Banach Space-valued Functions Research Articles

Talagrand’s influence inequality revisited

Let $\mathscr{C}_n=\{-1,1\}^n$ be the discrete hypercube equipped with the uniform probability measure $\sigma_n$. Talagrand's influence inequality (1994) asserts that there exists $C\in(0,\infty)$ such that for every $n\in\mathbb{N}$, every function $f:\mathscr{C}_n\to\mathbb{C}$ satisfies $$\mathrm{Var}_{\sigma_n}(f) \leq C \sum_{i=1}^n \frac{\|\partial_if\|_{L_2(\sigma_n)}^2}{1+\log\big(\|\partial_if\|_{L_2(\sigma_n)}/\|\partial_i f\|_{L_1(\sigma_n)}\big)}.$$ In this work, we undertake a systematic investigation of this and related inequalities via harmonic analytic and stochastic techniques and derive applications to metric embeddings. We prove that Talagrand's inequality extends, up to an additional doubly logarithmic factor, to Banach space-valued functions under the necessary assumption that the target space has Rademacher type 2 and that this doubly logarithmic term can be omitted if the target space admits an equivalent 2-uniformly smooth norm. These are the first vector-valued extensions of Talagrand's influence inequality. We also obtain a joint strengthening of results of Bakry-Meyer (1982) and Naor-Schechtman (2002) on the action of negative powers of the hypercube Laplacian on functions $f:\mathscr{C}_n\to E$, whose target space $E$ has nontrivial Rademacher type via a new vector-valued version of Meyer's multiplier theorem (1984). Inspired by Talagrand's influence inequality, we introduce a new metric invariant called Talagrand type and estimate it for Banach spaces with prescribed Rademacher or martingale type, Gromov hyperbolic groups and simply connected Riemannian manifolds of pinched negative curvature. Finally, we prove that Talagrand type is an obstruction to the bi-Lipschitz embeddability of nonlinear quotients of the hypercube $\mathscr{C}_n$, thus deriving new nonembeddability results for these finite metrics.

Existence Results for Fractional Integral Equations in Frechet Spaces

The objective of this paper is to present results on the existence of solutions for a class of fractional integral equations in Fr\'{e}chet spaces of Banach space-valued functions on the unbounded interval. Our main tool is the technique of measures of noncompactness and fixed points theorems.

Dunford–Henstock–Kurzweil and Dunford–McShane Integrals of Vector-Valued Functions Defined on m-Dimensional Bounded Sets

In this paper, we define the Dunford–Henstock–Kurzweil and the Dunford–McShane integrals of Banach space-valued functions defined on a bounded Lebesgue measurable subset of m-dimensional Euclidean space $${\mathbb {R}}^{m}$$ . We will show that the new integrals are “natural” extensions of the McShane and the Henstock–Kurzweil integrals from m-dimensional closed non-degenerate intervals to m-dimensional bounded Lebesgue measurable sets. As applications, we will present full descriptive characterizations of the McShane and Henstock–Kurzweil integrals in terms of our integrals. Moreover, a relationship between new integrals will be proved in terms of the Dunford integral.

Elliptic problems and holomorphic functions in Banach spaces

In the first part, we show that a Banach space-valued function f is holomorphic (harmonic) if and only if it is dominated by an L loc 1 function and there exists a separating set W ⊂ X ' such that 〈 f , x ' 〉 is holomorphic (harmonic) for all x ' ∈ W . This improves a known result which requires f to be locally bounded. In the second part, we consider classical results in the L p theory for elliptic differential operators of second order. In the vector-valued setting, these results are shown to be equivalent to the UMD property.

Maximal regularity with weights for parabolic problems with inhomogeneous boundary conditions

In this paper, we establish weighted L^{q}–L^{p}-maximal regularity for linear vector-valued parabolic initial-boundary value problems with inhomogeneous boundary conditions of static type. The weights we consider are power weights in time and in space, and yield flexibility in the optimal regularity of the initial-boundary data and allow to avoid compatibility conditions at the boundary. The novelty of the followed approach is the use of weighted anisotropic mixed-norm Banach space-valued function spaces of Sobolev, Bessel potential, Triebel–Lizorkin and Besov type, whose trace theory is also subject of study.

On Choquet-Pettis Expectation of Banach-Valued Functions: A Counter Example

In probability theory, mathematical expectation of a random variable is very important. Choquet expectation (integral), as a generalization of mathematical expectation, is a powerful tool in various areas, mainly in generalized probability theory and decision theory. In vector spaces, combining Choquet expectation and Pettis integral has led to a challenging and an interesting subject for researchers. In this paper, we indicate and discuss a failure in the previous definition of Choquet-Pettis integral of Banach space-valued functions. To obtain a correct definition of Choquet-Pettis integral, an open problem concerning the linearity of the Choquet integral is stated.

Iterative Methods and their Applications to Banach Space Valued Functions in Abstract Fractional Calculus

Iterative Methods and their Applications to Banach Space Valued Functions in Abstract Fractional Calculus

Iterative Convergence with Banach Space Valued Functions in Abstract Fractional Calculus

Abstract The goal of this paper is to present a semi-local convergence analysis for some iterative methods under generalized conditions. The operator is only assumed to be continuous and its domain is open. Applications are suggested including Banach space valued functions of fractional calculus, where all integrals are of Bochner-type.

Bounded convergence theorem for abstract Kurzweil–Stieltjes integral

In the theories of Lebesgue integration and of ordinary differential equations, the Lebesgue Dominated Convergence Theorem provides one of the most widely used tools. Available analogy in the Riemann or Riemann–Stieltjes integration is the Bounded Convergence Theorem, sometimes called also the Arzelà or Arzelà–Osgood or Osgood Theorem. In the setting of the Kurzweil–Stieltjes integral for real valued functions its proof can be obtained by a slight modification of the proof given for the $$\sigma $$ -Young–Stieltjes integral by T.H. Hildebrandt in his monograph from 1963. However, it is clear that the Hildebrandt’s proof cannot be extended to the case of Banach space-valued functions. Moreover, it essentially utilizes the Arzelà Lemma which does not fit too much into elementary text-books. In this paper, we present the proof of the Bounded Convergence Theorem for the abstract Kurzweil–Stieltjes integral in a setting elementary as much as possible.

A Banach–Zarecki Theorem for functions with values in Banach spaces

A Banach–Zarecki Theorem for a Banach space-valued function \(F : [0,1] \rightarrow X\) with compact range is presented. We define the strong absolute continuity (\(sAC_{||.||_{F}}\)) and the bounded variation (\(BV_{||.||_{F}}\)) of \(F\) with respect to the Minkowski functional \(||.||_{F}\) associated to the closed absolutely convex hull \(C_{F}\) of \(F([0,1])\). It is proved that \(F\) is \(sAC_{||.||_{F}}\) if and only if \(F\) is \(BV_{||.||_{F}}\), weak continuous on \([0,1]\) and satisfies the weak property \((N)\).

A convergence theorem for the improper Riemann integral of Banach space-valued functions

A convergence theorem for the improper Riemann integral of Banach space-valued functions

The differentiability of Banach space-valued functions of bounded variation

In this paper we consider Banach space-valued functions with the compact range. It is shown that if a Banach space-valued function \(F:[0,1] \rightarrow X\) is of bounded variation with respect to the Minkowski functional \(||.||_{F}\) associated to the closed absolutely convex hull \(C_{F}\) of \(F([0,1])\), then \(F\) is differentiable almost everywhere on \([0,1]\).

On Lineability in Vector Integration

We prove the existence of infinite-dimensional linear spaces of Banach space-valued functions whose non-zero elements witness that two given notions of integrability are different: Bochner, Birkhoff, McShane, Pettis and Dunford integrability are considered.

SCALAR BOUNDEDNESS OF VECTOR-VALUED FUNCTIONS

AbstractWe provide sufficient conditions for a Banach space-valued function to be scalarly bounded, which do not require to test on the whole dual space. Some applications in vector integration are also given.

Integration in Hilbert generated Banach spaces

We prove that McShane and Pettis integrability are equivalent for functions taking values in a subspace of a Hilbert generated Banach space. This generalizes simultaneously all previous results on such equivalence. On the other hand, for any super-reflexive generated Banach space having density character greater than or equal to the continuum, we show that Birkhoff integrability lies strictly between Bochner and McShane integrability. Finally, we give a ZFC example of a scalarly null Banach space-valued function (defined on a Radon probability space) which is not McShane integrable.

Pointwise limits of Birkhoff integrable functions

We study the Birkhoff integrability of pointwise limits of sequences of Birkhoff integrable Banach space-valued functions, as well as the conver gence of the corresponding integrals. Both norm and weak convergence are considered. We discuss the roles that equi-Birkhoff integrability and the Bour gain property play in these problems. Incidentally, a convergence theorem for the Pettis integral with respect to the norm topology is presented.

Spaces of vector functions that are integrable with respect to vector measures

AbstractWe study the normed spaces of (equivalence classes of) Banach space-valued functions that are Dobrakov,S* or McShane integrable with respect to a Banach space-valued measure, where the norm is the natural one given by the total semivariation of the indefinite integral. We show that simple functions are dense in these spaces. As a consequence we characterize when the corresponding indefinite integrals have norm relatively compact range. On the other hand, we also determine when these spaces are ultrabornological. Our results apply to conclude, for instance, that the spaces of Birkhoff (respectively McShane) integrable functions defined on a complete (respectively quasi-Radon) probability space, endowed with the Pettis norm, are ultrabornological.

On Henstock–Kurzweil and McShane integrals of Banach space-valued functions

This paper deals with the relation between the McShane integral and the Henstock–Kurzweil integral for the functions mapping a compact interval I 0 ⊂ R m into a Banach space X and some other questions in connection with the McShane integral and the Henstock–Kurzweil integral of Banach space-valued functions. We prove that if a Banach space-valued function f is Henstock–Kurzweil integrable on I 0 and satisfies Property (P), then I 0 can be written as a countable union of closed sets E n such that f is McShane integrable on each E n when X contains no copy of c 0 . We further give an answer to the Karták's question.

On integration of vector functions with respect to vector measures

We study integration of Banach space-valued functions with respect to Banach space-valued measures. We focus our attention on natural extensions to this setting of the Birkhoff and McShane integrals. The corresponding generalization of the Birkhoff integral was first considered by Dobrakov under the name S*-integral. Our main result states that S*-integrability implies McShane integrability in contexts in which the later notion is definable. We also show that a function is measurable and McShane integrable if and only if it is Dobrakov integrable (i.e. Bartle *-integrable).

On Integration by Parts for Stieltjes-Type Integrals of Banach Space-Valued Functions

In this paper we discuss integration by parts for several generalizations of the Riemann-Stieltjes integral. In addition, we obtain new results on integration by parts for the Henstock-Stieltjes integral and its interior modification for Banach space-valued functions.