Weighted Sobolev Spaces on Metric Measure Spaces

1998

There have been recent attempts to develop the theory of Sobolev spaces W1;pon metric spaces that do not admit any differentiable structure. We prove thatcertain definitions are equivalent. We also define the spaces in the limiting casep = 1.1. Introduction.Let\Omegaae IRnbe an open set. By the classical Sobolev spaceW1;p(\Omega\Gamma we mean the Banach space of those p-summable functions whose distributionalgradients

Indiana University Mathematics Journal, 2005

§0. Introduction and main results Let p ∈ [1, +∞), k ∈ N, and let µ = (µ 0 , µ 1 ,. .. , µ k) be a (k + 1)-tuple of positive finite Borel measures on the unit circle T = {z : |z| = 1} in the complex plane. Consider the continuous mapping Π : C k (T) → k j=0 C(T), given by Πf = f, f ,. .. , f (k) , where f (z) = df dz (all spaces of functions that we consider are complex-valued). Note that df dz (e iθ) = −ie −iθ d dθ f (e iθ). Definition. The abstract Sobolev space W k,p (µ) = W k,p (µ 0 ,. .. , µ k) is the closure of ΠC k (T) in the space k j=0 L p (T, µ j). We refer to [17] for the classical theory of Sobolev spaces in domains of R n. We refer to [13, 18, 5, 12] for the theory of weighted Sobolev spaces in domains of R n ; in [11, 14], one can find applications of this topic to partial differential equations. We consider in W k,p (µ) the usual norm f k,p,µ = k j=0 f j p p,µ j 1/p , f = (f 0 ,. .. , f k). Each function f in C k (T) has its image Πf in W k,p (µ), and these images are dense in W k,p (µ). In many cases, an element g = (g 0 ,. .. , g k) in W k,p (µ) is completely determined by its first component g 0 , so that W k,p (µ) can be identified with a certain space of measurable functions g 0 , and the components g 1 ,. .. , g k can be thought of as a kind of generalized derivatives of g 0. In general, however, elements of W k,p (µ) cannot be identified with scalar functions on T. This setting of abstract Sobolev spaces is the most natural for us. (See [2], [22]-[26] in order to know when W k,p (µ) is a space of functions.) This space plays a central role in the theory of orthogonal polynomials with respect to Sobolev inner products (see [2], [15], [16] and [23]; in [2] and [16], the authors consider measures supported in compact sets in the complex plane). In fact, if the multiplication operator (M f)(z) = zf (z) is bounded in W k,2 (µ), then every 1 The research of the first author was partially supported by two grants from DGI (BFM 2003-06335-C03-02 and BFM 2003-04870), Spain. 2 The research of the second author was partially supported by the Ramón y Cajal Programme by the Ministry of Science and Technology of Spain.

2014

We prove that the density of locally Lipschitz functions in a global Sobolev space based on a Banach function space implies the density of Lipschitz functions, with compact support in a given open set, in the corresponding Sobolev space with zero boundary values. In the case, when the Banach function space is a Lebesgue space, we recover some density results of Björn, Björn and Shanmugalingam (2008). Our results require neither a doubling measure nor the validity of a Poincaré inequality in the underlying metric measure space.

Commentationes Mathematicae Universitatis Carolinae, 1984

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz COMMENTATIONES MATH EM ATI CAE UNIVERSITATIS CAfcOLINAE 25(3) 1984 HOW TO DEFINE REASONABLY WEIGHTED SOBOLEV SPACES Alois KUFNER, Bohumír OPIC Dedicated to the memory of Svatopluk FUCIK

We consider general weighted Lipschitz spaces defined by a Banach space. Under certain assumptions on the weight and the space, we find a Littlewood-Paley type formulation for such spaces. This allows us to give a formulation for the predual space as a generalized Besov space. We also prove that operators acting on certain weighted Besov spaces corresponds to vector valued functions in a natural way.

Journal of Mathematical Analysis and Applications, 2007

For the Riesz potential operator I α there are proved weighted estimates I α f L q(•) (Ω,w q p) C f L p(•) (Ω,w) , Ω ⊆ R n , 1 q(x) ≡ 1 p(x) − α n within the framework of weighted Lebesgue spaces L p(•) (Ω, w) with variable exponent. In case Ω is a bounded domain, the order α = α(x) is allowed to be variable as well. The weight functions are radial type functions "fixed" to a finite point and/or to infinity and have a typical feature of Muckenhoupt-Wheeden weights: they may oscillate between two power functions. Conditions on weights are given in terms of their Boyd-type indices. An analogue of such a weighted estimate is also obtained for spherical potential operators on the unit sphere S n ⊂ R n .

Journal of Approximation Theory, 2001

In this paper we make a survey of some recent developments of the theory of Sobolev spaces W 1,q (X, d, m), 1 < q < ∞, in metric measure spaces (X, d, m). In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on Γ-convergence; this result extends Cheeger's work because no Poincaré inequality is needed and the measure-theoretic doubling property is weakened to the metric doubling property of the support of m. We also discuss the lower semicontinuity of the slope of Lipschitz functions and some open problems.

Mathematische Zeitschrift, 2006

In this article we study metric measure spaces with variable dimension. We consider Lebesgue spaces on these sets, and embeddings of the Riesz potential in these spaces. We also investigate Hajłasz-type Sobolev spaces, and prove Sobolev and Trudinger inequalities with optimal exponents. All of these questions lead naturally to function spaces with variable exponents.

Annales- Academiae Scientiarum Fennicae Mathematica

We develop a capacity theory based on the definition of Sobolev functions on metric spaces with a Borel regular outer measure. Basic properties of capacity, including monotonicity, countable subadditivity and several convergence results, are studied. As an application we prove that each Sobolev function has a quasicontinuous representative. For doubling measures we provide sharp estimates for the capacity of balls. Capacity and Hausdorff measures are related under an additional regularity assumption on the measure.

Nonlinear Analysis: Theory, Methods &amp; Applications, 2015

We give short simple proofs of Uspenskii's results characterizing Besov spaces as trace spaces of weighted Sobolev spaces. We generalize Uspenskii's results and prove the optimality of these generalizations. We next show how classical results on the functional calculus in the Besov spaces can be obtained as straightforward consequences of the theory of weighted Sobolev spaces.

Bulletin of Mathematical Sciences, 2017

Weighted Sobolev spaces play a main role in the study of Sobolev orthogonal polynomials. The aim of this paper is to prove several important properties of weighted Sobolev spaces: separability, reflexivity, uniform convexity, duality and Markov-type inequalities.

Transactions of the American Mathematical Society, 2009

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 operators for weighted Sobolev spaces. With the help of these homeomorphism classes the embedding problem for non-smooth domains is reduced to the corresponding classical embedding problem for smooth domains. Examples of domains with anisotropic Hölder singularities demonstrate sharpness of our machinery comparatively with known results.

1994

Introduction and statement of main resultsIn this note we deal with the problem of the density of smooth functions in weightedSobolev spaces (for general results and references on this topic see, for instance, [14], [10],[2], and the bibliography therein). In order to introduce some definitions, let us fix abounded openset\Omega` IRn, a real number p ? 1, and a function

2016

In the thesis we discuss several questions related to the study of degenerate, possibly nonlinear PDEs of elliptic type. At first we discuss the equivalent conditions between the validity of weighted Poincaré inequalities, structure of the functionals on weighted Sobolev spaces, isoperimetric inequalities and the existence and uniqueness of solutions to the degenerate nonlinear elliptic PDEs with nonhomogeneous boundary condition, having the form: { div (ρ(x)|∇u|p−2∇u) = x∗, u− w ∈ W 1,p ρ,0 (Ω), (0.0.1) involving any given x∗ ∈ (W 1,p ρ,0 (Ω))∗ and w ∈ W 1,p ρ (Ω), where u ∈ W 1,p ρ (Ω) and W 1,p ρ (Ω) denotes certain weighted Sobolev space, W 1,p ρ,0 (Ω) is the completion of C∞ 0 (Ω). As a next step, we undertake a natural question how to interpret the nonhomogenous boundary conditions in weighted Sobolev spaces, when the natural analytical tools, like trace embedding theorems, are missing. Our further goal is to contribute to solvability and uniqueness for degenerate elliptic PDE...