A. A. Vasil'eva
Here we obtain order estimates for widths of weighted Sobolev classes in the weighted Lebesgue space where parameters of the second weight satisfy some limiting conditions.
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In this paper we prove that Kolmogorov widths of weighted Sobolev classes with restrictions $f(a)=\dots=f^{(k-1)}(a)=f^{(k)}(b)=\dots=f^{(r-1)}(b)=0$ in a weighted Lebesgue space and spectral numbers of some non-linear differential equation coincide. Here we suppose that the embedding is compact.
Let $L_{q,μ}$, $1\leq q\leq\infty$, denotes the weighted $L_q$ space of functions on the unit ball $\Bbb B^d$ with respect to weight $(1-\|x\|_2^2)^{μ-\frac12},\,μ\ge 0$, and let $W_{2,μ}^r$ be the weighted Sobolev space on $\Bbb B^d$ with a Gaussian measure $ν$. We investigate the probabilistic linear $(n,δ)$-widths $λ_{n,δ}(W_{2,μ}^r,ν,L_{q,μ})$ and the $p$-average linear $n$-widths $λ_n^{(a)}(W_{2,μ}^r,μ,L_{q,μ})_p$, and obtain their asymptotic orders for all $1\le q\le \infty$ and $0<p<\infty$.
We give a sufficient condition for limiting Sobolev and Hardy inequalities to hold on stratified homogeneous groups. In the Euclidean case, this condition reduces to the known cancelling necessary and sufficient condition. We obtain in particular endpoint Korn--Sobolev and Korn--Hardy inequalities on stratified homogeneous groups.
Estimates for Kolmogorov and linear widths of some weighted Sobolev classes on a John domain are obtained.
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.
Kolmogorov widths of anisotropic Sobolev classes
本文研究了周期各向异性 Sobolev 和 Nikol'skii 类以及各向异性有限维球的 Kolmogorov 宽度，获得了相应的阶估计。
In this article, we obtain the order estimates for the Kolmogorov widths of sets with conditions on the norm in the weighted Sobolev space $W^r_{p_1}$ and in the weighted space $L_{p_0}$.
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.
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。
We prove sharp embedding inequalities for certain reduced Sobolev spaces that arise naturally in the context of Dirichlet problems with $L^1$ data. We also find the optimal target spaces for such embeddings, which in dimension 2 could be considered as limiting cases of the Hansson-Brezis-Wainger spaces, for the optimal embeddings of borderline Sobolev spaces $W_0^{k,n/k}$.
Kolmogorov widths of a Sobolev class with restrictions on the derivatives in different metrics
本文研究了具有不同度量导数限制的周期Sobolev类的Kolmogorov宽度，并获得了其阶估计。
In this paper order estimates for the linear widths of some function classes are obtained; these classes are defined by restrictions on the weighted $L_{p_1}$-norm of the r-th derivative and the weighted $L_{p_0}$-norm of zero derivative.
For many known non-compact embeddings of two Banach spaces $E\hookrightarrow F$, every bounded sequence in $E$ has a subsequence that takes form of a profile decomposition - a sum of clearly structured terms with asymptotically disjoint supports plus a remainder that vanishes in the norm of $F$. In this note we construct a profile decomposition for arbitrary sequences in the Sobolev space $H^{1,2}(M)$ of a compact Riemannian manifold, relative to the embedding of $H^{1,2}(M)$ into $L^{2^*}(M)$, generalizing the well-known profile decomposition of Struwe ([Proposition 2.1]{Struwe}) to the case …
We determine upper asymptotic estimates of Kolmogorov and linear $n$-widths of unit balls in Sobolev and Besov norms in $L_{p}$-spaces on smooth compact Riemannian manifolds. For compact homogeneous manifolds, we establish estimates which are asymptotically exact, for the natural ranges of indices. The proofs heavily rely on our previous results such as: estimates for the near-diagonal localization of the kernels of elliptic operators, Plancherel-Polya inequalities on manifolds of bounded geometry, cubature formulas with positive coefficients and uniform estimates on Clebsch-Gordon coefficient…
Limiting Sobolev estimates for vector fields and cancelling differential operators
本文研究齐次向量微分算子的端点Sobolev估计，用可消条件刻画满足估计的算子类，结合对偶估计与调和分析技术，统一已知不等式并推广到Hardy不等式情形。
We collect and extend results on the limit of $σ^{1-k}(1-σ)^k |v|_{l+σ,p,Ω}^p$ as $σ$ tends to $0^+$ or $1^-$, where $Ω$ is $\mathbb{R}^n$ or a smooth bounded domain, $k$ is 0 or 1, $l$ is a nonnegative integer, $p\in[1,\infty)$, and $|.|_{l+σ,p,Ω}$ is the intrinsic semi-norm of order $l+σ$ in the Sobolev space $W^{l+σ,p}(Ω)$. In general, the above limit is equal to $c[v]^p$, where $c$ and $[.]$ are, respectively, a constant and a semi-norm that we explicitly provide. The particular case $p=2$ for $Ω=\mathbb{R}^n$ is also examined and the results are then proved by using the Fourier transform.
For limiting non-compact Sobolev embeddings into continuous functions we study behavior of Approximation, Gelfand, Kolmogorov, Bernstein and Isomorphism s-numbers. In the one dimensional case the exact values of the above-mentioned strict s-numbers were obtained and in the higher dimensions sharp estimates for asymptotic behavior of strict s-numbers were established. As all known results for s-numbers of Sobolev type embeddings are studied mainly under the compactness assumption then our work is an extension of existing results and reveal an interesting behavior of s-numbers in the limiting ca…
Maximal noncompactness of limiting Sobolev embeddings
本文提出一种新方法，用于建立作用于一般拟范数函数空间上的算子的球测度非紧性的下界，并证明Brezis-Wainger型空间中的极限Sobolev嵌入是最大非紧的。
In this paper we completely solve the problem of finding the (upper) approximation order with respect to the Kolmogorov, Gel'fand, and linear widths for the embedding of the Sobolev spaces $W^{α,p}$ and $W_{0}^{α,p}$ in the euclidian measure spaces $L_ν^{q}$ for an arbitrary Borel probability measure $ν$ with support contained in the open $m$-dimensional unit cube and for all possible choices of $1\leq p,q\leq\infty$. We will determine the exact values for the various upper approximation orders in terms of the $L^{q}$-spectrum of $ν$ only and finally give sufficient conditions imposed on the r…
Widths of embeddings of Gaussian Sobolev spaces
本文研究了高斯Sobolev空间$W^s_p(\mathbb{R}^d, γ)$中函数的逼近问题，分析了不同逼近方法的最优性能，确定了Kolmogorov、线性及采样宽度的渐进行为。
In this paper we obtain order estimates for entropy numbers of embeddings of weighted Sobolev spaces into weighted Lebesgue spaces and of weighted summation operators on trees. Here we consider some critical conditions on the parameters.
We obtain the sharp constant for the Hardy-Sobolev inequality involving the distance to the origin. This inequality is equivalent to a limiting Caffarelli-Kohn-Nirenberg inequality. In three dimensions, in certain cases the sharp constant coincides with the best Sobolev constant.
Optimal Sobolev inequalities in the hyperbolic space
本文研究了超几何空间中第m阶Sobolev不等式最佳函数范数的确定，提供了多种具体实例并改进了某些极限情况下的不等式。
We present a Gagliardo-Nirenberg inequality which bounds Lorentz norms of the function by Sobolev norms and homogeneous Besov quasinorms with negative smoothness. We prove also other versions involving Besov or Triebel-Lizorkin quasinorms. These inequalities can be considered as refinements of Sobolev type embeddings. They can also be applied to obtain Gagliardo-Nirenberg inequalities in some limiting cases. Our methods are based on estimates of rearrangements in terms of heat kernels. These methods enable us to cover also the case of Sobolev norms with p = 1.
Affine Logarithmic HLS and Beckner-Type Logarithmic Sobolev Inequalities
本文研究Haddad和Ludwig近期仿射HLS不等式的两个极限情形，建立了仿射对数HLS不等式和仿射Beckner对数Sobolev不等式，它们分别强于经典版本。
The classical Hardy inequality holds in Sobolev spaces $W_0^{1,p}$ when $1\le p< N$. In the limiting case where $p=N$, it is known that by adding a logarithmic function to the Hardy potential, some inequality which is called the critical Hardy inequality holds in $W_0^{1,N}$. In this note, in order to give an explanation of appearance of the logarithmic function at the potential, we derive the logarithmic function from the classical Hardy inequality with the best constant via some limiting procedure as $p \nearrow N$. And we show that our limiting procedure is also available for the classical …
We discuss new proofs, and new forms, of a reverse logarithmic Sobolev inequality, with respect to the standard Gaussian measure, for low complexity functions, measured in terms of Gaussian-width. In particular, we provide a dimension-free improvement for a related result given in [Eldan '18].
We develop a general method to calculate entropy numbers of standard Sobolev's classes on an arbitrary compact homogeneous Riemannian manifold. Our method is essentially based on a detailed study of geometric characteristics of norms induced by subspaces of harmonics. The method's possibilities are not confined to the statements proved but can be applied in studying more general problems such as entropy of multiplier operators. As an application, we establish sharp orders of entropy of Sobolev's classes and respective n-widths.
We prove structure theorems for measures on the discrete cube and on Gaussian space, which provide sufficient conditions for mean-field behavior. These conditions rely on a new notion of complexity for such measures, namely the Gaussian-width of the gradient of the log-density. On the cube $\{-1,1\}^n$, we show that a measure $ν$ which exhibits low complexity can be written as a mixture of measures $\{ν_θ\}_{θ\in \mathcal{I}}$ such that: i. for each $θ$, the measure $ν_θ$ is a small perturbation of $ν$ such that $\log \tfrac{d ν_θ}{d ν}$ is a linear function whose gradient is small and, ii. $ν…
In this paper, estimates for Kolmogorov, Gelfand and linear widths of function classes on sets with a tree-like structure are obtained. As examples we consider weighted Sobolev classes on a John domain, as well as some function classes on a metric and combinatorial tree.
Gamma-liminf estimate for a class of non-local approximations of Sobolev and BV norms
本文研究一类非局部非凸泛函，证明其Gamma-liminf可由Sobolev范数或全变差的正倍数下界控制，并借助离散化分析回答了相关极限行为的未解决问题。
Gelfand and Kolmogorov numbers of Sobolev embeddings of weighted function spaces
本文研究了带多项式权重的Besov和Triebel-Lizorkin型加权函数空间之间Sobolev嵌入的Gelfand数和Kolmogorov数，在非极限情形下得到了精确的渐近估计。
In this paper we propose a unified approach, based on limiting interpolation, to investigate the embeddings for the Sobolev space $(\dot{W}^k_p(\mathcal{X}))_0, \, \mathcal{X} \in \{\mathbb{R}^d, \mathbb{T}^d, Ω\}$, in the subcritical case ($k < d/p$), critical case ($k = d/p$) and supercritical case ($k > d/p$). We characterize the Sobolev embeddings in terms of pointwise inequalities involving rearrangements and moduli of smoothness/derivatives of functions and via extrapolation theorems for corresponding smooth function spaces. Applications include Ulyanov-Kolyada type inequalities for rear…
On Choquet integrals and Sobolev type inequalities
本文研究n维欧氏空间开连通集上连续可微函数关于δ维Hausdorff容度的Choquet积分，证明了p=δ/n极限情形与p>δ情形的Sobolev型不等式，补充了已有的Poincaré-Sobolev和Morrey不等式。
Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs
本文为分数次临界Sobolev嵌入建立了最佳常数的精细界，覆盖边界与极限情形，并应用于非局部PDE解的存在性与不存在性证明。
Interpolation inequalities in Triebel-Lizorkin-Lorentz spaces and Besov-Lorentz spaces are studied for both inhomogeneous and homogeneous cases. First we establish interpolation inequalities under quite general assumptions on the parameters of the function spaces. Several results on necessary conditions are also provided. Next, utilizing the interpolation inequalities together with some embedding results, we prove Gagliardo-Nirenberg inequalities for fractional derivatives in Lorentz spaces, which do hold even for the limiting case when one of the parameters is equal to 1 or $\infty$.
Kolmogorov widths of the class $W_1^1$
研究类$W_1^1$的柯尔莫哥洛夫宽度，证明当$2<q<\boldsymbol{\u221e}$时其衰减阶为$n^{-1/2}\boldsymbol{\u221e}\boldsymbol{\u2061}n$，完成整数光滑性单变量索伯列夫类经典情形下该宽度衰减阶的研究。
We determine upper asymptotic estimates of Kolmogorov and linear $n$-widths of unit balls in Sobolev and Besov norms in $L_{p}$-spaces on compact Riemannian manifolds. The proofs rely on estimates for the near-diagonal localization of the kernels of elliptic operators. We also summarize some of our previous results about approximations by eigenfunctions of elliptic operators on compact homogeneous manifolds.
J. Bourgain and H. Brezis have obtained in 2002 some new and surprising estimates for systems of linear differential equations, dealing with the endpoint case $\mathrm{L}^1$ of singular integral estimates and the critical Sobolev space $\mathrm{W}^{1, n} (\mathbf{R}^n)$. This paper presents an overview of the results, further developments over the last ten years and challenging open problems.
We generalize the theorems of Stein--Tomas and Strichartz about surface restrictions of Fourier transforms to systems of orthonormal functions with an optimal dependence on the number of functions. We deduce the corresponding Strichartz bounds for solutions to Schrödinger equations up to the endpoint, thereby solving an open problem of Frank, Lewin, Lieb and Seiringer. We also prove uniform Sobolev estimates in Schatten spaces, extending the results of Kenig, Ruiz, and Sogge. We finally provide applications of these results to a Limiting Absorption Principle in Schatten spaces, to the well-pos…
For one-dimensional interval and integrable weight function $w$ we define via completion a weighted Sobolev space $H^{m,p}_{μ_w}$ of arbitrary integer order $m$. The weights in consideration may suffer strong degeneration so that, in general, functions $u$ from $H^{m,p}_{μ_w}$ do not have weak derivatives. This contribution is focussed on studying the continuity properties of functions $u$ at a chosen internal point $x_0$ to which we attribute a notion of criticality of order $k$ and with respect to the weight $w$. For non-critical points $x_0$ we formulate a local embedding result that guaran…
In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: Sharp embeddings between the Besov spaces defined by differences and by Fourier-analytical decompositions as well as between Besov and Sobolev/Triebel-Lizorkin spaces; Various new characterizations for Besov norms in terms of different K-functionals. For instance, we derive characterizations via ball averages, approximation methods, heat kernels, and Bianchini-type norms; Sharp estimates for Besov norms of derivatives and pote…
Non-strict singularity of optimal Sobolev embeddings
研究最优Sobolev嵌入在r.i.空间中的非严格奇异性，构造尖峰函数序列并证明其非严格奇异性质。
Gelfand and Kolmogorov numbers of Sobolev embeddings of weighted function spaces II
本文研究非极限情形下带多项式权重的Besov和Triebel-Lizorkin型加权函数空间之间紧嵌入的Gelfand数和Kolmogorov数，补充了拟Banach空间情形下的先前结果，并给出了Skrzypczak (JAT, 2005)中未解决的若干情形的逼近数的精确估计。
We derive the sharp constants for the inequalities on the Heisenberg group H^n whose analogues on Euclidean space R^n are the well known Hardy-Littlewood-Sobolev inequalities. Only one special case had been known previously, due to Jerison-Lee more than twenty years ago. From these inequalities we obtain the sharp constants for their duals, which are the Sobolev inequalities for the Laplacian and conformally invariant fractional Laplacians. By considering limiting cases of these inequalities sharp constants for the analogues of the Onofri and log-Sobolev inequalities on H^n are obtained. The m…
In this work we investigate $n$-widths of multiplier operators $Λ_*$ and $Λ$, defined for functions on the complex sphere $Ω_d$ of $\mathbb{C}^d$, associated with sequences of multipliers of the type $\{λ_{m,n}^*\}_{m,n\in \mathbb{N}}$, $λ_{m,n}^*=λ(m+n)$ and $\{λ_{m,n}\}_{m,n\in \mathbb{N}}$, $λ_{m,n}=λ(\max\{m,n\})$, respectively, for a bounded function $λ$ defined on $[0,\infty)$. If the operators $Λ_{*}$ and $Λ$ are bounded from $L^p(Ω_d)$ into $L^q(Ω_d)$, $1\leq p,q\leq\infty$, and $U_p$ is the closed unit ball of $L^p(Ω_d)$, we study lower and upper estimates for the $n$-widths of Kolmog…
We study volumes of sections of convex origin-symmetric bodies in $\mathbb{R}% ^{n}$ induced by orthonormal systems on probability spaces. The approach is based on volume estimates of \ John-Löwner ellipsoids and expectations of norms induced by the respective systems. The estimates obtained allows us to establish lower bounds for the radii of sections which gives lower bounds for Gelfand widths (or linear cowidths). As an application we offer a new method of evaluation of Gelfand and Kolmogorov widths of multiplier operators. In particular, we establish sharp orders of widths of standard Sobo…
Boundary ellipticity and limiting $L^1$-estimates on halfspaces
本文利用边界 $H$ 上的尖锐迹定理，刻画半空间中保证极限 Gagliardo–Nirenberg–Sobolev $L^1$ 估计成立的 $k$ 阶算子边界椭圆性充要条件。
Optimality of embeddings in Orlicz spaces
本文研究欧氏域Mazya类中Orlicz-Sobolev嵌入的最优性问题，证明了临界状态下部分此类嵌入不存在最优Orlicz空间，其关键特例为John域上的Brezis-Wainger嵌入。
A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II
本文定义了BV^0空间并证明其与Hardy空间相等，且S^{α,p}等于Bessel位势空间，进而证明分数阶梯度在α趋近0时强收敛于Riesz变换，并建立了稳定的分数阶插值不等式。
Optimal approximation and Kolmogorov widths estimates for certain singular classes related to equations of mathematical physics
针对定义在超立方体上的两类奇异函数类，估计其Kolmogorov和Babenko宽度，并构造局部样条实现最优阶逼近。
Comparison Theorems of Kolmogorov Type for Classes Defined by Cyclic Variation Diminishing Operators and Their Application
本文通过统一方法，对由具有特定振荡性质的算子定义的 $2\pi$-周期函数类建立了 Kolmogorov 型比较定理，并应用该结果证明了 Taikov 型不等式，计算了该类函数在 $L_q$ 空间中的 Kolmogorov、Gel'fand、线性及信息 $n$-宽度的精确值。
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