Lipschitz boundaries, including Besov's Theorem and other characterizations of traces on planar domains, polygons in particular, in the spirit of the work of P. Grisvard. Finally, we present a recent approach, originally developed by G. Auchmuty in the case of the Sobolev space $H^1(Ω)$ on a Lipschitz domain $Ω$, and which we have further developed for the trace spaces of $H^k(Ω)…
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, 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 prove a logarithmic Sobolev trace inequality in a gaussian space and we study the trace operator in the weighted Sobolev space W^{1,p}(Ω,γ) for sufficiently regular domain. We exhibit examples to show the sharpness of the results. Applications to PDE are also considered.
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 $Ω\subset R^n$.
Optimal higher-order Sobolev type embeddings are shown to follow via isoperimetric inequalities. This establishes a higher-order analogue of a well-known link between first-order Sobolev embeddings and isoperimetric inequalities. Sobolev type inequalities of any order, involving arbitrary rearrangement-invariant norms, on open sets in $\rn$, possibly endowed with a measure density, are reduced to much simpler one-dimensional inequalities for suitable integral operators depending on the isoperimetric function of the relevant sets. As a consequence, the optimal target space in the relevant Sobol…
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.
We consider a version of the fractional Sobolev inequality in domains and study whether the best constant in this inequality is attained. For the half-space and a large class of bounded domains we show that a minimizer exists, which is in contrast to the classical Sobolev inequalities in domains.
We show that in a bounded simply connected planar domain $Ω$ the smooth Sobolev functions $W^{k,\infty}(Ω)\cap C^\infty(Ω)$ are dense in the homogeneous Sobolev spaces $L^{k,p}(Ω)$.
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.
We find a new sharp trace Gagliardo-Nirenberg-Sobolev inequality on convex cones, aswell as a weighted sharp trace Sobolev inequality on epigraphs of convex functions. This is done by using a generalized Borell-Brascamp-Lieb inequality, coming from the Brunn-Minkowski theory.
We show that Sobolev space $W^1_1(Ω)$ of any planar one-connected domain $Ω$ has the Bounded Approximation property. The result holds independently from the properties of the boundary of $Ω$. The prove is based on a new decomposition of a planar domain.
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 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.
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 $|\partialΩ|>0$.
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.
In this article, we develop the theory of weighted $L^2$ Sobolev spaces on unbounded domains in $\mathbb R^n$. As an application, we establish the elliptic theory for elliptic operators and prove trace and extension results analogous to the bounded, unweighted case.
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 reveal relations between the duality of capacities and the duality between Sobolev extendability of Jordan domains in the plane, and explain how to read the curve conditions involvd in the Sobolev extendability of Jordan domains via the duality of capacities. Finally as an application, we give an alternative proof of the necessary condition for a Jordan planar domain to be $W^{1, q}$-extension domain when $2 < q < \infty$.
Logarithmic Sobolev Inequalities for Bounded Domains and Applications to Drift-Diffusion Equations
本文提出新型 Gagliardo-Nirenberg 型插值不等式，据此建立高维有界光滑区域上的对数 Sobolev 不等式，并用于证明非线性非局部漂移-扩散模型解指数收敛至常数稳态。
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。
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