Real interpolation of Sobolev spaces
Nadine Badr
我们证明了$W^{1}_{p}$是$W^{1}_{p_{1}}$和$W^{1}_{p_{2}}$之间的插值空间，$p>q_{0}$和$1\leq p_{1}<p<p_{2}\leq \infty$在某些类流形和一般度量空间，其中$q_{0}$取决于我们的假设。
We prove that $W^{1}_{p}$ is an interpolation space between $W^{1}_{p_{1}}$ and $W^{1}_{p_{2}}$ for $p>q_{0}$ and $1\leq p_{1}<p<p_{2}\leq \infty$ on some classes of manifolds and general metric spaces, where $q_{0}$ depends on our hypotheses.
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Sobolev inequalities, rearrangements, isoperimetry and interpolation spaces
利用重排不等式刻画了度量空间中的Poincaré不等式
Korevaar-Schoen Energy and Interpolations of Fractional Sobolev mappings on Carnot groups
该研究将欧氏空间的弱接触方程结果推广到次黎曼框架下的Carnot群，探究分数阶Sobolev映射的Korevaar-Schoen能量与插值性质，证明满足弱接触方程的映射属于特定水平分数阶Sobolev空间，建立插值定理并刻画Korevaar-Schoen能量的极小化子。
Abstract Hardy-Sobolev spaces and interpolation
本文的目的是通过原子分解描述二重黎曼流形上Hardy-索伯列夫空间的抽象理论。研究了这些空间与索伯列夫空间的实插值，最后给出了Riesz不等式的应用.
Approximation in Sobolev spaces by piecewise affine interpolation
索伯列夫空间中的函数直接用空间范数中的分段仿射插值逼近。证明是基于估计插值和不依赖于密度的光滑函数。
Interpolation, embeddings and traces of anisotropic fractional Sobolev spaces with temporal weights
研究了一类加权向量值$L_p$ -空间和相应的迷向Sobolev-Slobodetskii空间的性质.这些空间自然产生的上下文中的最大$L_p$ -正则抛物型初边值问题。我们的主要工具是有界$\calH^\infty$演算，插值理论和运营商的总和。
Maximal inequalities for dual Sobolev spaces $W^{-1,p}$ and applications to interpolation
首先给出了对偶索伯列夫空间W^{-1，p}的一个极大不等式。这一个对应于一个“索伯列夫版本”的Hardy-Littlewood极大算子在Lebesgue空间的通常性质。即使在欧几里得空间，这似乎是一个新的，我们发展的黎曼流形的一般框架的参数。然后给出了一个应用，得到了索伯列夫空间的插值结果。
Real interpoaltion of Sobolev spaces associated to a weight
研究了由具有正位势的薛定谔算子所引起的一阶索伯列夫空间的插值性质，记作$W^{1}_{p,V}$.证明了在某些流形和李群上，对具有$p>s_0$的$1\leq p_1<p<p_2<q_{0}$，$W^{1}_{p,V}$是介于$W_{p_1,V}^{1}$和$W_{p_2,V}^{1}$之间的实插值空间.常数$s_{0}, q_{0}$取决于我们的假设。
We explicitly describe all Hilbert function spaces that are interpolation spaces with respect to a given couple of Sobolev inner product spaces considered over $\mathbb{R}^{n}$ or a half-space in $\mathbb{R}^{n}$ or a bounded Euclidean domain with Lipschitz boundary. We prove that these interpolation spaces form a subclass of isotropic Hörmander spaces. They are parametrized with a radial function parameter which is OR-varying at $+\infty$ and satisfies some additional conditions. We give explicit examples of intermediate but not interpolation spaces.
This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert space case is that, with the right normalisations, all the key results hold with equality of norms.) In the final section we apply the Hilbert space results to the Sobolev spaces $H^s(Ω)$ and $\widetilde{H}^s(Ω)$, for $s\in \mathbb{R}$ and an open $Ω\subset \mathbb{R}^n$. We exhibit examples in one and two dimensions of sets $Ω$ for which these scales of Sobo…
Fractional Sobolev spaces via interpolation, and applications to mixed local-nonlocal operators
本文介绍分数阶Sobolev空间与插值理论的关联，给出其作为Lebesgue与整数阶Sobolev空间间实/复插值空间的等价刻画，并用插值理论证明混合局部-非局部算子的谱定理，为早期研究者提供该领域简洁入门内容。
In this paper we prove structural and topological characterizations of the screened Sobolev spaces with screening functions bounded below and above by positive constants. We generalize a method of interpolation to the case of seminormed spaces. This method, which we call the truncated method, generates the screened Sobolev subfamily and a more general screened Besov scale. We then prove that the screened Besov spaces are equivalent to the sum of a Lebesgue space and a homogeneous Sobolev space and provide a Littlewood-Paley frequency space characterization.
Real Interpolation method, Lorentz spaces and refined Sobolev inequalities
本文给出了Lorentz空间与Besov空间之间精化不等式的一个直接证明，推广了H. Bahouri和A.科恩我们的方法是基于洛伦兹空间作为真正的插值空间的特征。我们还将研究这些不等式的尖锐性和最优性。
The real interpolation spaces between $L^{p}({\mathbb{R}}^{n})$ and $\dot {H}^{t,p}({\mathbb{R}}^{n})$ (resp. $H^{t,p}({\mathbb{R}}^{n})$), $t>0,$ are characterized in terms of fractional moduli of smoothness, and the underlying seminorms are shown to be " the correct" fractional generalization of the classical Gagliardo seminorms. This is confirmed by the fact that, using the new spaces combined with interpolation and extrapolation methods, we are able to extend the Bourgain-Brezis-Mironescu-Maz'ya-Shaposhnikova limit formulae, as well as the Bourgain-Brezis-Mironescu convergence theorem, to …
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…
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 $\infty$.
Generalized Sobolev-Orlicz spaces based on the Riesz fractional gradient as interpolation and potential spaces
本文证明基于Riesz分数阶梯度的Musielak-Orlicz函数分数阶Sobolev空间与广义Orlicz框架下的Bessel位势空间一致，并识别为复插值空间，进而获得空间的结构性质。
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。
Logarithmic Sobolev and interpolation inequalities on the sphere: constructive stability results
本文研究球面上插值于 Poincaré 与 Sobolev 不等式之间的 Gagliardo–Nirenberg 不等式，并结合谱分解、熵及 carré du champ 方法证明次临界情形下的显式稳定性。
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。
The Refined Sobolev Scale, Interpolation, and Elliptic Problems
本文对广义光滑Hilbert空间中椭圆问题的近期结果作了详细的综述。后者是各向同性的霍曼德空间$H^{s,φ}:=B_{2,μ}$，其中$μ(ξ)=<ξ>^{s}φ(<ξ>)$是$ξ\in\mathbb{R}^{n}$。它们由实数$s$和在Karamata意义上缓慢变化的$+\infty$的正函数$φ$两者参数化。这些空间形成了精化的Sobolev尺度，它比Sobolev尺度${H^{s}}\equiv{H^{s,1}}$精细得多，并且相对于具有函数参数的插值是封闭的。椭圆算子和椭圆边值问题的Fredholm性质在这个新尺度下得以保持。给出了椭圆问题可解性的各种类型的定理。研究了椭圆型方程解的局部加细光滑性。得到了解具有连续导数的新的充分条件，并给出了它在椭圆算子谱理论中的一些应用。
Interpolation of subspaces and applications to exponential bases in Sobolev spaces
当Y_0是X_0的余维1的闭子空间时，给出了实插值空间[Y_0，X_1]_{s，p}与相应的插值空间[X_0，X_1]_{s，p}的闭子空间重合的精确条件.应用这一结果研究了区间上索伯列夫空间H^s中非调和傅里叶级数的基的性质，当0<s<1时.主要结果是：设E是指数族exp（i λ_n t），且E在区间L ^2中形成一个无条件基.则存在两个数s_0，s_1使得E在H^s中形成一个无条件基，其中s<s_0; E在H^s中形成一个余维数为1的无条件基，其中s_1<s。对于[s_0，s_1]中的s，指数族在其跨度上不是无条件基.
Fractional Sobolev Inequalities: Symmetrization, Isoperimetry and Interpolation
利用Peetre $K-$泛函和等周轮廓得到了度量空间中新的振动不等式.所提供的应用包括分数Sobolev不等式和Morrey-Sobolev嵌入定理在不同情况下的详细研究。特别是，我们包括高斯措施以及高斯和指数之间的概率措施的详细研究。我们展示了一种反向Polya-Szego原理，它允许我们使用对称化不等式从有界性获得连续性作为自我改进。我们的方法也允许精确估计广义Sobolev和Besov空间的增长包络的度量空间。我们还考虑嵌入到$BMO$和他们的连接到Sobolev嵌入。
On any complete Riemannian manifold $M$ and for all $p\in [2,\infty)$, we prove a family of second order $L^{p}$-interpolation inequalities that arise from the following simple $L^{p}$-estimate valid for every $u \in C^{\infty}(M)$: $$ \|\nabla u\|_{p}^p \leq \|u Δ_{p} u\|_1\in [0,\infty], $$ where $Δ_p$ denotes the $p$-Laplace operator. We show that these inequalities, in combination with abstract functional analytic arguments, allow to establish new global Sobolev regularity results for $L^p$-solutions of the Poisson equation for all $p\in (1,\infty)$, and new global Sobolev regularity resul…
We study the Sobolev spaces $H^σ(K)$ and $H^σ_0(K)$ on p.c.f. self-similar sets in terms of the boundary behavior of functions. First, for $σ\in \mathbb{R}^+$, we make an exact description of the tangents of functions in $H^σ(K)$ at the boundary. Second, we characterize $H_0^σ(K)$ as the space of functions in $H^σ(K)$ with zero tangent of an appropriate order depending on $σ$. Last, we extend $H^σ(K)$ to $σ\in\mathbb{R}$, and obtain various interpolation theorems with $σ\in\mathbb{R}^+$ or $σ\in\mathbb{R}$. We illustrate that there is a countable set of critical orders, that arises naturally i…
Parabolic problems and interpolation with a function parameter
本文给出了带函数参数的插值在抛物型微分算子中的一个应用。引入了由广义光滑的Hilbert函数空间构成的各向异性精化Sobolev尺度。后者的特点是由一个实数和函数在无穷远处缓慢变化卡拉马塔的意义。这种尺度通过带函数参数的插值与各向异性的索伯列夫空间相联系。本文研究了一类广义初边值抛物问题。我们证明，该问题对应的运营商集同构适当的空间之间属于这个规模。
Scattered Data Interpolation on Embedded Submanifolds with Restricted Positive Definite Kernels: Sobolev Error Estimates
研究通过限制正定核（如径向基函数）在光滑紧致嵌入子流形上构造核插值，并给出本征空间的完整刻画及Sobolev型误差估计。
We consider a homogeneous fractional Sobolev space obtained by completion of the space of smooth test functions, with respect to a Sobolev--Slobodecki\uı norm. We compare it to the fractional Sobolev space obtained by the $K-$method in real interpolation theory. We show that the two spaces do not always coincide and give some sufficient conditions on the open sets for this to happen. We also highlight some unnatural behaviors of the interpolation space. The treatment is as self-contained as possible.
This paper is devoted to Gaussian interpolation inequalities with endpoint cases corresponding to the Gaussian Poincaré and the logarithmic Sobolev inequalities, seen as limits in large dimensions of Gagliardo-Nirenberg-Sobolev inequalities on spheres. Entropy methods are investigated using not only heat flow techniques but also nonlinear diffusion equations as on spheres. A new stability result is established for the Gaussian measure, which is directly inspired by recent results for spheres.
A \Riesz-basis sequence for $L_2[-π,π]$ is a strictly increasing sequence $X:=(x_j)_{j\in\mathbb{Z}}$ in $\mathbb{R}$ such that the set of functions $\left(e^{-ix_j(\cdot)}\right)_{j\in\mathbb{Z}}$ is a Riesz basis for $L_2[-π,π]$. Given such a sequence and a parameter $0<h\leq1$, we consider interpolation of functions $g\in W_2^k(\mathbb{R})$ at the set $(hx_j)_{j\in\mathbb{Z}}$ via translates of the Gaussian kernel. Existence is shown of an interpolant of the form $$I^{hX}(g)(x):=\underset{j\in\mathbb{Z}}{\sum}a_je^{-(x-hx_j)^2},\quad x\in\mathbb{R},$$ which is continuous and square-integrab…
Hörmander Spaces, Interpolation, and Elliptic Problems
该研究专著首次系统地阐述了椭圆（标量和矩阵）算子理论和椭圆边值问题的尺度希尔伯特空间的Hörmander的函数/分布的任意正或负光滑。这本书是基于方法的插值与功能参数的抽象和索博列夫酉空间。一些结果也是新的Sobolev尺度。专着是为研究人员，教授和博士生。
Sobolev spaces on multiple cones
本文讨论了定义在多锥上的各种索伯列夫空间关于光滑函数的密度、插值和对$\RR^n$的扩张/限制的性质。有趣的分析结合使用Poincaré不等式和一些Hardy型不等式。
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 prove results on complex interpolation of vector-valued Sobolev spaces over the half-line with Dirichlet boundary condition. Motivated by applications in evolution equations, the results are presented for Banach space-valued Sobolev spaces with a power weight. The proof is based on recent results on pointwise multipliers in Bessel potential spaces, for which we present a new and simpler proof as well. We apply the results to characterize the fractional domain spaces of the first derivative operator on the half line.
In this paper we prove that several natural approaches to Sobolev spaces coincide on the Vicsek fractal. More precisely, we show that the metric approach of Korevaar-Schoen, the approach by limit of discrete $p$-energies and the approach by limit of Sobolev spaces on cable systems all yield the same functional space with equivalent norms for $p>1$. As a consequence we prove that the Sobolev spaces form a real interpolation scale. We also obtain $L^p$-Poincaré inequalities for all values of $p \ge 1$.
On interpolation of cocompact imbeddnings
余紧性是研究函数空间之间嵌入的一个有用的弱紧性对应。本文证明了\mathbb{R}^{N}上分数索伯列夫空间和Besov空间的亚临界连续嵌入是关于格移的余紧的.我们利用插值空间的技巧从经典索伯列夫空间的已知余紧嵌入（Lieb和Lions的“消失”引理）中推出了我们的结果.我们给出了余紧性在某些径向子空间嵌入的紧性和某些等周问题极小解的存在性中的应用例子，我们的研究补充了一系列以前的结果，并回顾了有一个自然的概念框架来统一它们.
We refine and generalize several interpolation inequalities bounding the $L^p$ norm of a probability density with respect to the reference measure $μ$ by its Sobolev norm and the Kantorovich distance to $μ$ on a smooth weighted Riemannian manifold satisfying $CD(0, \infty)$ condition.
We present the fractional perimeter as a set-function interpolation between the Lebesgue measure and the perimeter in the sense of De Giorgi. Our motivation comes from a new fractional Boxing inequality that relates the fractional perimeter and the Hausdorff content and implies several known inequalities involving the Gagliardo seminorm of the Sobolev spaces $W^{α, 1}$ of order $0 < α< 1$.
Asymptotic expansions and extremals for the critical Sobolev and Gagliardo-Nirenberg inequalities on a torus
本文全面地研究了零均值周期函数的插值不等式，包括极值、最佳常数、各种余项等的存在性和渐近展开.主要关注的是二维情形下的临界（对数）Sobolev不等式，虽然也得到了一些关于代数情形和不同空间维数下的最佳常数的结果.
In this article the authors study complex interpolation of Sobolev-Morrey spaces and their generalizations, Lizorkin-Triebel-Morrey spaces. Both scales are considered on bounded domains. Under certain conditions on the parameters the outcome belongs to the scale of the so-called diamond spaces.
Tangential interpolation in weighted vector-valued H^p spaces, with applications
本文在加权H^p空间中得到了向量值解析函数的最小范数切插值问题的范数估计，用相关标量测度的Carleson常数表示.给出了线性半群系统的p-能控性和函数能控性概念在某些索伯列夫空间中的应用。
Bessel Potential Spaces and Complex Interpolation: Continuous embeddings
本文基于抽象插值理论研究贝塞尔位势空间作为复插值空间的性质，给出范数等价性、连续嵌入及其与Gagliardo空间关系的原创证明。
Sobolev spaces and averaging I
对Lebesgue空间$L_{loc}(R^n)$中的函数定义了一个新的概念--极大平均差商.我们的定义对于一般测度度量空间上的向量值函数也很有意义，并且似乎可以导出最自然的度量索伯列夫空间类。在广义Taylor-Whitney喷注的背景下，也可以讨论高阶索伯列夫空间和欧几里得空间的正则子集上的高阶平均差导数。本文是B. Bojarski，Taylor展开和索伯列夫空间，Bull. Georgian Natl. Acad. Sci.（未注明）5（2011），no. 2，5-10. B. Bojarski，索伯列夫空间和Lagrange插值，Proc. A. Razmadze Math.Inst.158（2012），1-12.
In this paper we investigate the Besov spaces on compact Lie groups in a subelliptic setting, that is, associated with a family of vector fields, satisfying the Hörmander condition, and their corresponding sub-Laplacian. Embedding properties between subelliptic Besov spaces and Besov spaces associated to the Laplacian on the group are proved. We link the description of subelliptic Sobolev spaces with the matrix-valued quantisation procedure of pseudo-differential operators in order to provide subelliptic Sobolev and Besov estimates for operators in the Hörmander classes. Interpolation properti…
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。
This book provides a gentle introduction to fractional Sobolev spaces, which play a central role in the calculus of variations, partial differential equations, and harmonic analysis. The first part deals with fractional Sobolev spaces of one variable. It covers the definition, standard properties, extensions, embeddings, Hardy inequalities, and interpolation inequalities. The second part deals with fractional Sobolev spaces of several variables. The author studies completeness, density, homogeneous fractional Sobolev spaces, embeddings, necessary and sufficient conditions for extensions, Gagli…
Parameter-elliptic operators on the extended Sobolev scale
在推广的Sobolev尺度下研究了闭光滑流形上的参数椭圆型伪微分算子.这个尺度由所有相对于希尔伯特索伯列夫尺度是插值空间的希尔伯特空间组成。我们证明，这些运营商之间的规模提供了适当的空间同构的参数是模足够大。对于相应的参数--椭圆型方程的解，我们建立了双边的先验估计，其中常数与参数无关。
On the Sierpinski gasket $\mathcal{SG}$, we consider Sobolev spaces $L^2_σ(\mathcal{SG})$ associated with the standard Laplacian $Δ$ with order $σ\geq 0$. When $σ\in\mathbb{Z}^+$, $L^2_σ(\mathcal{SG})$ consists of functions equipped with $L^2$ norms of the function itself and its Laplacians up to $σ$ order; when $σ\notin \mathbb{Z}^+$, we fill up the gaps between integer orders by using complex interpolation. Let $L^2_{σ, D}(\mathcal{SG})=(I-Δ_D)^{-σ}L^2(\mathcal{SG})$ where $Δ_D$ is the Dirichlet Laplacian associated with $Δ$. Let $\{p_n\}_{n\geq 0}$ be a collection of countably many points l…
Green's function asymptotics and sharp point-wise interpolation inequalities
本文提出了一种寻找n维Riemann流形上m阶Hilbert索伯列夫空间空间到有界连续函数空间（其中m>n/2）的嵌入中的锐常数的一般方法.该方法是基于对2 m阶椭圆算子的格林函数的谱参数的渐近性的分析，其平方根的域定义了相应的索伯列夫空间的范数。详细讨论了n维环面和n维球面的情形，以及一些有边界的流形。在某些情形下，当流形是紧的时，得到了各种类型的带余项的乘法不等式。带修正项的周期函数不等式是著名的Carlson不等式的改进。
We introduce an extended Sobolev scale on a smooth compact manifold with boundary. The scale is formed by inner-product Hörmander spaces for which an RO-varying radial function serves as a regularity index. These spaces do not depend on a choice of local charts on the manifold. The scale consists of all Hilbert spaces that are interpolation ones for pairs of inner-product Sobolev spaces, is obtained by the interpolation with a function parameter of these pairs, and is closed with respect to this interpolation. As an application of the scale introduced, we give a theorem on the Fredholm propert…
We consider Banach spaces equipped with a set of strongly continuous bounded semigroups satisfying certain conditions. Using these semigroups we introduce an analog of a modulus of continuity and define analogs of Besov norms. A generalization of a classical interpolation theorem is proven in which the role of Sobolev spaces is played by subspaces defined in terms of infinitesimal operators of these semigroups. We show that our assumptions about a given set of semigroups are satisfied in the case of a strongly continuous bounded representation of a Lie group. In the case of a unitary represent…
It is a widely used method, for instance in perturbation theory, to associate with a given C0-semigroup its so-called interpolation and extrapolation spaces. In the model case of the shift semigroup acting on L^2(R), the resulting chain of spaces recovers the classical Sobolev scale. In 2014, the second named author defined the universal interpolation space as the projective limit of the interpolation spaces and the universal extrapolation space as the completion of the inductive limit of the extrapolation spaces, provided that the latter is Hausdorff. In this note we use the notion of the dua…
We prove compactness of the embeddings in Sobolev spaces for fractional super and sub harmonic functions with radial symmetry. The main tool is a pointwise decay for radially symmetric functions belonging to a function space defined by finite homogeneous Sobolev norm together with finite $L^2$ norm of the Riesz potentials. As a byproduct we prove also existence of maximizers for the interpolation inequalities in Sobolev spaces for radially symmetric fractional super and sub harmonic functions.
This paper deals with the fractional Sobolev spaces W^[s,p]. We analyze the relations among some of their possible definitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results. Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.
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…
A Functorial Approach to Multi-Space Interpolation with Function Parameters
本文提出了一种基于函数参数的多空间插值方法，通过构建广义中间空间并确保稳定性，展示了其在广义Sobolev空间插值中的应用及理论价值。
We propose a new viewpoint on Hilbert scales extending them by means of all Hilbert spaces that are interpolation ones between spaces on the scale. We prove that this extension admits an explicit description with the help of $\mathrm{OR}$-varying functions of the operator generating the scale. We also show that this extended Hilbert scale is obtained by the quadratic interpolation (with function parameter) between the above spaces and is closed with respect to the quadratic interpolation between Hilbert spaces. We give applications of the extended Hilbert scale to interpolational inequalities,…
Given a metric measure space $X$, we consider a scale of function spaces $T^{p,q}_s(X)$, called the weighted tent space scale. This is an extension of the tent space scale of Coifman, Meyer, and Stein. Under various geometric assumptions on $X$ we identify some associated interpolation spaces, in particular certain real interpolation spaces. These are identified with a new scale of function spaces, which we call $Z$-spaces, that have recently appeared in the work of Barton and Mayboroda on elliptic boundary value problems with boundary data in Besov spaces. We also prove Hardy--Littlewood--Sob…
Explicit traces of functions on Sobolev spaces and quasi-optimal linear interpolators
设$Λ\subset R$为严格递增序列。对于$r = 1,2$，我们给出了非齐性和齐性索伯列夫空间的迹空间$W_p^r(R)|_Λ$，$L_p^r(R)|_Λ$上具有$r$导数$W_p^r(R)$，$L_p^r(R)$的等价范数的一个简单的显式表达式.我们还构造了一个低次插值样条，它具有最优范数，直到一个常数因子。文中还给出了对所有$r \geq 1$，在$L^r_p(R)$和$W^r_p(R)$中插值的一般结果.
Minimal Norm Interpolation in Harmonic Hilbert Spaces and Wiener Amalgam Spaces on Locally Compact Abelian Groups
调和希尔伯特空间族是由连续函数构成的经典$L^{2}$ - Sobolev空间在$\R^d$上的自然扩展.在本文中，我们证明了使用的基本结果，从理论的维纳汞齐空间允许建立调和希尔伯特空间的基本性质，即使它们被定义在一个任意的局部紧阿贝尔群$\G$。即使对于$\G = \R^{d}$，这种新方法也改进了先前已知的结果。本文给出了格上极小范数插值的一些结果，证明了无限极小范数插值是有限极小范数插值的极限。此外，新方法为研究局部紧阿贝尔群上调和Hilbert空间和Banach空间中范数插值的稳定性问题和误差分析铺平了道路。
We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on $\mathbb{R}^n$, using sizes of superlevel sets of suitable difference quotients. This provides an alternative point of view to the BBM formula by Bourgain, Brezis and Mironescu, and complements in the case of BV some results of Cohen, Dahmen, Daubechies and DeVore about the sizes of wavelet coefficients of such functions. An application towards Gagliardo-Nirenberg interpolation inequalities is then given. We also establish a related one-parameter family of formulae for the $L^p$ norm of functions in …
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