Klaus Gansberger
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.
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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.
The aim in the present paper is to study removable sets for weighted Orlicz-Sobolev spaces. We generalize the definition of porous sets and show that the porous sets lying in a hyperplane are removable.
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 $L_{p_0}$.
New weighted Riesz-type pointwise inequalities and applications to generalized Sobolev estimates
本文提出新的加权里斯型点wise不等式，并将其应用于广义索伯列夫估计中。
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.
In this paper, we study the traces and the extensions for weighted Sobolev spaces on upper half spaces when the weights reach to the borderline cases. We first give a full characterization of the existence of trace spaces for these weighted Sobolev spaces, and then study the trace parts and the extension parts between the weighted Sobolev spaces and a new kind of Besov-type spaces (on hyperplanes) which are defined by using integral averages over selected layers of dyadic cubes.
We extend the classical Hardy-Sobolev-Poincare-Wirtinger inequalities from the ordinary Lebesgue-Riesz spaces into the Grand Lebesgue ones, with exact constants evaluation.
Stability of weighted composition strongly continuous semigroups acting on Lebesgue and Sobolev spaces is studied, without the use of spectral conditions on the generator of the semigroup. Applications to the generalized von Foerster - Lasota semigroup and a comparison with hypercylicity conditions are presented.\
We establish sharp affine weighted $L^p$ Sobolev type inequalities by using the $L_p$ Busemann-Petty centroid inequality proved by Lutwak, Yang and Zhang. Our approach consists in combining in a convenient way the latter one with a suitable family of sharp weighted $L^p$ Sobolev type inequalities obtained by Nguyen and allows to characterize all extremizers in some cases. The new inequalities don't rely on any euclidean geometric structure.
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.
In this paper we focus our attention on an embedding result for a weighted Sobolev space that involves as weight the distance function from the boundary taken with respect to a general smooth gauge function $F$. Starting from this type of inequalities we prove some refined Hardy-type inequalities.
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.
There are known trace and extension theorems relating functions in a weighted Sobolev space in a domain U to functions in a Besov space on the boundary bU. We extend these theorems to the case where the Sobolev exponent p is less than one by modifying our Sobolev spaces to consider averages of functions in Whitney balls. Averaged Sobolev spaces are also of interest in the applications in the case where p>1, and so we also provide trace and extension results in that case. Finally, we provide some comparable results for Neumann traces and extensions.
In this paper, we define weighted relative $p(.)$-capacity and discuss properties of capacity in the space $W_{\vartheta }^{1,p(.)}(\mathbb{R}^{n}).$ Also, we investigate some properties of weighted variable Sobolev capacity. It is shown that there is a relation between these two capacities. Moreover, we introduce a thinness in sense to this new defined relative capacity and prove an equivalence statement for this thinness.
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 $Ω\subset \Real^n$.
In this paper, we study a family of general fractional Sobolev spaces $\MsqpOm$ when $\Om=\Rn$ or $\Om$ is a bounded domain, having a compact, Lipschitz boundary $\Bdy$, in $\Rn$ for $n\geq2$. Among other results, some compact embedding results of $\MVsqpRn\hookrightarrow\LqRn$ and $\MVsqpRn\hookrightarrow\LlRn$ for suitable potential functions $V(x)$ are described.
We study associate and double associate spaces of two-weighted Sobolev spaces of the first order on real half-line and we show that unlike the notion of duality the associativity is divided into two cases which we call "strong" and "weak" ones with the division of the second associativity into four cases. On the way we prove that the Sobolev space of compactly supported functions possess weak associated reflexivity and the double weak-strong associate space is vacuous. The case of power weights was recently characterized by reduction to Cesàro or Copson type spaces [18].
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.
On Removable Sets for Weighted Sobolev Functions
本文给出了不涉及容量的几何条件，利用立方体覆盖定义的孔隙性条件，证明了在满足加倍和 Poincaré 不等式的权重下，加权 Sobolev 函数的紧零集可去性，并运用度量空间分析理论进行证明。
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.
Optimal function spaces in weighted Sobolev embeddings with $α$-homogeneous weights
本文研究带 $\alpha$-齐次权重的加权 Sobolev 不等式，建立约化原理并刻画最优重排不变目标与定义域空间，通过 Lorentz--Karamata 空间类给出具体例子。
By adapting the mass transportation technique of Cordero-Erausquin, Nazaret and Villani, we obtain a family of sharp Sobolev and Gagliardo-Nirenberg (GN) inequalities on the half space $\mathbf{R}^{n-1}\times\mathbf{R}_+$, $n\geq 1$ equipped with the weight $ω(x) = x_n^a$, $a\geq 0$. It amounts to work with the fractional dimension $n_a = n+a$. The extremal functions in the weighted Sobolev inequalities are fully characterized. Using a dimension reduction argument and the weighted Sobolev inequalities, we can reproduce a subfamily of the sharp GN inequalities on the Euclidean space due to Del …
We investigate the asymptotic behaviour of entropy and approximation numbers of the compact embedding $E^m_{p,σ}(B)\hookrightarrow L_p(B)$, $1\leq p<\infty,$ defined on the unit ball $B$ in $\mathbb{R}^n$. Here $E^m_{p,σ}(B)$ denotes a Sobolev space with a power weight perturbed by a logarithmic function. The weight contains a singularity at the origin. Inspired by Evans and Harris, we apply a bracketing technique which is an analogue to that of Dirichlet-Neumann-bracketing used by Triebel if $p=2$.
A complete description of traces on $\mathbb{R}^{n}$ of functions from the weighted Sobolev space $W^{l}_{1}(\mathbb{R}^{n+1},γ)$, $l \in \mathbb{N}$, with weight $γ\in A^{\rm loc}_{1}(\mathbb{R}^{n+1})$ is obtained. In the case $l=1$ the proof of the trace theorems is based on a~special nonlinear algorithm for constructing a~system of tilings of the space~$\mathbb R^n$. As the trace of the space $W^1_1(\mathbb R^{n+1},γ)$ we have the new function space $Z(\{γ_{k,m}\})$.
In this work we present a newly developed study of the interpolation of weighted Sobolev spaces by the complex method. We show that in some cases, one can obtain an analogue of the famous Stein-Weiss theorem for weighted $L^{p}$ spaces. We consider an example which gives some indication that this may not be possible in all cases. Our results apply in cases which cannot be treated by methods in earlier papers about interpolation of weighted Sobolev spaces. They include, for example, a proof that $\left[W^{1,p}(\mathbb{R}^{d},ω_{0}),W^{1,p}(\mathbb{R}^{d},ω_{1})\right]_θ=W^{1,p}(\mathbb{R}^{d},ω…
Let $p>n$ and let $L^1_p(R^n)$ be a homogeneous Sobolev space. For an arbitrary Borel measure $μ$ on $R^n$ we give a constructive characterization of the space $L^1_p(R^n)+L_p(R^n;μ)$. We express the norm in this space in terms of certain oscillations with respect to the measure $μ$. This enables us to describe the $K$-functional for the couple $(L_p(R^n;μ),L^1_p(R^n))$ in terms of these oscillations, and to prove that this couple is quasi-linearizable.
On the compactness of embeddings for a class of weighted Orlicz-Sobolev sequence spaces
本文引入一类加权Orlicz-Sobolev序列空间，证明其连续性与紧性准则，连续性源于Orlicz函数逐点比较，紧性结合Schauder基与局部条件，并给出实例与开放问题。
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.
We give here a simple proof of weighted logarithmic Sobolev inequality, for example for Cauchy type measures, with optimal weight, sharpening results of Bobkov-Ledoux. Some consequences are also discussed.
Compact embeddings and Pitt's property for weighted sequence spaces of Sobolev type
本文引入一类新的Sobolev型加权序列空间，证明其紧嵌入定理，并推广了非平衡统计力学中主方程谱分析的紧嵌入结果，同时证明了Pitt型定理：满足条件时，该类空间之间的线性有界变换均为紧算子。
In this paper order estimates for entropy numbers of embeddings of weighted Sobolev spaces on a John domain are obtained. In addition, we obtain order estimates for entropy numbers of summation operators on trees.
We study compactness and boundedness of embeddings from Sobolev type spaces on metric spaces into $L^q$ spaces with respect to another measure. The considered Sobolev spaces can be of fractional order and some statements allow also nondoubling measures. Our results are formulated in a general form, using sequences of covering families and local Poincaré type inequalities. We show how to construct such suitable coverings and Poincaré inequalities. For locally doubling measures, we prove a self-improvement property for two-weighted Poincaré inequalities, which applies also to lower-dimensional m…
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…
In the setting of the Euclidean space equipped with an arbitrary Radon measure, we prove the equivalence between several notions of function of bounded variation present in the literature. We also study the relation between various definitions of 1-Sobolev function.
We study the removability of a singular set for elliptic equations involving weight functions and variable exponents. We consider the case where the singular set satisfies conditions related to some generalization of upper Minkowski content or a net measure, and give sufficient conditions for removability of this singularity for equations in the weighted variable exponent Sobolev spaces.
In this paper we study weighted Hardy-Sobolev spaces of vector valued functions analytic on double-napped cones of the complex plane. We introduce these spaces as a tool for complex scaling of linear ordinary differential equations with dilation analytic unbounded operator coefficients. As examples we consider boundary value problems in cylindrical domains and domains with quasicylindrical ends.
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.
Complex interpolation of power-weighted Sobolev spaces with boundary conditions
本文研究了带有边界条件的加权Sobolev空间的复插值理论，推导了迹定理和密度结果。
In the present paper we study embedding operators for weighted Sobolev spaces whose weights satisfy the well-known Muckenhoupt A_p-condition. Sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities. The proposed method is based on the concept of 'generalized' quasiconformal homeomorphisms (homeomorphisms with bounded mean distortion.) The choice of the homeomorphism type depends on the choice of the corresponding weighted Sobolev space. Such classes of homeomorphisms induce bounded composition oper…
We describe a new, short proof of some facts relating the gap lengths of the spectrum of a potential of Hill's equation to its regularity. For example, a real potential is in a weighted Gevrey-Sobolev space if and only if its gap lengths belong to a similarly weighted sequence space. An extension of this result to complex potentials is proven as well. We also recover Trubowitz results about analytic potentials. The proof essentially employs the implicit function theorem.
In this study, we consider weighted stochastic field exponent function spaces $L_{\vartheta }^{p(.,.)}\left( D\times Ω\right) $ and $W_{\vartheta }^{k,p(.,.)}\left( D\times Ω\right) $. Also, we investigate some basic properties and embeddings of these spaces. Finally, we present an application of these spaces to the stochastic partial differential equations with stochastic field growth.
In this study, we define double weighted variable exponent Sobolev spaces $W^{1,q(.),p(.)}\left( Ω,\vartheta _{0},\vartheta \right) $ with respect to two different weight functions. Also, we investigate the basic properties of this spaces. Moreover, we discuss the existence of weak solutions for weighted Dirichlet problem of $p(.)$-Laplacian equation \begin{equation*} \left\{ \begin{array}{cc} -\text{div}\left( \vartheta (x)\left\vert \nabla f\right\vert ^{p(x)-2}\nabla f\right) =\vartheta _{0}(x)\left\vert f\right\vert ^{q(x)-2}f & x\in Ω\\ f=0 & x\in \partial Ω\end{array} \right. \end{equati…
We consider the Sobolev space over $\mathbb{R}^d$ of square integrable functions whose gradient is also square integrable with respect to some positive weight. Tt is well known that smooth functions are dense in the weighted Sobolev space when the weight is uniformly bounded from below and above. This may not be the case when the weight is unbounded. In this paper, we focus on a class of two dimensional weights where the density of smooth functions does not hold. This class was originally introduced by V.V. Zhikov; such weights have a unique singularity point of non-zero capacity. Following V.…
In this paper we develop elements of the global calculus of Fourier integral operators in $R^n$ under minimal decay assumptions on phases and amplitudes. We also establish global weighted Sobolev $L^2$ estimates for a class of Fourier integral operators that appears in the analysis of global smoothing problems for dispersive partial differential equations. As an application, we exhibit a new type of smoothing estimates for hyperbolic equations, where the decay of data in space is quantitatively translated into the time decay of solutions.
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 $α,s\to0^+$.
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 $0<p<\infty$.
Fractional weighted Sobolev spaces associated to the Riesz fractional gradient
本文引入了新的函数空间——加权分数Sobolev空间X^{s,p}_{0,w}(Ω)，通过Riesz分数梯度扩展到加权Lebesgue空间，研究其结构性质及嵌入，并探讨退化分数椭圆偏微分方程。
We characterize all the real numbers a,b,c and 1<= p,q,r<infty such that the weighted Sobolev space W_{a,b}^(q,p)(R^N\{0}) with power weights |x|^a and |x|^b is continuously embedded into L^{r}(R^N;|x|^cdx). Furthermore, we show that this embedding is (almost) always characterized by a multiplicative norm inequality. When a,b,c>-N and attention is confined to smooth functions with compact support, these inequalities coincide with the Caffarelli-Kohn-Nirenberg inequalities. Variants for radially symmetric functions, or when r<=min{p,q}, are also obtained along the way.
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。
Novel global weighted parabolic Sobolev estimates, weighted mixed-norm estimates and a.e. convergence results of singular integrals for evolution equations are obtained. Our results include the classical heat equation, the harmonic oscillator evolution equation $$\partial_tu=Δu-|x|^2u+f,$$ and their corresponding Cauchy problems. We also show weighted mixed-norm estimates for solutions to degenerate parabolic extension problems arising in connection with the fractional space-time nonlocal equations $(\partial_t-Δ)^su=f$ and $(\partial_t-Δ+|x|^2)^su=f$, for $0<s<1$.
We introduce a class of weak solutions to the quasilinear equation $-Δ_p u = σ|u|^{p-2}u$ in an open set $Ω\subset\mathbf{R}^n$. Here $p>1$, and $Δ_p u$ is the $p$-Laplacian operator. Our notion of solution is tailored to general distributional coefficients $σ$ satisfying a certain weighted Sobolev-Poincare inequality. We also study weak solutions of the closely related equation $-Δ_p v = (p-1)|\nabla v|^p + σ$, under the same conditions on $σ$. Our results for this latter equation will allow us to characterize the class of distributions $σ$ which satisfy the Sobolev-Poincare inequality, there…
Global solvability of the Laplace equation in weighted Sobolev spaces
本文研究了拉普拉斯方程在加权Sobolev空间中的弱解与强解的可解性，证明了弱解在特定条件下满足强解条件，并指出此类问题需特殊技术处理。
We prove trace theorems for weighted mixed norm Sobolev spaces in the upper-half space where the weight is a power function of the vertical variable. The results show the differentiability order of the trace functions depends only on the power in the weight function and the integrability power for the integration with respect to the vertical variable but not on the integrability powers for the integration with respect to the horizontal ones. They are new even in the un-weighted case and they recover classical results in the case of un-mixed norm spaces. The work is motivated by the study of re…
In this paper we study singular integral operators which are hyper or weak over Lipschitz or Holder spaces and over weghted Sobolev spaces defined on unbounded domains in the standard $n$-D space $R^n$ for $n>0$. The $π$-operator in this case is one of the hyper integral operators which has been studied extensively than other hyper singular integral operators. It will be shown the control of singularity of such integral operators that are defined interms of Cauchy generating kernels by working on weghted Sobolev spaces $W^{p,k}(Ω,|x|^{ζ+epsilon}dx)$ for some $ε>0$ and $ζ$ some positive integer…
Log-Sobolev and Beckner inequalities and stability of Poincaré inequality with weighted Gaussian measures
本文基于马尔可夫半群与Γ-微分算子，推导加权高斯测度下的广义贝克纳不等式及皮卡德不等式稳定性，进一步研究其梯度和L^2稳定性，并应用至海森堡不确定性原理分析及对数Sobolev不等式构建。
Functional calculus on weighted Sobolev spaces for the Laplacian on the half-space
本文研究了半空间上带 Dirichlet 和 Neumann 边界条件的 Laplace 算子，证明其在测距边界的幂权重 Sobolev 空间上具有有界 $H^\infty$-calculus，并导出了热半群的多项式增长及热方程的极大正则性结果。
Sharp weighted CR trace Sobolev inequalities
本文在Siegel域上建立了涉及加权范数的尖锐Sobolev迹不等式，并将其与Heisenberg群上次Laplace算子分数幂的广义Dirichlet-to-Neumann实现联系起来，推广了已有结果。
Optimal semigroup estimates and functional calculus for the Laplacian on weighted Sobolev spaces
该研究推广了Lindemulder等人的成果，证明半空间上带Dirichlet和Neumann边界条件的Laplace算子在幂权Sobolev空间上的最优半群估计，及边界满足相容性条件时具有有界$H^\infty$-泛函演算，且该条件不可省略。
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