Showing papers on "Birnbaum–Orlicz space published in 2000"

Newtonian spaces: An extension of Sobolev spaces to metric measure spaces

Abstract: This paper studies a possible definition of Sobolev spaces in abstract metric spaces, and answers in the affirmative the question whether this definition yields a Banach space. The paper also explores the relationship between this definition and the Hajlasz spaces. For specialized metric spaces the Sobolev embedding theorems are proven. Different versions of capacities are also explored, and these various definitions are compared. The main tool used in this paper is the concepto of moduli of path families.

Optimized Tensor-Product Approximation Spaces

Abstract: This paper is concerned with the construction of optimized grids and approximation spaces for elliptic differential and integral equations. The main result is the analysis of the approximation of the embedding of the intersection of classes of functions with bounded mixed derivatives in standard Sobolev spaces. Based on the framework of tensor-product biorthogonal wavelet bases and stable subspace splittings, the problem is reduced to diagonal mappings between Hilbert sequence spaces. We construct operator adapted finite element subspaces with a lower dimension than the standard full-grid spaces. These new approximation spaces preserve the approximation order of the standard full-grid spaces, provided that certain additional regularity assumptions are fulfilled. The form of the approximation spaces is governed by the ratios of the smoothness exponents of the considered classes of functions. We show in which cases the so-called curse of dimensionality can be broken. The theory covers elliptic boundary value problems as well as boundary integral equations.

A Proximal Point Method for the Variational Inequality Problem in Banach Spaces

Abstract: In this paper we prove well-definedness and weak convergence of the generalized proximal point method when applied to the variational inequality problem in reflexive Banach spaces. The proximal version we consider makes use of Bregman functions, whose original definition for finite dimensional spaces has here been properly extended to our more general framework.

Functional Analysis: An Introduction to Banach Space Theory

Abstract: Preface. Introduction. Basic Definitions and Examples. Basic Principles with Applications. Weak Topologies and Applications. Operators on Banach Spaces. Bases in Banach Spaces. Sequences, Series, and a Little Geometry in Banach Spaces. Bibliography. Author/Name Index. Subject Index Symbol Index.

Some sequence spaces defined by Orlicz functions

Abstract: The object of this paper is to introduce a new concept of lacunary strong convergence with respect to an Orlicz function and examine some properties of the resulting sequence spaces. We establish some elementary connections between lacunary strong convergence and lacunary strong convergence with respect to an Orlicz function which satisfies ¿^-condition. It is also shown that if a sequence is lacunary strongly convergent with respect to an Orlicz function then it is lacunary statistically convergent. In addition, lacunary strong convergence with respect to an Orlicz function is compared to other summability methods.

Linear extension operators for restrictions of function spaces to irregular open sets

Abstract: Let an open set R n satisfy for some 0 d n and \" > 0 the condition: the d-Hausdorr content H d (\\ B) \"jBj d=n for any ball B centered at of volume jBj 1. Let H s p denote the Bessel potential space on R n (1 < p < 1, s > 0), and let H s p ] be the linear space of restrictions of elements of H s p to endowed with the quotient space norm. In this paper we nd suucient conditions for the existence of a linear extension operator for H s p ], i.e., a linear bounded operator ext : H s p ] ! H s p such that ext fj = f for all f. The main result is that such an operator exists if (i) d > n ? 1 and s > (n ? d)= min(p; 2), or (ii) d n ? 1 and s ? s] > (n ? d)= min(p; 2). It is an open problem whether these assumptions are sharp.

Weighted spaces of holomorphic functions on Banach spaces

Abstract: In this paper we study composition operators between weighted spaces of holomorphic functions defined on the open unit ball of a Banach space. Necessary and sufficient conditions are given for composition operators to be compact. We show that new phenomena appear in the infinite-dimensional setting different from the ones of the finite-dimensional one. 1. Weights. Weighted spaces The starting idea of composition operators is a simple and very natural question. Consider D the open unit disc of C and a holomorphic map φ: D −→ D . If f : D −→ C is a holomorphic function, we can compose f ◦ φ and try to analyze what happens when we let the f vary; in other words we define an operator between spaces of holomorphic functions and we want to study what properties does this operator have (continuity, compactness, . . .). This obviously depends on which are the spaces considered. First candidates are the Hardy spaces and a full study of the situation in this case can be found in [18]. In the last few years a lot of research has been done studying the behavior of operators between weighted spaces of holomorphic functions Hw(B) whenever B is the unit disk of C or, more in general, an open subset of C (see below for definitions and notation). Among the operators between these spaces particular attention has been paid to composition operators. We refer to [4], [5], [7], [8], [9], [10], [15], [19] and particularly to the recent surveys [3], [6] for information about the subject. But also some interest has been given to the more general case where X is a Banach space and BX is its open unit ball (see e.g. [1], [2], [13], [14], [17]). In this paper, strongly influenced by the work of Bonet, Domański, Lindström and Taskinen [8], we study composition operators between Hw(BX) and Hv(BY ) and we find that new phenomena appear in the infinite-dimensional setting different from the ones of the finite-dimensional one. In Section 2 we make an introductory 2000 Mathematics Subject Classification: Primary 47B38; Secondary 46E15, 46G20. The authors were supported by the MCYT and FEDER Project BFM2002-01423. 82 Domingo Garćıa, Manuel Maestre, and Pablo Sevilla-Peris study of composition operators. In Section 3 we study the compactness of a composition operator giving necessary and sufficient conditions for such an operator to be compact. Finally, in Section 4 we show that Hilbert spaces are a natural setting to extend [8, Theorem 2.3], a result that gives conditions on the weight v such that all composition operators from Hv(BX) into itself are continuous. We fix the notation to be used in the rest of the article. Let X be a complex Banach space and BX its open unit ball. Any continuous bounded mapping v: BX →]0,∞[ is called a weight. We denote by H(BX) the space of all holomorphic functions f : BX −→ C . A set A ⊂ BX is said to be BX -bounded if there exists 0 < r < 1 such that A ⊂ rBX . The subspace of H(BX) of those functions that are bounded on the BX -bounded sets is denoted by Hb(BX). Following [8] and [17] we consider Hv(BX) = { f ∈ H(BX) : ‖f‖v = sup x∈BX v(x)|f(x)| <∞ } . With the norm ‖ · ‖v , the space Hv(BX) is a Banach space. We denote Bv the closed unit ball of Hv(BX). It is well known that in Hv(BX) the τv (norm) topology is finer than the τ0 (compact-open) topology ([17, Section 3]) and that Bv is τ0 -compact ([17, p. 349]). Following [4], [6], we say that a weight is radial if v(λx) = v(x) for every λ ∈ C with |λ| = 1 and every x ∈ BX . A weight v satisfies Condition I if infx∈rBX v(x) > 0 for every 0 < r < 1 ([14]). If v satisfies Condition I, then Hv(BX) ⊆ Hb(BX) ([14, Proposition 2]). If X is finite-dimensional, then all weights on BX satisfy Condition I. From now on, unless otherwise stated, every weight is assumed to satisfy Condition I. Given any weight v , following [5], we consider an associated growth condition u: BX −→]0,+∞[ defined by u(x) = 1/v(x). With this new function we can rewrite Bv = { f ∈ Hv(BX) : |f | ≤ u } . From this, ũ: BX −→]0,+∞[ is defined by ũ(x) = sup f∈Bv |f(x)| and a new associated weight ṽ = 1/ũ . All these functions were defined by Bierstedt, Bonet and Taskinen for open subsets of C in [5]. In [5, Proposition 1.2], the following relations between weights for open sets on C are proved. The same arguments work for the unit ball of a Banach space. Proposition 1.1 Let X be a Banach space and v a weight defined on BX . The following hold: Composition operators 83 (i) 0 < v ≤ ṽ and ṽ is bounded and continuous; i.e., ṽ is a weight. (ii) ũ (respectively ṽ ) is radial and decreasing or increasing whenever u (respectively v ) is so. (iii) ‖f‖v ≤ 1 ⇔ ‖f‖ṽ ≤ 1 . (iv) For each x ∈ BX there exists fx ∈ Bv such that ũ(x) = |fx(x)| . As an immediate consequence of (iii) we have Corollary 1.2 ([5, Observation 1.12]). If X is a Banach space and v is any weight defined on BX , then Hv(BX) = Hṽ(BX) holds isometrically. Since the constant function 1 belongs to Hv(BX) we have sup x∈BX v(x) = ‖1‖v = ‖1‖ṽ = sup x∈BX ṽ(x). Definition 1.3 ([19]). A weight v is said to be essential if there exists C > 0 such that v(x) ≤ ṽ(x) ≤ C v(x) for all x ∈ BX . We say that a weight v is norm-radial if v(x) = v(y) for every x , y such that ‖x‖ = ‖y‖ . If v is norm-radial and non-increasing (with respect to the norm) then ṽ is also norm-radial. Indeed, if v is such a weight and T : X → X is a linear mapping, T 6= 0, with ‖T‖ ≤ 1, then for any f ∈ Hv(BX) we can consider fT = f ◦ T . Then ‖fT ‖v = sup z∈BX v(z) ∣f ( T (z) )∣∣ ≤ sup z∈BX v ( T (z) ∣f ( T (z) )∣∣ ≤ ‖f‖v. Hence, for any x ∈ BX we have sup‖f‖v≤1 |f(x)| ≥ sup‖f‖v≤1 |f(T (x))| . Now if y ∈ BX with ‖x‖ = ‖y‖ we can take T such that T (x) = y to get that ṽ(x) ≤ ṽ(y). The converse inequality is proved in the same way. Note that given a Banach space X such that for any two x, y ∈ BX with ‖x‖ = ‖y‖ there exists a holomorphic isometry T : BX −→ BX with T (x) = y then any norm-radial weight v satisfies that ṽ is also norm-radial. This happens if X is a Hilbert space. 2. Composition operators Let X , Y be Banach spaces. We denote by BX , BY their open unit balls. Let φ: BX → BY be a holomorphic mapping. The composition operator associated to φ is defined by Cφ: H(BY ) −→ H(BX), f Cφ(f) = f ◦ φ. Cφ is clearly linear and (τ0, τ0)-continuous. Given any two weights v , w we consider the restriction Cφ: Hv(BY ) → Hw(BX) whenever this is well defined. If h: BX −→ Y is bounded we denote as usual ‖h‖∞ = sup { ‖h(x)‖ : ‖x‖ < 1 } . 84 Domingo Garćıa, Manuel Maestre, and Pablo Sevilla-Peris Remark 2.1. Let H , E be two Banach spaces of holomorphic functions whose topologies are stronger than the pointwise convergence topology. If Cφ: H → E is well defined then, by the closed graph theorem, Cφ is continuous. As a consequence, to find out if the composition operator Cφ is continuous it is enough to find out if Cφ is well defined. Proposition 2.2. If there is some 0 < r < 1 such that φ(BX) ⊆ rBY , then Cφ: Hv(BY ) → Hw(BX) is well defined (and then continuous) for any two weights v with Condition I and w . Proof. Since φ(BX) ⊆ rBY , then for each f ∈ Hv(BY ) there is K > 0 such that supy∈φ(BX) |f(y)| ≤ K . Hence sup x∈BX w(x)|Cφ(f)(x)| = sup x∈BX w(x) ∣f ( φ(x) )∣∣ ≤ sup x∈BX w(x) sup x∈BX ∣f ( φ(x) )∣∣ <∞. Therefore Cφ(f) ∈ Hw(BX) and Cφ is well defined. The following proposition extends some of the results in [8, Proposition 2.1] (see also [7, Theorem 4]). Proposition 2.3. Let v , w be two weights satisfying Condition I and φ: BX −→ BY holomorphic. Then the following are equivalent: (i) Cφ: Hv(BY ) −→ Hw(BX) is well defined and continuous. (ii) supx∈BX ( w(x)/ṽ(φ(x)) ) <∞ . (iii) supx∈BX ( w̃(x)/ṽ(φ(x)) ) <∞ . (iv) sup‖φ(x)‖>r0 ( w(x)/ṽ(φ(x)) ) <∞ for some 0 < r0 < 1 . Proof. The implication (iii) ⇒ (ii) is trivial, since w ≤ w̃ . Let us assume now (ii). Let f ∈ Hv(BY ); we have w(x) ∣f ( φ(x) )∣∣ = w(x) ṽ ( φ(x) ) ṽ ( φ(x) ∣f ( φ(x) )∣∣ ≤M‖f‖ṽ = M‖f‖v for all x . Hence Cφ is continuous. Suppose now that Cφ is continuous. If (iii) does not hold there exists (xn)n∈N ⊆ BX such that lim n→∞ w̃(xn) ṽ ( φ(xn) ) = ∞. For each n ∈ N we can take fn ∈ Bv so that ∣fn ( φ(xn) )∣∣ = ũ ( φ(xn) ) = 1/ṽ ( φ(xn) ) . Hence ∣fn ( φ(xn) ∣w̃(xn) = w̃(xn) ṽ ( φ(xn) ) which is a contradiction with the fact that Cφ(Bv) is bounded. Composition operators 85 Clearly (ii) implies (iv). Conversely, if (iv) holds, let M = sup ‖φ(x)‖>r0 w(x)

Traces of anisotropic Besov-Lizorkin-Triebel spaces---a complete treatment of the borderline cases

Abstract: Including the previously untreated borderline cases, the trace spaces (in the distributional sense) of the Besov-Lizorkin-Triebel spaces are determined for the anisotropic (or quasi-homogeneous) version of these classes. The ranges of the traces are in all cases shown to be approximation spaces, and these are shown to be different from the usual spaces precisely in the cases previously untreated. To analyse the new spaces, we carry over some real interpolation results as well as the refined Sobolev embeddings of J. Franke and B. Jawerth to the anisotropic scales.

Some Properties of the Anisotropic Riesz-Bessel Potential

Abstract: The Hardy-Littlewood-Bessel maximal functions (B-maximal functions), Morrey-Bessel and BMO-Bessel spaces were introduced and studied in [6]. In the present paper, we study the anisotropic Riesz-Bessel potential (B-potential) in the Morrey-Bessel and BMO-Bessel spaces. We obtain a theorem analogous to the Sobolev theorem, for the anisotropic Riesz-Bessel potential in Morrey-Bessel spaces. We introduce a metric characteristic Ω p,γ in the space of locally integrable functions and establish estimates connecting the characteristics of the image and preimage of the corresponding integral transform. These estimates are of independent interest. Moreover, they are used for the investigation of integral operators in different scales of Banach function spaces, in particular, in weighted L p γ-spaces. The results seem to be new even in the isotropic case.

Representation theorems of the dual of Lebesgue-Bochner function spaces

Abstract: By representing random conjugate spaces a general representation theorem on classical duals is proved. for application,we unify and improve many known imjportant representation theorems of the dual of Lebesgue-Bichner function spaces.

On More General Lipschitz Spaces

Abstract: The present paper deals with (logarithmic) Lipschitz spaces of type Lip(1,−α) p,q (1 ≤ p ≤ ∞, 0 < q ≤ ∞, α > 1 q ). We study their properties and derive some (sharp) embedding results. In that sense this paper can be regarded as some continuation and extension of our papers [8, 9], but there are also connections with some recent work of Triebel concerning Hardy inequalities and sharp embeddings. Recall that the nowadays almost ‘classical’ forerunner of investigations of this type is the Brézis-Wainger result [6] about the ‘almost’ Lipschitz continuity of elements of the Sobolev spaces H 1+ n p p (R) when 1 < p < ∞.

On Decomposition in Exponential Orlicz Spaces

Abstract: Various characterizations are given of the exponential Orlicz space L and the Orlicz-Lorentz space L. By way of application we give a simple proof of the celebrated theorem of Brezis and Wainger concerning a limiting case of a Sobolev imbedding theorem.

On convergence of sections of sequences in Banach spaces

Abstract: An elementary proof of the (known) fact that each element of the Banach spacelwp(X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element oflwp(X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications to spaces of compact operators on Banach sequence spaces are considered.

Ray Hölder-continuity for fractional Sobolev spaces in infinite dimensions and applications

Abstract: We prove Holder-continuity on rays in the direction of vectors in the (generalized) Cameron-Martin space for functions in Sobolev spaces in L p of fractional order α∈ ( , 1) over infinite dimensional linear spaces. The underlying measures are required to satisfy some easy standard structural assumptions only. Apart from Wiener measure they include Gibbs measures on a lattice and Euclidean interacting quantum fields in infinite volume. A number of applications, e.g., to the two-dimensional polymer measure, are presented. In particular, irreducibility of the Dirichlet form associated with the latter measure is proved without restrictions on the coupling constant.

Al- and am-spaces of integrable scalar functions with respect to a fr#233chet-valued measure

Abstract: We study the question of determining conditions for the space of R-valued integrable functions with respect to a vector measure taking its values in a real Fréchet space to be an AL- or an AM-space*. Keywords: Fréchet lattice, generalized AL-spaces, generalized AM-spaces, Köthe sequences spaces. Quaestiones Mathematicae 23(2000), 247–258

Boundedness of the fractional integral on weighted lebesgue and lipschitz spaces

Abstract: Sufficient conditions are given for fractional integral operater I α to be bounded from weighted weak Legesgue spaces with some range p into another suitable weighted BMO and Lipschitz spaces of order β . Moreover, sufficient and necessary conditions for the boundedness of I α between these spaces are obtained.

Unconditional Haar bases for Lebesgue spaces on spaces of homogeneous type

Abstract: We show that spaces of homogeneous type are adequate structures on which the unbalanced wavelet of Girabardi and Sweldens, can be constructed with an additional geometric control for the size of the nested partitions, given by the underlying quasi-distance. Moreover, we show that if a non- degeneracy condition is satisfied, we can still apply the Calderon-Zygmund theory in order to get the characterization of Lp spaces.© (2000) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Unconditional Convergence of Random Series and the Geometry of Banach Spaces

Abstract: Abstract The a.s. unconditionally convergent random series are investigated. The connection of the a.s. unconditionally convergence with the geometry of spaces is established.

Multipliers of banach valued weighted function spaces

Abstract: We generalize Banach valued spaces to Banach valued weighted function spaces and study the multipliers space of these spaces We also show the relationship between multipliers and tensor product of Banach valued weighted function spaces 1 Introduction We generalize Banach valued function spaces and its multipliers to the Banach valued weighted function spaces and its multipliers We also get some elementary results Throughout we let G be a locally compact abelian group with Haar measure, X be a Banach space and A be a commutative Banach algebra with identity of norm 1 A weight, Ψ , is a measurable and strictly positive function on G L p (G, X) denote the space of equivalence classes of X-valued strongly measurable functions f on G such that Ψ (t)f (t) ∈ L p (G, X) The space L p (G, X) is a Banach space normed by

On Equivalent Normings in the Spaces H p M on Compact Homogeneous Spaces

Abstract: Let M be an arbitrary compact Riemannian symmetric space of rank 1. The function spaces $$H_p^r$$ of Nikol'skii type were introduced earlier by means of averaged differences along geodesics. In the present paper we give an equivalent description of these spaces and norms in them by using the Laplace--Beltrami operator. The results obtained generalize the results of Nikol'skii and Lizorkin on the spaces $$H_p^r$$ over the sphere.

Function spaces and applications

Abstract: Preface / Limiting Behaviour of Solutions of a Sequence of Non-Homogeneous Boundary Value Problems / Some Separation Criteria and Inequalities Associated with Linear Second Order Differential Operators / Weak Type Estimates for Averaging Operators / Norms of Interpolation Operators Controlled by the Dicesar Function / On the Garcia-Falset Coefficient in Orlicz Sequence Spaces Equipped with the Orlicz Norm / On Some Fundamental Properties of the Maximal Operator / Nontangential Approach Regions on Groups / Stability of Sobolev Spaces with Zero Boundary Values / Imbeddings of Weighted Sobolev Spaces / From Hardy to Carleman and General Mean - Type Inequalities / One Dimensional Approximation of Eigenvalue Problems in Thin Rods / Some Comments to the Hardy Inequalities / On Asymptotic Behavior of the Approximation Numbers and Estimates of Schatten-Von Neumann Norms of the Hardy-Type Integral Operators / Expansions in Series of Legendre Functions / Over determined Weighted Hardy Inequalities on Semiaxis / Hankel Convolution on Some Ultra Differentiable Function Spaces / Embedding Theorems in Functional Analysis / Four Questions Related to Hardy's Inequality/ Optimal Inequalities on Quasinormed Function Spaces

Multiwavelet characterization of function spaces adapted to the Navier-Stokes equations

Abstract: We use wavelets based ona modification of the Geronimo- Hardin-Massopust construction to define localized extension/restriction operators form half-spaces to their full spaces/boundaries respectively. These operations are continuous in Sobolev and Morrey space norms. We also prove estimates for multiresolution projections of pointwise products of functions in these spaces. These are two of the key steps in extending results of Federbush and of Cannone and Meyer concerning solutions of Navier-Stokes with initial data in Sobolev and Morrey spaces to the case of half spaces and, ultimately, to more general domains with boundary.