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 in the boundary behavior of functions, such that $H^σ_0(K)$ presents a critical phenomenon if $σ$ is critical. These orders will play a crucial role in our study. They are just the values in $\frac 12+\mathbb{Z}_+$ in the classical case, but are much more complicated in the fractal case.
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.
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…
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…
The purpose of this work is to describe an abstract theory of Hardy-Sobolev spaces on doubling Riemannian manifolds via an atomic decomposition. We study the real interpolation of these spaces with Sobolev spaces and finally give applications to Riesz inequalities.
Functions in a Sobolev space are approximated directly by piecewise affine interpolation in the norm of the space. The proof is based on estimates for interpolations and does not rely on the density of smooth functions.
Sobolev extensions, interpolation inequalities and consequences
Ludwig-Maximilians Universität München（慕尼黑大学）；Munich Center for Quantum Science and Technology（慕尼黑量子科学与技术中心）；Caltech（加州理工学院）；University of Jyväskylä（于韦斯屈莱大学）
We firstly describe a maximal inequality for dual Sobolev spaces W^{-1,p}. This one corresponds to a "Sobolev version" of usual properties of the Hardy-Littlewood maximal operator in Lebesgue spaces. Even in the euclidean space, this one seems to be new and we develop arguments in the general framework of Riemannian manifold. Then we present an application to obtain interpolation results for Sobolev spaces.
The paper gives a detailed survey of recent results on elliptic problems in Hilbert spaces of generalized smoothness. The latter are the isotropic Hörmander spaces $H^{s,φ}:=B_{2,μ}$, with $μ(ξ)=<ξ>^{s}φ(<ξ>)$ for $ξ\in\mathbb{R}^{n}$. They are parametrized by both the real number $s$ and the positive function $φ$ varying slowly at $+\infty$ in the Karamata sense. These spaces form the refined Sobolev scale, which is much finer than the Sobolev scale ${H^{s}}\equiv{H^{s,1}}$ and is closed with respect to the interpolation with a function parameter. The Fredholm property of elliptic operators a…
We give precise conditions under which the real interpolation space [Y_0,X_1]_{s,p} coincides with a closed subspace of the corresponding interpolation space [X_0,X_1]_{s,p} when Y_0 is a closed subspace of X_0 of codimension one. This result is applied to study the basis properties of nonharmonic Fourier series in Sobolev spaces H^s on an interval when 0<s<1. The main result: let E be a family of exponentials exp(i λ_n t) and E forms an unconditional basis in L^2 on an interval. Then there exist two number s_0, s_1 such that E forms an unconditional basis in H^s for s<s_0, E forms an uncondit…
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.
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.
In this article we give a straightforward proof of refined inequalities between Lorentz spaces and Besov spaces and we generalize previous results of H. Bahouri and A. Cohen. Our approach is based in the characterization of Lorentz spaces as real interpolation spaces. We will also study the sharpness and optimality of these inequalities.
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 $\infty$.
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…
We obtain new oscillation inequalities in metric spaces in terms of the Peetre $K-$functional and the isoperimetric profile. Applications provided include a detailed study of Fractional Sobolev inequalities and the Morrey-Sobolev embedding theorems in different contexts. In particular we include a detailed study of Gaussian measures as well as probablity measures between Gaussian and exponential. We show a kind of reverse Polya-Szego principle that allows us to obtain continuity as a self improvement from boundedness, using symetrization inequalities. Our methods also allow for precise estimat…
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 give an application of interpolation with a function parameter to parabolic differential operators. We introduce the refined anisotropic Sobolev scale that consists of some Hilbert function spaces of generalized smoothness. The latter is characterized by a real number and a function varying slowly at infinity in Karamata's sense. This scale is connected with anisotropic Sobolev spaces by means of interpolation with a function parameter. We investigate a general initial--boundary value parabolic problem in the refined Sobolev scale. We prove that the operator corresponding to this problem se…
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 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.
We study removable sets for the Orlicz-Sobolev space $W^{1,Ψ},$ for functions of the form $Ψ(t)=t^p\log^λ(e+t).$ We show that $(p,λ)$-porous sets lying in a hyperplane are removable and that this result is essentially sharp.
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.
The research monograph gives the first systematic exposition of the elliptic (scalar and matrix) operators theory and elliptic boundary-value problems in the scales of Hilbert spaces of Hörmander of the functions/distributions of arbitrary positive or negative smoothness. The book is based on the method of interpolation with a functional parameter for the abstract and Sobolev unitary spaces. Some results are also new for the Sobolev scales. The monograph is intended for the researches, professors and PhD students.
The properties of the compactness of interpolation sets of algebras of generalized analytic functions are investigated and convenient sufficient conditions for interpolation are given.
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 $0 < α< 1$.
We give a comprehensive study of interpolation inequalities for periodic functions with zero mean, including the existence of and the asymptotic expansions for the extremals, best constants, various remainder terms, etc. Most attention is paid to the critical (logarithmic) Sobolev inequality in the two-dimensional case, although a number of results concerning the best constants in the algebraic case and different space dimensions are also obtained.
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.
We consider the Sobolev type spaces $H^σ(K)$ with $σ\geq 0$, where $K$ is a post-critically finite self-similar set with the natural boundary. Firstly, we compare different classes of Sobolev spaces $H^σ_N(K),H^σ_D(K)$ and $H^σ(K)$, {and observe} a sequence of critical orders of $σ$ in our comparison theorem. Secondly, We present a general atomic decomposition theorem of Sobolev spaces $H^σ(K)$, where the same critical orders play an important role. At the same time, we provide purely analytic approaches for various Besov type characterizations of Sobolev spaces $H^σ(K)$.
An apparently new concept of maximal mean difference quotient is defined for functions in the Lebesgue space $L_{loc}(R^n)$. Our definitions are meaningful for vector valued functions on general measure metric spaces as well and seem to lead to the most natural class of metric Sobolev spaces. The discussion of higher order Sobolev spaces and higher order mean difference quotients on regular subsets of Euclidean spaces is also possible in the context of the generalized Taylor-Whitney jets. This paper is a direct sequel to the papers: B. Bojarski, Taylor expansions and Sobolev spaces, Bull. Geor…
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…
We study conditions on closed sets $C,F \subset \mathbb{R}$ making the product $C \times F$ removable or non-removable for $W^{1,p}$. The main results show that the Hausdorff-dimension of the smaller dimensional component $C$ determines a critical exponent above which the product is removable for some positive measure sets $F$, but below which the product is not removable for another collection of positive measure totally disconnected sets $F$. Moreover, if the set $C$ is Ahlfors-regular, the above removability holds for any totally disconnected $F$.
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…
Let $E$ be a Sidon subset of the integers and suppose $X$ is a Banach space. Then Pisier has shown that $E$-spectral polynomials with values in $X$ behave like Rademacher sums with respect to $L_p-$norms. We consider the situation when $X$ is a quasi-Banach space. For general quasi-Banach spaces we show that a similar result holds if and only if $E$ is a set of interpolation ($I_0$-set). However for certain special classes of quasi-Banach spaces we are able to prove such a result for larger sets. Thus if $X$ is restricted to be ``natural'' then the result holds for all Sidon sets. We also cons…
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.
Cocompactness is a useful weaker counterpart of compactness in the study of imbeddings between function spaces. In this paper we show that subcritical continuous imbeddings of fractional Sobolev spaces and Besov spaces over \mathbb{R}^{N} are cocompact relative to lattice shifts. We use techniques of interpolation spaces to deduce our results from known cocompact imbeddings for classical Sobolev spaces ("vanishing" lemmas of Lieb and Lions). We give examples of applications of cocompactness to compactness of imbeddings of some radial subspaces and to existence of minimizers in some isoperimetr…
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…
This paper concerns the following question: given a subset $E$ of $\mathbb{R}^n$ with empty interior and an integrability parameter $1<p<\infty$, what is the maximal regularity $s\in\mathbb{R}$ for which there exists a non-zero distribution in the Bessel potential Sobolev space $H^{s,p}(\mathbb{R}^n)$ that is supported in $E$? For sets of zero Lebesgue measure we apply well-known results on set capacities from potential theory to characterise the maximal regularity in terms of the Hausdorff dimension of $E$, sharpening previous results. Furthermore, we provide a full classification of all poss…
Sobolev wavefront sets and $2$-microlocal spaces play a key role in describing and analyzing the singularities of distributions in microlocal analysis and solutions of partial differential equations. Employing the continuous shearlet transform to Sobolev spaces, in this paper we characterize the microlocal Sobolev wavefront sets, the $2$-microlocal spaces, and local Hölder spaces of distributions/functions. We then establish the connections among Sobolev wavefront sets, $2$-microlocal spaces, and local Hölder spaces through the continuous shearlet transform.
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.
A set $E \subset \mathbb{N}$ is an interpolation set for nilsequences if every bounded function on $E$ can be extended to a nilsequence on $\mathbb{N}$. Following a theorem of Strzelecki, every lacunary set is an interpolation set for nilsequences. We show that sublacunary sets are not interpolation sets for nilsequences. Furthermore, we prove that the union of an interpolation set for nilsequences and a finite set is an interpolation set for nilsequences. Lastly, we provide a new class of interpolation sets for Bohr almost periodic sequences, and as the result, obtain a new example of interpo…
In this paper, norm estimates are obtained for the problem of minimal-norm tangential interpolation by vector-valued analytic functions in weighted H^p spaces, expressed in terms of the Carleson constants of related scalar measures. Applications are given to the notion of p-controllability properties of linear semigroup systems and controllability by functions in certain Sobolev spaces.
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.
The purpose of this note is to discuss how various Sobolev spaces defined on multiple cones behave with respect to density of smooth functions, interpolation and extension/restriction to/from $\RR^n$. The analysis interestingly combines use of Poincaré inequalities and of some Hardy type inequalities.
Parameter--elliptic pseudodifferential operators given on a closed smooth manifold are investigated on the extended Sobolev scale. This scale consists of all Hilbert spaces that are interpolation spaces with respect to the Hilbert Sobolev scale. We prove that these operators set isomorphisms between appropriate spaces of the scale provided the parameter is modulo large enough. For solutions to the corresponding parameter--elliptic equations, we establish two-sided a priori estimates, in which the constants are independent of the parameter.
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…
