A. A. Vasil'eva
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.
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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 $L_{p_0}$.
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.
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。
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…
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…
Estimates for Kolmogorov and linear widths of some weighted Sobolev classes on a John domain are obtained.
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。
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.
In this paper we establish some new Kolmogorov type inequalities for the Marchaud and Hadamard fractional derivatives of functions defined on a real axis or semi-axis. Simultaneously we solve two related problems: the Stechkin problem on the best approximation of unbounded operators by bounded ones on a given class of elements and the problem of optimal recovery of operator on elements from some class given with prescribed error. Keywords: inequalities for derivatives, fractional derivatives, approx- imation of unbounded operators by bounded ones, optimal recovery of operators, ideal lattice.
We prove that every probability measure $μ$ satisfying the stationary Fokker-Planck-Kolmogorov equation obtained by a $μ$-integrable perturbation $v$ of the drift term $-x$ of the Ornstein-Uhlenbeck operator is absolutely continuous with respect to the corresponding Gaussian measure $γ$ and for the density $f=dμ/dγ$ the integral of $f |\log (f+1)|^α$ against $γ$ is estimated via $\|v\|_{L^1(μ)}$ for all $α<1/4$, which is a weakened $L^1$-analog of the logarithmic Sobolev inequality. This means that stationary measures of diffusions whose drifts are integrable perturbations of $-x$ are absolute…
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.
Equivalence of Sobolev norms for Kolmogorov operators with scaling-critical drift
本文研究了Kolmogorov算子中具有缩放临界漂移项的Sobolev范数等价性，指出需对可比Sobolev空间范围施加额外限制，因相关半群缺乏梯度界。
In this paper we show different inequalities for fractional order differential operators. In particular, the Sobolev, Hardy, Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenberg type inequalities for the Caputo, Riemann-Liouville and Hadamard derivatives are obtained. In addition, we show some applications of these inequalities.
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。
On some sharp Landau--Kolmogorov--Nagy type inequalities in Sobolev spaces of multivariate functions
本文研究开凸锥上 Sobolev 空间中的尖锐 Landau–Kolmogorov–Nagy 型不等式，并将其推广到电荷导数及多元函数混合导数的范数估计。
A chain rule for power product is studied with fractional differential operators in the framework of Sobolev spaces. The fractional differential operators are defined by the Fourier multipliers. The chain rule is considered newly in the case where the order of differential operators is between one and two. The study is based on the analogy of the classical chain rule or Leibniz rule.
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。
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。
A note on the Kolmogorov-type inequalities for more than three norms
本文通过标准技术从已知的柯尔莫哥洛夫问题解中获得精确的柯尔莫哥洛夫型不等式，用于估计函数导数的统一范数。
An approach to metric space valued Sobolev maps via weak* derivatives
本文提出用弱*导数刻画度量空间值Sobolev映射，并修正了Hajłasz和Tyson的先前结果。
In this paper we solve Kolmogorov problem about existence of a function with given norms of derivatives for classes of multiple monotone functions and absolute monotone functions in the case of arbitrary number of norms. We also show the connection of Kolmogorov problem with Markov moment problem.
The paper describes the splines $Q_{n,k}(x,a)$, which for an arbitrary point $a\in(0;1)$ and an arbitrary function $y\in\mathring{W}^n_p[0;1]$ set the relations $y^{(k)}(a)=\int_0^1 y^{(n)}(x)Q^{(n)}_{n,k}(x,a)dx$. The relation of the $L_{p'}[0;1]$ norm minimization for $Q^{(n)}_{n,k}$ ($1/ p+1/p'=1$) with the problem of the best estimates of derivatives of $y^{(k)}(a)\leqslant A_{n,k,p}(a)\|y^{(n)}\|_{L_p[0;1]}$, and also with the problem of finding the exact embedding constants of the Sobolev space $\mathring{W}^n_p[0;1]$ into the space $\mathring{W}^k_\infty[0;1]$, $n\in\mathbb{N}$, $k=0,1,…
In this note we show that the general theory of vector valued singular integral operators of Calderón-Zygmund defined on general metric measure spaces, can be applied to obtain Sobolev type regularity properties for solutions of the dyadic fractional Laplacian. In doing so, we define partial derivatives in terms of Haar multipliers and dyadic homogeneous singular integral operators.
Fractional Sobolev spaces related to an ultraparabolic operator
本文为满足弱 Hörmander 条件的超抛物 Kolmogorov 型算子构建分数阶 Sobolev 空间框架，通过实插值与内蕴 Taylor 余项估计证明其到 L^p 和内蕴 Hölder 空间的连续嵌入。
We define the notion of colocally weakly differentiable maps from a manifold $M$ to a manifold $N$. If $p \ge 1$ and $M$ and $N$ are endowed with a Riemannian metric, this allows us to define intrinsically the homogeneous Sobolev space $\dot{W}^{1, p} (M, N)$. This new definition is equivalent with the definition by embedding in the Euclidean space and with the definition of Sobolev maps into a metric space. The colocal weak derivative is an approximate derivative. The colocal weak differentiability is stable under the suitable weak convergence. The Sobolev spaces can be endowed with various i…
On Landau -- Kolmogorov type inequalities for charges and their applications
本文针对绝对连续的电荷证明了锐利的Landau--Kolmogorov型不等式，并解决了Stechkin逼近问题及相关极值问题，应用于Sobolev空间中的本质有界函数。
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.
In this paper, we study the sharp Poincaré inequality and the Sobolev inequalities in the higher order Lorentz--Sobolev spaces in the hyperbolic spaces. These results generalize the ones obtained in \cite{Nguyen2020a} to the higher order derivatives and seem to be new in the context of the Lorentz--Sobolev spaces defined in the hyperbolic spaces.
This paper presents a self-contained new theory of weak fractional differential calculus and fractional Sobolev spaces in one-dimension. The crux of this new theory is the introduction of a weak fractional derivative notion which is a natural generalization of integer order weak derivatives; it also helps to unify multiple existing fractional derivative definitions and characterize what functions are fractionally differentiable. Various calculus rules including a fundamental theorem of calculus, product and chain rules, and integration by parts formulas are established for weak fractional deri…
Using the concept of fractional derivatives of Riemann$-$Liouville on time scales, we first introduce right fractional Sobolev spaces and characterize them. Then, we prove the equivalence of some norms in the introduced spaces, and obtain their completeness, reflexivity, separability and some imbeddings. Finally, as an application, we propose a recent method to study the existence of weak solutions of fractional boundary value problems on time scales by using variational method and critical point theory, and by constructing an appropriate variational setting, we obtain two existence results of…
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。
This article contains a characterization of when certain weighted Sobolev spaces on $\Bbb R^n$ embed compactly into $L^2(\mathbb R^n, φ)$. The characterization is in terms of derivatives of the weight function $φ$ and involves the Wiener capacity, as it is obtained from reformulating the problem in terms of resolvent properties of Schrödinger operators. This reformulation also works for general domains.
We show that Sobolev maps with values in a dual Banach space can be characterized in terms of weak derivatives in a weak* sense. Since every metric space embeds isometrically into a dual Banach space, this implies a characterization of metric space valued Sobolev maps in terms of such derivatives. Furthermore, we investigate for which target spaces Sobolev maps are weak* differentiable almost everywhere.
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。
In this paper we investigate the modulus of continuity of differential operator of order $k$, $k\in\mathbb{N}$, on the classes of functions defined on half-line that have positive non-increasing continuous majorants of functions and their higher derivatives.
Under appropriate assumptions on the $N(Ω)$-fucntion, Morrey estimate is presented in the Musielak-Orlicz-Sobolev space.The assumptions include a new increasing condition on the $x$-derivative of the Young complementary function of the $N(Ω)$-fucntion. The conclusion applies to several important nonlinear examples frequently appeared in mathematical literature.
We first prove the equivalence of two definitions of Riemann-Liouville fractional integral on time scales, then by the concept of fractional derivative of Riemann-Liouville on time scales, we introduce fractional Sobolev spaces, characterize them, define weak fractional derivatives, and show that they coincide with the Riemann-Liouville ones on time scales. Next, we prove equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, separability and some imbeddings. Finally, as an application, by constructing an appropriate variational setting, using the mounta…
We characterize one-sided weighted Sobolev spaces $W^{1,p}(\mathbb{R},ω)$, where $ω$ is a one-sided Sawyer weight, in terms of a.e.~and weighted $L^p$ limits as $α\to1^-$ of Marchaud fractional derivatives of order $α$. Similar results for weighted Sobolev spaces $W^{2,p}(\mathbb{R}^n,ν)$, where $ν$ is an $A_p$-Muckenhoupt weight, are proved in terms of limits as $s\to1^-$ of fractional Laplacians $(-Δ)^s$. These are Bourgain--Brezis--Mironescu-type characterizations for weighted Sobolev spaces. We also complement their work by studying a.e.~and weighted $L^p$ limits as $α,s\to0^+$.
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.
A Short Note on the Comparison of Interpolation Widths, Entropy Numbers, and Kolmogorov Widths
本文比较了紧算子从希尔伯特空间到巴拿赫空间的科莫戈洛夫和熵数，并将其应用于再生核希尔伯特空间与L_∞(μ)之间的嵌入，给出了插值宽度与科莫戈洛夫n-宽度之间n^{1/2}级差距的充分条件，并证明在多维索伯列夫情况下，科莫戈洛夫与近似宽度之间确实存在此差距。
We investigate the 1D Riemann-Liouville fractional derivative focusing on the connections with fractional Sobolev spaces, the space $BV$ of functions of bounded variation, whose derivatives are not functions but measures and the space $SBV$, say the space of bounded variation functions whose derivative has no Cantor part. We prove that $SBV$ is included in $W^{s,1} $ for every $s \in (0,1)$ while the result remains open for $BV$. We study examples and address open questions.
A classical first-order Hardy-Sobolev inequality in Euclidean domains, involving weighted norms depending on powers of the distance function from their boundary, is known to hold for every, but one, value of the power. We show that, by contrast, the missing power is admissible in a suitable counterpart for higher-order Sobolev norms. Our result complements and extends contributions by Castro and Wang [CW], and Castro, Dávila and Wang [CDW1, CDW2], where a surprising canceling phenomenon underling the relevant inequalities was discovered in the special case of functions with derivatives in $L^1…
We show that in a bounded Gromov hyperbolic domain $Ω$ smooth functions with bounded derivatives $C^\infty(Ω)\cap W^{k,\infty}(Ω)$ are dense in the homogeneous Sobolev spaces $L^{k,p}(Ω)$.
Sobolev embeddings for kinetic Fokker-Planck equations
本文研究满足弱 Hörmander 条件的超抛物 Kolmogorov 型算子，构造内在 Sobolev-Slobodeckij 空间，并证明其到 Lorentz 与内在 Hölder 空间的连续嵌入及相关逼近、插值不等式。
We study global regularity properties of Sobolev homeomorphisms on $n$-dimensional Riemannian manifolds under the assumption of $p$-integrability of its first weak derivatives in degree $p\geq n-1$. We prove that inverse homeomorphisms have integrable first weak derivatives. For the case $p>n$ we obtain necessary conditions for existence of Sobolev homeomorphisms between manifolds. These necessary conditions based on Poincaré type inequality: $$ \inf_{c\in \mathbb R} \|u-c\mid L_{\infty}(M)\|\leq K \|u\mid L^1_{\infty}(M)\|. $$ As a corollary we obtain the following geometrical necessary condi…
Approximation order of Kolmogorov diameters via $L^{q}$-spectra and applications to polyharmonic operators
本文建立了 Borel 测度 $L^{q}$-谱与 Kolmogorov 直径逼近阶的联系，改进了奇异测度的经典结果，并应用于多调和算子获得了特征值衰减率的改进上界。
The existence and decay rates of strong solutions for Navier-Stokes Equations Sobolev
研究Sobolev空间中Navier-Stokes方程Cauchy问题局部与全局解的高阶导数存在性及衰减性质。
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.
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。
Riesz-Kolmogorov type compactness criteria in function spaces with applications
本文提出适用于多种函数空间的Riesz-Kolmogorov型紧性准则，并用于刻画Bergman空间上的紧Toeplitz算子及推导Hardy空间上Hankel算子的紧性。
Higher-order derivatives of radially symmetric functions
该研究推导了径向对称函数n阶偏导数张量Frobenius范数的简洁逐点公式，规避高阶径向导数的技术困难，刻画了径向对称函数在高阶Sobolev空间中的子空间，并推广到Banach格上的Sobolev型空间。
In this article, motivated by a work of Caffarelli and Cordoba in phase transitions analysis, we prove new weighted anisotropic Sobolev type inequalities, that is Sobolev type inequalities where different derivatives have different weight functions. The inequalities we are dealing with, are also intimately connected to weighted Sobolev inequalities for Grushin type operators, the weights being not necessarily Muckenhoupt. For example we consider here Sobolev inequalities on finite cylinders, the weights being different powers of the distance function from the top and the bottom of the cylinder…
Building on the inequalities for homogeneous tetrahedral polynomials in independent Gaussian variables due to R. Latała we provide a concentration inequality for non-necessarily Lipschitz functions $f\colon \R^n \to \R$ with bounded derivatives of higher orders, which hold when the underlying measure satisfies a family of Sobolev type inequalities $\|g- \E g\|_p \le C(p)\|\nabla g\|_p.$ Such Sobolev type inequalities hold, e.g., if the underlying measure satisfies the log-Sobolev inequality (in which case $C(p) \le C\sqrt{p}$) or the Poincaré inequality (then $C(p) \le Cp$). Our concentration …
Universal Approximation of Nonlinear Operators and Their Derivatives
本文提出通过运算学习架构证明非线性算子及其导数的通用逼近定理，扩展了经典结果到无限维空间，并探讨了其在高阶精度、约束优化和无限维PDE数值方法中的应用。
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