On the Structure of L 1 of a Vector Measure via its Integration Operator

Geometric and summability properties of the integration operator associated to a vector measure m can be translated in terms of structure properties of the space L¹(m). In this paper we study the cases of the integration operator being: (i) p-concave on L^p(m), or (ii) positive p-summing on L¹(m) (where ). We prove that (i) is equivalent to saying that L¹(m) contains continuously the L^p space of a (non-negative scalar) control measure for m. On the other hand, we show that (ii) holds if and only if L¹(m) is order isomorphic to the L¹ space of a non-negative scalar measure. J.M. Calabuig was supported by MEC and FEDER (MTM2005-08350-C03-03) and Generalitat Valenciana (GV/2007/191). J. Rodríguez was supported by MEC and FEDER (MTM2005-08379) and Generalitat Valenciana (GVPRE/2008/312). E.A. Sánchez-Pérez was supported by MEC and FEDER (MTM2006-11690-C02-01).     

Multiplication operators on vector measure Orlicz spaces

Let m be a countably additive vector measure with values in a real Banach space X, and let L¹(m) and L_w(m) be the spaces of functions which are, correspondingly, integrable and weakly integrable with respect to m. Given a Young's function Φ, we consider the vector measure Orlicz spaces L^Φ(m) and L^Φ_w(m) and establish that the Banach space of multiplication operators going from W = L^Φ(m) into Y = L¹ (m) is M = L^Ψ_w (m) with an equivalent norm; here Ψ is the conjugated Young's function for Φ. We also prove that when W = L^Φ_w(m), Y = L¹(m) we have M = L^Ψ_w (m), and when W = L^Φ_w(m), Y = L¹(m) we have M = L^Ψ (m).     

Sobolev–Morrey spaces related to an ultraparabolic equation

Let us consider the class of hypoelliptic operators

SOME ASPECTS OF THE LIFTING PROPERTY OF THE SPACE £∞ (μ, X)

Abstract

Using a lifting of £_∞ (μ, X) ([5],[6]), we construct a lifting ρ x of the seminormed vector space £_∞ (μ, X) of measurable, essentially bounded X-valued functions. We show that in a certain sense such a lifting always exists. If μ is Lebesgue measure on (0, 1) we show that ρ x exists as map from £_∞ ((O, 1), X) → £_∞,((0, l), X) if and only if X is reflexive. In general the lifted function takes its values in X **. Therefore we investigate the question, when f ε £_∞ (μ, X) is strictly liftable in the sense that the lifted function is a map with values even in X.

As an application we introduce the space £_∞ ^strong (μ, L (X, Y**)), a subspace of the space of strongly measurable, essentially bounded L (X, Y, **)-valued functions, and the associated quotient space £_∞ ^strong (μ, L (X,Y**)). We show that this space is a Banach space because there is a kind of a Dunford-Pettis Theorem for a subspace of L (X, £_∞(μ Y**)). Finally we investigate the measurability property of functions in £_∞(μ Y**)) und see that there exists a connection to the Radon-Nikodym property of the space L (X, Y).     

On the ideal centre of the space of vector valued integrable functions

Let E be a Banach lattice and L¹(μ, E) be the space of E-valued Bochner integrable functions. Some order properties of L¹(μ, E) are given. It is shown that L_s^∞(μ, Z(E)) is the ideal centre of L¹(μ, E) and it is obtained a Radon-Nikodym type theorem for B -integrable functions.      

Ordered representations of spaces of integrable functions

Let X be a Banach space, (Ω,Σ) a measurable space and let m : Σ → X be a (countably additive) vector measure. Consider the corresponding space of integrable functions L¹(m). In this paper we analyze the set of (countably additive) vector measures n satisfying that L¹(n) = L¹(m). In order to do this we define a (quasi) order relation on this set to obtain under adequate requirements the simplest representation of the space L¹(m) associated to downward directed subsets of the set of all the representations. This research has been partially supported by La Junta de Andalucía. The support of D.G.I. under project MTM2006–11690–C02 (M.E.C. Spain) and FEDER is gratefully acknowledged.     

Nonlinear integration on Lp-spaces

For a Banach space E and for 1 ? p < ∞ let ?p<∞ let L_E^p(μ) = L_E^p(S,B,μ) denote all Bochner p-integrable E-valued functions on a measure space (S,B,μ). Under study are convergence theorems for integrals of functions in L_E^p(μ) with respect to Nemytskii measures. Weak integrals are then denoted to Hammerstein operators, and a study of topologies generated by vector measures leads to a characterization of compact Hammerstein operators.     

von Neumann��s mean ergodic theorem on complete random inner product modules

We first prove two forms of von Neumann’s mean ergodic theorems under the framework of complete random inner product modules. As applications, we obtain two conditional mean ergodic convergence theorems for random isometric operators which are defined on L _ℱ ^p (ℰ, H) and generated by measure-preserving transformations on Ω, where H is a Hilbert space, L ^p (ℰ, H) (1 ⩽ p < ∞) the Banach space of equivalence classes of H-valued p-integrable random variables defined on a probability space (Ω, ℰ, P), F a sub σ-algebra of ℰ, and L _ℱ ^p (ℰ(E,H) the complete random normed module generated by L ^p (ℰ, H).     

On the Structure of Spaces of Polyanalytic Functions

Suppose that A_mL_p(D,) is the space of all m-analytic functions on the disk D={z:|z| < 1} which are pth power integrable over area with the weight (1-|z|²), > -1. In the paper, we introduce subspaces A_kL_p ⁰(D,), k=1,2,...,m, of the space A _mL_p(D,) and prove that A _mL_p(D,) is the direct sum of these subspaces. These results are used to obtain growth estimates of derivatives of polyanalytic functions near the boundary of arbitrary domains.     

Optimal Domains for <Emphasis Type="Italic">L</Emphasis><Superscript>0</Superscript>-valued Operators Via Stochastic Measures

We study extension of operators T: E→ L⁰([0, 1]), where E is an F–function space and L⁰([0, 1]) the space of measurable functions with the topology of convergence in measure, to domains larger than E, and we study the properties of such domains. The main tool is the integration of scalar functions with respect to stochastic measures and the corresponding spaces of integrable functions. Partially supported by D.G.I. #MTM2006-13000-C03-01 (Spain).     

Hölder type quasicontinuity

It is proved that a functionuL ^m,p (R ⁿ ) (which coincides with the Sobolev spaceW ^1,p (R ⁿ ) ifm=1) coincides with a Hölder continuous functionw outside a set of smallm,q-capacity, whereq<p. Moreover, ifm=1, then the functionw can be chosen to be close tou in theW ^1,p -norm.     

A multiplier theorem using the Schechter's method of interpolation

Let m be a measurable bounded function and let us assume that there exists a bounded functions S so that m(ξ)S(ξ)^it−1 is a Fourier multiplier on L^p uniformly in . Then, using the analytic interpolation theorem of Stein, one can show that necessarily m is a L^p multiplier. The purpose of this work is to show that under the above conditions, it holds that, for every , m(log S)^kM_p. The technique is based on the Schechter's interpolation method.     

On the Fréchet space L p −

This paper contains a study of the structure of the Fréchet space L _p ⁻, 1< p ≤∞, defined as the intersection of L _q [0,1] for q<p, and endowed with the projective topology. The main topics covered are: normable, Schwartz and nuclear subspaces of L _p ⁻; construction of uncomplemented copies of ?₂ inside L _p ⁻ for p<2; construction of Montel non-Schwartz subspaces; the space L _p ⁻ is primary. Received: 30 October 1996 / Revised version: 1 February 1998     

Tangent Sets of Order One and Two to the Positive Cones of Some Functional Spaces

We compute the tangent cone and the tangent set of order two to the positive cone of some important functional spaces such as the space C(K) of continuous functions on a compact space and the space L_p(Q) of p-integrable functions on a measured space. The results are applied to the calculus of derivatives of order one and two of marginal functions. Accepted 19 April 1996     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-Cohomology and Normal Solvability

We consider some problems concerning the L _p,q -cohomology of Riemannian manifolds. In the first part, we study the question of the normal solvability of the operator of exterior derivation on a surface of revolution M considered as an unbounded linear operator acting from L_p^k (M) into L^k+1_q (M). In the second part, we prove that the first L _p,q-cohomology of the general Heisenberg group is nontrivial, provided that p < q. Received: 17 January 2006 Supported by INTAS (Grant 03–51–3251) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grants NSh 311.2003.1, NSh 8526.2006.1).     

Optimal Extension of Fourier Multiplier Operators in L p (G)

Given 1 ≤ p < ∞, a compact abelian group G and a p-multiplier ${\psi : \Gamma \to {\mathbb C}}Given 1 ≤ p < ∞, a compact abelian group G and a p-multiplier y: G? \mathbb C{\psi : \Gamma \to {\mathbb C}} (with Γ the dual group), we study the optimal domain of the multiplier operator T^(p)_y : L^p (G) ? L^p (G){T^{(p)}_\psi : L^p (G) \to L^p (G)}. This is the largest Banach function space, denoted by L¹(m^(p)_y){L^1(m^{(p)}_\psi)}, with order continuous norm into which L ^p (G) is embedded and to which T^(p)_y{ T^{(p)}_\psi} has a continuous L ^p (G)-valued extension. Compactness conditions for the optimal extension are given, as well as criteria for those ψ for which L¹(m^(p)_y) = L^p (G){L^1(m^{(p)}_\psi) = L^p (G)} is as small as possible and also for those ψ for which L¹(m^(p)_y) = L¹ (G){L^1(m^{(p)}_\psi) = L^1 (G)} is as large as possible. Several results and examples are presented for cases when L^p (G) \subsetneqq L¹(m^(p)_y) \subsetneqq L¹ (G){L^p (G) \subsetneqq L^1(m^{(p)}_\psi) \subsetneqq L^1 (G)}.     

Algebraic Equivalence and Homology Classes of Real Algebraic Cycles

In this note we study the spectral properties of a multiplication operator in the space L_p(X)^m which is given by an m by m matrix of measurable functions. Our particular interest is directed to the eigenvalues and the isolated spectral points which turn out to be eigenvalues. We apply these results in order to investigate the spectrum of an ordinary differential operator with so called “floating singularities”.     

The Estimates for Sharp Maximal Functions of Multilinear Strongly Singular Integral Operators

In this paper, the author obtains that the multilinear operators of strongly singular integral operators and their dual operators are bounded from some L^p（R^n） to L^p（R^n） when the m-th order derivatives of A belong to L^p（R^n） for r large enough. By this result, the author gets the estimates for the Sharp maximal functions of the multilinear operators with the m-th order derivatives of A being Lipschitz functions. It follows that the multilinear operators are （L^p, L^p）-type operators for 1 〈 p 〈 ∞.     

Sampling theorem of Hermite type and aliasing error on the Sobolev class of functions

Denote by B _2σ,p (1 < p < ∞) the bandlimited class p-integrable functions whose Fourier transform is supported in the interval [−σ, σ]. It is shown that a function in B _2σ,p can be reconstructed in L _p(ℝ) by its sampling sequences {f (κπ / σ)} _κ∈ℤ and {f’ (κπ / σ)} _κ∈ℤ using the Hermite cardinal interpolation. Moreover, it will be shown that if f belongs to L _p ^r (ℝ), 1 < p < ∞, then the exact order of its aliasing error can be determined. Project supported by the Scientific Research Common Program of Beijing Municipal Commission of Education under grant number KM 200410009010 and by the Natural Science Foundation of China under grant number 10071006     

Overcompleteness of Sequences of Reproducing Kernels in Model Spaces

We give necessary conditions and sufficient conditions for sequences of reproducing kernels (k_Θ(·, λ_n))_{n ≥ 1} to be overcomplete in a given model space K_Θ^p where Θ is an inner function in H^∞, p ∈ (1, ∞), and where (λ_n)_{n ≥ 1} is an infinite sequence of pairwise distinct points of Under certain conditions on Θ we obtain an exact characterization of overcompleteness. As a consequence we are able to describe the overcomplete exponential systems in L² (0, a).