Showing papers on "Birnbaum–Orlicz space published in 1981"

Banach Spaces of Distributions of Wiener’s Type and Interpolation

Abstract: In the parallel paper [9] we have introduced “spaces of Wiener’s type”, a family of Banach spaces of (classes of) measurable functions, measures or distributions on locally compact groups. The elements of these spaces are characterized by — what we call — the global behaviour of certain of their local properties. In the present paper it is to be shown that interpolation methods can be applied to these spaces in a very natural way. Using the results on interpolation it is not difficult to extend various theorems of analysis to the setting of Wiener-type spaces. As illustration we present a version of the Hausdorff — Young inequality for locally compact abelian groups. As a consequence, one obtains a sharpened version of Soboley’s embedding theorem.

Multivariate rearrangements and Banach function spaces with mixed norms

Abstract: Multivariate nonincreasing rearrangement and averaging functions are defined for functions defined over product spaces. An investigation is made of Banach function spaces with mixed norms and using multivariate rearrangements. Particular emphasis is given to the L(P, Q; *) spaces. These are Banach function spaces which are in terms of mixed norms, multivariate rearrangements and the Lorentz L(p, g) spaces. Embedding theorems are given for the various function spaces. Several well-known theorems are extended to the L(P, Q; *) spaces. Principal among these are the Strong Type (Riesz-Thorin) Interpolation Theorem and the Convolution (Young's inequality) Theorem.

Primitive spaces of matrices of bounded rank

Abstract: A weak canonical form is derived for vector spaces of m x n matrices all of rank at most r. This shows that the structure of such spaces is controlled by the structure of an associated 'primitive' space. In the case of primitive spaces it is shown that m and n are bounded by functions of r and that these bounds are tight.

A characterization of minimal homogeneous Banach spaces

Abstract: Let G be a locally compact group. It is shown that for a homogeneous Banach space B on G satisfying a slight additional condition there exists a minimal space fimm in the family of all homogeneous Banach spaces which contain all elements of B with compact support. Two characterizations of Bm¡1¡ aie given, the first one in terms of "atomic" representations. The equivalence of these two characterizations is derived by means of certain (bounded) partitions of unity which are of interest for themselves. Notations. In the sequel G denotes a locally compact group. \M\ denotes the (Haar) measure of a measurable subset M G G, or the cardinality of a finite set. ®(G) denotes the space of continuous, complex-valued functions on G with compact support (supp). For y G G the (teft) translation operator ly is given by Lyf(x) = f(y ~ xx). A translation invariant Banach space B is called a homogeneous Banach space (in the sense of Katznelson [9]) if it is continuously embedded into the topological vector space /^(G) of all locally integrable functions on G, satisfies \\Lyf\\B = 11/11, for all y G G, and lim^iy-f\\B 0 for all / G B (hence, as usual, two measurable functions coinciding l.a.e. are identified). If, furthermore, B is a dense subspace of LX(G) it is called a Segal algebra (in the sense of Reiter [15], [16]). Any homogeneous Banach space is a left L'((7)-Banachmodule with respect to convolution, i.e./ G B, h G LX(G) implies h */ G B, and \\h */\\b < ll*Uill/IJi» m particular any Segal algebra is a Banach ideal of LX(G). In the sequel we shall write Bm for the space {/|/ e B, supp/compact}. Lemma 1. Let V = V~x be an open, relatively compact subset of G. Then there exists a subset Y = (.y,),e/ Ç G such that

Hardy spaces of vector-valued functions

Abstract: The purpose of this paper is to consider the relations among the various definitions of the spaces Hp of analytic functions with values in a Banach space and to investigate the problem of the structure of the conjugates of these spaces. In particular, one constructs an example of a reflexive separable Banach space χ, for which the equality Hp(X)*=Hp′(X*) (1

Isometries between Banach spaces of Lipschitz functions

Abstract: The Banach spaces Lip a (S, Δ), lip a (S, Δ), Lip a (S, Δ;s 0) and lip a (S, Δ;s 0) of Lipschitz functions are defined. We shall identify the extreme points of the unit balls in their corresponding dual spaces and make use of them to present a complete characterization of the isometries between these function spaces.

Points de namioka espaces normants applications à la théorie isométrique de la dualité

Abstract: We study in this work various aspects of the isometric theory of duality. We show that in wide classes of Banach spaces, dual spaces are characterized by the existence of a retraction fromE″ ontoE. The predual of such spaces is then unique. We study the imbedding of regularly normed spaces into dual spaces. We better the known results on loss of regularity of the norm of dual spaces. We characterize the dual norms on an Asplund space in terms of “bad differentiability”.

Norm inequalities of product form in weighted Lp spaces

Abstract: The classical inequality ‖y′‖2≦K‖y‖ ‖y″‖ and its higher order analogues are extended from the Lp spaces to the weighted spaces, for appropriate w.

The asymptotic-norming and Radon-Nikodým properties for Banach spaces

Abstract: A Banach space has the Radon--Nikod3~m property (RNP) if it is isomorphic to a subspace of a separable du~d. Until very recently, it was thought this might be a necessary condition for RNP. The asymptotic-norming property (ANP) is introduced. It is shown that ANP is satisfied by a larger class of Banach spaces than those that are isomorphic to subspaces of separable duals, and that ANP implies RNP. For product spaces and subspaces of duals, there are significant similarities between ANP and RNP. Different formulations of ANP are studied, as well as relations between A N P and Kaded--Klee-type properties. A Banach space X is said to have the Radon--Nikodj:m property if for each finite-measure space (S, S, #) and each #-continuous, X-valued measure m with finite norm, there is a Bochner-integrable function f from S to X such that rn(E)= fEfd# if E belongs to S. A Banach space has RN P if it is isomorphic to a subspace of a separable dual ([8] and [7, Theorem 1, page 79]). Until recently, it was thought this might be a necessary condition for RNP. However, there are now examples of Banach spaces with RNP that are not isomorphic to a subspace of any separable dual ([4] and [12]). In Section 1, we will introduce ANP and show it to be a sufficient condition for RNP. Also, ANP is satisfied by all subspaces of separable duals. In Section 2, it will be shown that ANP is satisfied by some spaces not isomorphic to a subspace of any separable dual, and is preserved for certain product-type spaces. However, it remains unknown whether ANP is necessary for RNP. In Section 3, ANP will be studied for subspaces of duals. The spaces Co and LI[0, 1] do not have RNP, but /1 does have RNP. A great deal is known about conditions for RNP (e.g., see [7, pages 217 219]). Frequently,

Facial characterizations of complex Lindenstrauss spaces

Abstract: We characterize complex Banach spaces A whose Banach dual spaces are L 1 (μ) spaces in terms of L-ideals generated by certain extremal subsets of the closed unit ball K of A ∗ . Our treatment covers the case of spaces A containing constant functions and also spaces not containing constants. Separable spaces are characterized in terms of w ∗ -compact sets of extreme points of K, whereas the nonseparable spaces necessitate usage of the w ∗ -closed faces of K. Our results represent natural extensions of known characterizations of Choquet simplexes. We obtain also a characterization of complex Lindenstrauss spaces in terms of boundary annihilating measures, and this leads to a characterization of the closed subalgebras of C c (X) which are complex Lindenstrauss spaces.

Limits of banach spaces. imbedding theorems. applications to sobolev spaces of infinite order

Abstract: For a sequence of Banach spaces , a concept of limit is introduced that is a natural generalization of the concept of the limit of a monotonically decreasing numerical sequence. Necessary and sufficient conditions are obtained for an imbedding and for a compact imbedding. Applications are given to the Sobolev spaces of infinite order . Necessary and sufficient conditions bearing an algebraic character are established for the imbedding . Sufficient algebraic imbedding conditions are obtained for the spaces for any .Bibliography: 8 titles.

Exponential bases and free interpolation in banach spaces with the Lp-norm

Abstract: We investigate symmetric bases of exponential functions in the closure of their linear span in the spaces Lp(μβ) and lp(πβ) where αμβ=tβαt, t<0, and νβ({K})=(K+1)β, K∈{0,1,2,} We also consider the dual problem of free interpolation in the corresponding Banach Lp-spaces of analytic functions In many cases one gives the complete description of bases of exponential functions (in the closure of its linear span) in the above-indicated spaces and the solution of the corresponding problems of free interpolation by analytic functions