Nadine Badr
We study the interpolation property of Sobolev spaces of order 1 denoted by $W^{1}_{p,V}$, arising from Schrödinger operators with positive potential. We show that for $1\leq p_1<p<p_2<q_{0}$ with $p>s_0$, $W^{1}_{p,V}$ is a real interpolation space between $W_{p_1,V}^{1}$ and $W_{p_2,V}^{1}$ on some classes of manifolds and Lie groups. The constants $s_{0}, q_{0}$ depend on our hypotheses.
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We generalize in this article the classical Sobolev's and Sobolev's trace inequalities on the Grand Lebesgue Spaces under monomial weight instead the classical Lebesgue or grand Lebesgue Spaces. We will distinguish the classical Sobolev's inequality and the so-called trace Sobolev's 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.
We provide an extension operator for weighted Sobolev spaces on bounded polyhedral cones $K$ involving a mixture of weights, which measure the distance to the vertex and the edges of the cone, respectively. Our results are based on Stein's extension operator for Sobolev spaces.
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.
Morrey--Sobolev inequalities with power weights on the half-space
该研究建立了n维半空间上带边界距离幂权的Sobolev空间及n维双曲空间上Sobolev空间的Morrey--Sobolev不等式，且所有估计在相差乘法常数意义下最优。
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.
We derive some anisotropic Sobolev inequalities in $\mathbb{R}^{n}$ with a monomial weight in the general setting of rearrangement invariant spaces. Our starting point is to obtain an integral oscillation inequality in multiplicative form.
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.
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 extend the classical Hardy-Sobolev-Poincare-Wirtinger inequalities from the ordinary Lebesgue-Riesz spaces into the Grand Lebesgue ones, with exact constants evaluation.
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.
We give in this short paper the exact value for norms of two operators of Hardy-Sobolev type acting between two weight Grand Lebesgue Space (GLS) based on the whole multidimensional Euclidean space.
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.
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.\
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.
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 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 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.
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 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.
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。
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 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.
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 in this article the Improved Sobolev inequalities with Muckenhoupt weights within the framework of stratified Lie groups. This family of inequalities estimate the Lq norm of a function by the geometric mean of two norms corresponding to Sobolev spaces W(s;p) and Besov spaces B(-b, infty, infty). When the value p which characterizes Sobolev space is strictly larger than 1, the required result is well known in R^n and is classically obtained by a Littlewood-Paley dyadic blocks manipulation. For these inequalities we will develop here another totally different technique. When p = 1, thes…
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 $Z(\{γ_{k,m}\})$.
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].
We investigate the properties of a class of weighted vector-valued $L_p$-spaces and the corresponding (an)isotropic Sobolev-Slobodetskii spaces. These spaces arise naturally in the context of maximal $L_p$-regularity for parabolic initial-boundary value problems. Our main tools are operators with a bounded $\calH^\infty$-calculus, interpolation theory, and operator sums.
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 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 $p=2$.
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 …
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。
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 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.
We investigate the form of the closure of the smooth, compactly supported functions $C_{c}^{\infty}(Ω)$ in the weighted fractional Sobolev space $W^{s,p;\,w,v}(Ω)$ for bounded $Ω$. We focus on the weights $w,\,v$ being powers of the distance to the boundary of the domain. Our results depend on the lower and upper Assouad codimension of the boundary of $Ω$. For such weights we also prove the comparability between the full weighted fractional Gagliardo seminorm and the truncated one.
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.
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 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 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…
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.
Generalization for Poincaré--Sobolev inequalities with local weights
本文研究了不假设Muckenhoupt条件时，依赖区域立方体的局部权重下庞加莱-索伯列夫不等式的加权推广，并将其应用于齐次与非齐次型索伯列夫嵌入定理的加权推广。
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 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 $α,s\to0^+$.
The aim of this note is to explain in which sense an axiomatic Sobolev space over a general metric measure space (à la Gol'dshtein-Troyanov) induces - under suitable locality assumptions - a first-order differential structure.
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 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…
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.
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…
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.
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。
Using Gutzmer's formula, due to Lassalle, we characterise the image of Sobolev spaces under the Segal-Bargmann transform on compact Riemannian symmetric spaces.
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 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.
Sobolev spaces with mixed weights and the Poisson equation on angular domains
本文在角域上引入并分析了基于边界与顶点距离的混合权重 Sobolev 空间，并利用 Mellin 变换在可积性参数为2时分析了 Poisson 方程的正则性，为研究随机偏微分方程及退化抛物型方程提供基础。
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 prove that weakly differentiable weights $w$ which, together with their reciprocals, satisfy certain local integrability conditions, admit a unique associated first-order $p$-Sobolev space, that is \[H^{1,p}(\mathbb{R}^d,w\,\d x)=V^{1,p}(\mathbb{R}^d,w\,\d x)=W^{1,p}(\mathbb{R}^d,w\,\d x),\] where $d\in\N$ and $p\in [1,\infty)$. If $w$ admits a (weak) logarithmic gradient $\nabla w/w$ which is in $L^q_{\text{loc}}(w\,\d x;\R^d)$, $q=p/(p-1)$, we propose an alternative definition of the weighted $p$-Sobolev space based on an integration by parts formula involving $\nabla w/w$. We prove that …
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。
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…
We study properties of $W_0^{1,p}(\mathbb{R}_+,t^β)$ - the completion of $C_0^\infty(\mathbb{R}_+)$ in the power-weighted Sobolev spaces $W^{1,p}(\mathbb{R}_+,t^β)$, where $β\in\mathbb{R}$. Among other results, we obtain the analytic characterization of $W_0^{1,p}(\mathbb{R}_+,t^β)$ for all $β\in \mathbb{R}$. Our analysis is based on the precise study the two trace operators: $Tr^{0}(u):= \lim_{t\to 0} u(t)$ and $Tr^{\infty}(u):= \lim_{t\to \infty} u(t)$, which leads to the analysis of the asymptotic behavior of functions from $W_0^{1,p}(\mathbb{R}_+,t^β)$ near zero or infinity. The obtained s…
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