A new characterization of the Sobolev space (original) (raw)
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A new characterization of Sobolev spaces on {\mathbb{R}^{n}}$$
Mathematische Annalen, 2012
In this paper we present a new characterization of Sobolev spaces on R n . Our characterizing condition is obtained via a quadratic multiscale expression which exploits the particular symmetry properties of Euclidean space. An interesting feature of our condition is that depends only on the metric of R n and the Lebesgue measure, so that one can define Sobolev spaces of any order of smoothness on any metric measure space.
A new characterization of Sobolev spaces on Rn
2011
In this paper we present a new characterization of Sobolev spaces on Rn. Our characterizing condition is obtained via a quadratic multiscale expression which exploits the particular symmetry properties of Euclidean space. An interesting feature of our condition is that depends only on the metric of Rn and the Lebesgue measure, so that one can define Sobolev spaces of any order of smoothness on any metric measure space.
Rendiconti Lincei - Matematica e Applicazioni
We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on 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 L p (R n).
Characterization of Sobolev and BV Spaces
The main results of this paper are new characterizations of W 1,p (Ω), 1 < p < ∞, and BV (Ω) for Ω ⊂ R N an arbitrary open set. Using these results, we answer some open questions of Brezis and Ponce .
Master of ScienceDepartment of MathematicsMarianne KortenThe goal for this paper is to present material from Gilbarg and Trudinger’s Elliptic Partial Differential Equations of Second Order chapter 7 on Sobolev spaces, in a manner easily accessible to a beginning graduate student. The properties of weak derivatives and there relationship to conventional concepts from calculus are the main focus, that is when do weak and strong derivatives coincide. To enable the progression into the primary focus, the process of mollification is presented and is widely used in estimations. Imbedding theorems and compactness results are briefly covered in the final sections. Finally, we add some exercises at the end to illustrate the use of the ideas presented throughout the paper
Lipschitz spaces generated by the Sobolev-Poincar 'e inequality and extensions of Sobolev functions
arXiv:1310.0795, 2013
Let ddd be a metric on RnR^nRn and let Cm,(d)(Rn)C^{m,(d)}(R^n)Cm,(d)(Rn) be the space of CmC^mCm-function on RnR^nRn whose partial derivatives of order mmm belong to the space Lip(Rn;d)Lip(R^n;d)Lip(Rn;d). We show that the homogeneous Sobolev space Lm+1p(Rn),p>n,L^{m+1}_p(R^n),p>n,Lm+1p(Rn),p>n, can be represented as a union of Cm,(d)(Rn)C^{m,(d)}(R^n)Cm,(d)(Rn)-spaces where ddd belongs to a family of metrics on RnR^nRn with certain "nice" properties. This enables us in several important cases to give intrinsic characterizations of the restrictions of Sobolev spaces to arbitrary closed subsets of RnR^nRn. In particular, we generalize the classical Whitney extension theorem for the space Cm(Rn)C^m(R^n)Cm(Rn) to the case of the Sobolev space Lmp(Rn)L^m_p(R^n)Lmp(Rn) whenever mge1m\ge 1mge1 and p>np>np>n.
Sobolev-Poincaré inequalities for p<1
Indiana University Mathematics Journal
If Ω is a John domain (or certain more general domains), and |∇u| satisfies a certain mild condition, we show that u ∈ W 1,1 loc (Ω) satisfies a Sobolev-Poincaré inequality (∫ Ω |u − a| q )
Sobolev-Poincare inequalities for p < 1
1994
Abstract. If Ω is a John domain (or certain more general domains), and |∇u | satisfies a certain mild condition, we show that u ∈W 1;1loc (Ω) satisfies a Sobolev-Poincare ́ inequality(∫ Ω |u − a|q)1=q ≤ C (∫Ω |∇u|p)1=p for all 0 < p < 1, and appropriate q> 0. Our conclusion is new even when Ω is a ball.
Teubner Texte zur Mathematik, 1998
The book is based on the lecture course "Function spaces", which the author gave for more than 10 years in the People's Friendship University of Russia (Moscow). The idea to write this book was proposed by Professors H. Triebel and H.-J. Schmeisser in May-June 1993, when the author gave a short lecture course for post-graduate students in the Friedrich-Schiller University Jena. The initial plan to write a short book for postgraduate students was transformed to wider aims after the work on the book had started. Finally, the book is intended both for graduate and postgraduate students and for researchers, who are interested in applying the theory of Sobolev spaces. Moreover, the methods used in the book allow us to include, in a natural way, some recent results, which have been published only in journals. Nowadays there exist numerous variants and generalizations of Sobolev spaces and it is clear that this variety is inevitable since different problems in real analysis and partial dfferential equations give rise to different spaces of Sobolev type. However, it is more or less clear that an attempt to develop a theory, which includes all these spaces, would not be effective. On the other hand, the basic ideas of the investigation of such spaces have very much in common. For all these reasons we restrict ourselves to the study of Sobolev spaces themselves. However, we aim to discuss the main ideas in detail, and in such a way that, we hope, it will be clear how to apply them to other types of Sobolev spaces. We shall discuss the following main topics: approximation by smooth functions, integral representations, embedding and compactness theorems, the problem of traces and extension theorems. The basic tools of investigation will be mollffers with a variable step and integral representations. Molliffers with a variable step are used both for approximation by smooth functions and for extension of functions (from open sets in Rn in Chapter 6 and from manifolds of lower dimensions in Chapter 5). All approximation and extension operators constructed in these chapters are the best possible in the sense that the derivatives of higher orders of approximating and extending functions have the minimal possible growth on approaching the boundary. Sobolev's integral representation is discussed in detail in Chaper 3. It is used in the proofs of the embedding theorems (Chapter 4) and some essential estimates in Chapter 6. An alternative proof of the embedding theorems, without application of Sobolev's integral representation, is also given. The direct trace theorems (Chapter 5) are proved on the basis of some elementary identities for the differences of higher orders and the definition of Nikol'skii-Besov spaces in terms of differences only. The author pays particular attention to all possible \limiting" cases, including the cases p = 1 in approximation theorems, p = 1 in embedding theorems and p = 1;1 in extension theorems. There are no references to the literature in the main text (Chapters 1 - 6): all relevant references are to be found in Chapter 7, which consists of brief notes and comments on the results presented in the earlier chapters. The proofs of all statements in the book consist of two parts: the idea of the proof and the proof itself. In some simple or less important cases the proofs are omitted. On the other hand, the proofs of the main results are given in full detail and sometimes alternative proofs are also given or at least discussed. The one-dimensional case is often discussed separately to provide a better understanding of the origin of multi-dimensional statements. Also sharper results for this case are presented. It is expected that the reader has a sound basic knowledge of functional analysis, the theory of Lebesgue integration and the main properties of the spaces Lp. It is desirable, in particular, that he/she is accustomed to applying Holder's and Minkowski's inequalities for sums and integrals. The book is otherwise self-contained: all necessary references are given in the text or footnotes. Each chapter has its own numeration of theorems, corollaries, lemmas, etc. If you are reading, say, Chapter 4 and Theorem 2 is mentioned, then Theorem 2 of Chapter 4 is meant. If we refer to a theorem in another chapter, we give the number of that chapter, say, Theorem 2 of Chapter 3. For more than 30 years the author participated in the famous seminar "The theory of differentiable functions of several variables and applications" in the Steklov Institute of Mathematics (Moscow) headed at different times by Professors S.L. Sobolev, V.I. Kondrashov, S.M. Nikol'skii L.D. Kudryavtsev and O.V. Besov. He was much influenced by ideas discussed during its work and, in particular, by his personal talks with Professors S.M. Nikol'skii and S.L. Sobolev. It is a pleasure for the author to express his deepest gratitude to the participants of that seminar, to his friends and co-authors, with whom he discussed the general plan and different parts of the book. I am grateful to my colleagues in the University of Wales Cardiff: Professor W.D. Evans, with whom I have had many discussions, and Mr. D.J. Harris, who has thoroughly read the manuscript of the book. I would also like to mention Dr. A.V. Kulakov who has actively helped in typing the book in TEX. Finally, I express my deepest love, respect and gratitude to my wife Dr. T.V. Tararykova who not only typed in TEX a considerable part of the book but also encouraged me in all possible ways. Moscow/Cardi , November 1997, V.I. Burenkov