Showing papers on "Birnbaum–Orlicz space published in 2001"
Book•
01 Jan 2001
TL;DR: Banach spaces have been studied extensively in the literature, see as mentioned in this paper for a survey of some of the main aspects of the Banach spaces and its application in the analysis of finite dimensional subspaces.
Abstract: Basic concepts in the geometry of Banach spaces (W.B. Johnson, J. Lindenstrauss). Positive operators (Y.A. Abramovitch, C.D. Aliprantis). Lp spaces (D. Alspach, E. Odell). Convex geometry and functional analysis (K. Ball). A p-sets in analysis: Results, problems and related aspects (J. Bourgain). Martingales and singular integrals in Banach spaces (D.L. Burkholder). Approximation properties (P.G. Casazza). Local operator theory, random matrices and Banach spaces (K.R. Davidson, S.J. Szarek). Applications to mathematical finance (F. Delbaen). Perturbed minimization principles and applications (R. Deville, N. Ghoussoub). Operator ideals (J. Diestel, H. Jarchow, A. Pietsch). Special Banach lattices and their applications(S.J. Dilworth). Some aspects of the invariant subspace problem (P. Enflo,V. Lomonosov). Special bases in function spaces (T. Figel, P. Wojtaszczyk). Infinite dimensional convexity (V. Fonf, J. Lindenstrauss, R.R. Phelps). Uniform algebras as Banach spaces (T.W. Gamelin, S.V. Kisliakov). Euclidean structure in finite dimensional normed spaces (A.A. Giannopoulos, V.D. Milman). Renormings of Banach spaces (G. Godefroy). Finite dimensional subspaces of Lp (W.B. Johnson, G. Schechtman). Banach spaces and classical harmonic analysis (S.V. Kisliakov). Aspects of the isometric theory of Banach spaces (A. Koldobsky, H. Konig). Eigenvalues of operators and applications (H. Konig).
687 citations
DOI•
01 Jan 2001
TL;DR: A submitted manuscript is the version of the article upon submission and before peer-review as mentioned in this paper, while a published version is the final layout of the paper including the volume, issue and page numbers.
Abstract: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers.
546 citations
TL;DR: In this paper, it was shown that quasisymmetric homeomorphisms belong to a Sobolev space of borderline degree and that they are continuous in the sense that they respect the Cheeger differentials of Lipschitz functions on metric measure spaces.
Abstract: We give a definition for the class of Sobolev functions from a metric measure space into a Banach space. We give various characterizations of Sobolev classes and study the absolute continuity in measure of Sobolev mappings in the “borderline case”. We show under rather weak assumptions on the source space that quasisymmetric homeomorphisms belong to a Sobolev space of borderline degree; in particular, they are absolutely continuous. This leads to an analytic characterization of quasiconformal mappings between Ahlfors regular Loewner spaces akin to the classical Euclidean situation. As a consequence, we deduce that quasisymmetric maps respect the Cheeger differentials of Lipschitz functions on metric measure spaces with borderline Poincare inequality.
214 citations
TL;DR: In this article, it was shown that for smooth and bounded derivatives, the map f(u) is in W^{s,p} and W^{1,sp} spaces, respectively.
Abstract: Our main result is that, when $f$ is smooth and has bounded derivatives, and when $u$ belongs to the spaces $W^{s,p}$ and $W^{1,sp}$, the map $f(u)$ is in $W^{s,p}$.
212 citations
TL;DR: In this paper, existence and uniqueness theorems are presented for the heat equation in Lp spaces with or without weights allowing derivatives of solutions to blow up near the boundary, allowing for the powers of summability with respect to space and time variables to be different.
Abstract: Existence and uniqueness theorems are presented for the heat equation in Lp spaces with or without weights allowing derivatives of solutions to blow up near the boundary. It is allowed for the powers of summability with respect to space and time variables to be different.
88 citations
76 citations
TL;DR: In this article, an axiomatic approach to the theory of Sobolev spaces on metric measure spaces is developed, which covers the main known examples (Hajtasz Soboleov spaces, weighted Soboleve spaces, Upper-gradients, etc).
73 citations
01 Jan 2001
TL;DR: In this paper, the authors discuss the isometric theory of Banach spaces that was born and developed in inseparable connection with other areas of the Banach space theory, and discover connections between convexity and the Fourier transform.
Abstract: This chapter discusses the isometric theory of Banach spaces that was born and developed in inseparable connection with other areas of the Banach space theory. The classical direction, initiated by the work of Banach, is the characterization of the isometries of Banach spaces. Surjective isometries of Banach spaces are now well understood in a long sequence of the results culminating in complete characterization of such isometries for all rearrangement invariant spaces by Zaidenberg, Kalton, and Randrianantoanina. The results that belong to the intersection of the isometric theory, harmonic analysis, probability, and combinatorics are discussed in this chapter. Recently, discovered connections between convexity and the Fourier transform are also discussed and are explained through a complete analytic solution to the Busemann–Petty problem on the section of convex bodies that was considered one of the most important isometric questions in convexity.
54 citations
TL;DR: A survey of results about norm-one projections and 1-complemented subspaces in K-theoretic function spaces and Banach sequence spaces is given in this article.
Abstract: This is a survey of results about norm-one projections and 1-
complemented subspaces in K¨othe function spaces and Banach sequence spaces.
The historical development of the theory is presented from the 1930s to the
newest ideas. Proofs of the main results are outlined. Open problems are also
discussed. Every effort has been made to include as complete a bibliography
as possible.
53 citations
TL;DR: In this article, the invertible, compact and Fredholm multiplication operators on Orlicz spaces are characterized and the inversion and compactness of the Fredholm operator is analyzed.
Abstract: The invertible, compact and Fredholm multiplication operators on Orlicz spaces are characterized in this paper.
49 citations
TL;DR: Campanato, Morrey, BMO and Sobolev-type spaces for mappings from a space of homogeneous type into a complete metric space which possess properties comparable to their classical analogues were introduced in this article.
Abstract: We introduce Campanato, Morrey, BMO and Sobolev-type spaces for mappings from a space of homogeneous type into a complete metric space which possess properties comparable to their classical analogues. In particular we show integral characterizations, the validity of the John–Nirenberg theorem, Poincare and Sobolev inequalities, Sobolev's embedding theorem and estimates on the pointwise behavior of Sobolev-type mappings.
TL;DR: In this article, the minimal kernel of a p-Banach ideal of n-homogeneous polynomials between Banach spaces is defined as a composition ideal, characterized to be the smallest ideal which coincides with the given one on finite-dimensional spaces and represented through tensor products with appropriate norms.
Abstract: The minimal kernel of a p-Banach ideal of n-homogeneous polynomials between Banach spaces is defined as a composition ideal, characterized to be the smallest ideal which coincides with the given one on finite-dimensional spaces and represented through tensor products with appropriate norms.
01 Nov 2001
TL;DR: In this article, the authors consider the multiplication operator in Sobolev spaces with respect to general measures and give a characterization for M to be bounded, in terms of sequentially dominated measures.
01 Jan 2001
TL;DR: In this article, the authors present several well known orthogonal systems of functions and discuss their properties as bases in selected classical function spaces and show that in some function spaces there are no bases with additional special properties.
Abstract: We present several well known orthogonal systems of functions and discuss their properties as bases in selected classical function spaces. We show that in some function spaces there are no bases with additional special properties. We discuss non-explicit methods of constructing Schauder bases. On each compact, smooth manifold we construct a system of smooth functions which is a good basis in a wide range of spaces of Sobolev and Besov type. Finally we discuss systems of scalar valued functions which are bases (with vector coefficients) in some spaces of vector valued functions.
TL;DR: In this article, the proof of equivalence of certain L2-Sobolev spaces on manifolds with bounded curvature of all orders was extended to generalized compatible Dirac operators.
Abstract: We repair the proof of equivalence of certain L2-Sobolev spaces on manifolds with bounded curvature of all orders from [4]. The results are extended to generalized compatible Dirac operators, fractional order Sobolev spaces and weighted Sobolev spaces. A certain way of doing coordinate free computations is presented.
TL;DR: Spaces of functions which have prescribed degree of n-term approximation in terms of interpolation spaces are characterized in Besov and Triebel-Lizorkin spaces.
TL;DR: In this article, the Fourier transforms of functions in fractional anisotropic Sobolev-Liouville spaces are investigated for positive and positive Fourier transform functions, and estimates of norms in modified spaces of Lorentz and Besov kinds, defined in terms of iterative rearrangements, are established.
Abstract: Fractional anisotropic Sobolev-Liouville spaces are investigated for and positive . For functions in these spaces estimates of norms in modified spaces of Lorentz and Besov kinds, defined in terms of iterative rearrangements, are established. These estimates are used to prove inequalities for the Fourier transforms of functions in .This paper continues works of the author in which similar issues have been discussed for integer . The methods used in the paper are based on estimates of iterative rearrangements. This approach enables one to simplify proofs and at the same time to obtain stronger results. In particular, the analysis of the limit case becomes much easier.
TL;DR: In this paper, the Rademacher series in rearrangement invariant function spaces close to the space L ∞ is considered and a correspon- dence between such spaces and spaces of coefficients generated by them is proved.
Abstract: The Rademacher series in rearrangement invariant function spaces "close" to the space L∞ are considered. In terms of interpolation theory of operators, a correspon- dence between such spaces and spaces of coefficients generated by them is stated. It is proved that this correspondence is one-to-one. Some examples and applications are pre- sented.
17 Apr 2001
TL;DR: In this paper, the interpolation from Lebesgue spaces into Orlicz spaces is discussed, with special attention to interpolation constant $C. The authors give some estimates for the constant.
Abstract: The authors discuss the interpolation from Lebesgue spaces into Orlicz spaces, with special attention to the interpolation constant $C$. The authors give some estimates for $C$.
TL;DR: In this paper, it was shown that if u ∈ W k,p (Ω) with kp > N and Φ ∈ C k (R), then ∆ • u ∆ W k and p ⊂ L ∞ by the Sobolev embedding theorem.
Abstract: 1. Introduction. A classical result about composition in Sobolev spaces asserts that if u ∈ W k,p (Ω)∩L ∞ (Ω) and Φ ∈ C k (R), then Φ • u ∈ W k,p (Ω). Here Ω denotes a smooth bounded domain in R N , k ≥ 1 is an integer and 1 ≤ p < ∞. This result was first proved in [13] with the help of the Gagliardo-Nirenberg inequality [14]. In particular if u ∈ W k,p (Ω) with kp > N and Φ ∈ C k (R) then Φ • u ∈ W k,p since W k,p ⊂ L ∞ by the Sobolev embedding theorem. When kp = N the situation is more delicate since W k,p is not contained in L ∞. However the following result still holds (see [2],[3])
16 Apr 2001
TL;DR: In this article, it was shown that every separable Banach space X universal for the class of reflexive Hereditarily Indecomposable spaces contains C[O, 1] isomorphically and hence it is universal for all separable spaces.
Abstract: It is shown that every separable Banach space X universal for the class of reflexive Hereditarily Indecomposable space contains C[O, 1] isomorphically and hence it is universal for all separable spaces. This result shows the large variety of reflexive H.I. spaces.
01 Jan 2001
TL;DR: In this paper, a selection of extremal problems to do with constrained approximation in certain Banach spaces of holomorphic functions, including the classical Hardy spaces and Paley-Wiener spaces, are reviewed.
Abstract: We review a selection of extremal problems to do with constrained approximation in certain Banach spaces of holomorphic functions, including the classical Hardy spaces and Paley-Wiener spaces In many cases the solutions are best interpreted in terms of linear operators Applications of the problems under discussion to systems identification, signal processing, inverse problems for partial differential equations, and operator theory are presented
01 Jan 2001
TL;DR: Two-sided estimates are established for entropy numbers of embeddings between certain weighted Banach sequence spaces with mixed norms that are ''almost'' sharp and improve previous results by D. Edmunds and D. Haroske.
TL;DR: The results of Dragomir's inequality of Simpson's type are generalized using functions whose n th derivatives, n ∈ {2, 3, 4}, belong to L p spaces.
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01 Jan 2001
TL;DR: In this paper, the authors describe the structure of the L p -spaces and their subspaces, which are the simplest of the rearrangement invariant spaces and have been studied extensively in the literature.
Abstract: This chapter describes the structure of the L p -spaces and their subspaces. The L p -spaces have provided much fodder for the general theory of Banach spaces, because they appeared early in the theory, and the study of these spaces has motivated the definitions of many properties of more general Banach spaces. For example, with its usual norm L p is a Banach lattice under the pointwise almost everywhere ordering. The spaces naturally occur as interpolation spaces and are the simplest of the rearrangement invariant spaces. The study of the structure of the finite dimensional subspaces of L p paved the way for much of the extraordinary development of the local theory of Banach spaces in 1980. In the investigations of other Banach spaces and operators, the existence and classification of operators from, into, or factoring through L p -spaces provide fundamental information on the structure.
TL;DR: In this paper, it was shown that the pointwise inequality used by P. Hajlasz in his definition of Sobolev spaces on metric spaces is equivalent to an integral (Poincare-type) inequality.
Abstract: We prove that the pointwise inequality used by P. Hajlasz in his definition of Sobolev spaces on metric spaces is equivalent to an integral (Poincare-type) inequality.
TL;DR: In this article, the authors studied nearly regular-Lindelof, almost regular Lindelof and weakly regular-lindelof spaces and proposed characterizations and properties for these spaces.
Abstract: We study nearly regular-Lindelof, almost regular-Lindelof and weakly regular- Lindelof spaces. Characterizations and some properties for these spaces are proposed. Relations among them are also studied.