Carathéodory functions in the Banach space setting

2020

Let $ \mathcal{S}=\{T_{s}:s\in S\} $ be a representation of a semigroup $S$. We show that the mapping $T_{\mu}$ introduced by a mean on a subspace of $l^{\infty}(S)$ inherits some properties of $\mathcal{S}$ in Banach spaces and locally convex spaces. The notions of $Q$-$G$-nonexpansive mapping and $Q$-$G$-attractive point in locally convex spaces are introduced. We prove that $T_{\mu}$ is a $Q$-$G$-nonexpansive mapping when $T_{s}$ is $Q$-$G$-nonexpansive mapping for each $s\in S$ and a point in a locally convex space is $Q$-$G$-attractive point of $T_{\mu}$ if it is a $Q$-$G$-attractive point of $ \mathcal{S}$.

Journal of Mathematical Analysis and Applications

We study properties of functions with bounded variation in Carnot-Carathéodory spaces. We prove their almost everywhere approximate differentiability and we examine their approximate discontinuity set and the decomposition of their distributional derivatives. Under an additional assumption on the space, called property R, we show that almost all approximate discontinuities are of jump type and we study a representation formula for the jump part of the derivative.

Journal of Mathematical Analysis and Applications, 2003

We consider the classes of "Grothendieck-integral" (G-integral) and "Pietsch-integral" (P-integral) linear and multilinear operators (see definitions below), and we prove that a multilinear operator between Banach spaces is G-integral (resp. P-integral) if and only if its linearization is G-integral (resp. P-integral) on the injective tensor product of the spaces, together with some related results concerning certain canonically associated linear operators. As an application we give a new proof of a result on the Radon-Nikodym property of the dual of the injective tensor product of Banach spaces. Moreover, we give a simple proof of a characterization of the G-integral operators on C(K, X) spaces and we also give a partial characterization of P-integral operators on C(K, X) spaces.

1982

The fundamental link between prespectral measures and Banach function spaces is to be found in a theorem of T.A.Gillespie which relates cyclic spaces isornorphically to certain Banach function spaces. We obtain here an extension of this result to the wider class of precyclic spaces. We then consider the properties of weak sequential completeness and reflexivity in Banach function spaces: necessary and sufficient conditions are obtained which in turn, via the afore-mentioned isomorphisms., both extend and simplify analogously formulated existing results for cyclic spaces. Finally the concept of a homomorphism between pairs of Banach function spaces is examined.The class of such mappings is determined and a complete description obtained in the form of a (unique) disjoint sum of two mappings, one of which is always an isomorphism and the other of which is arbitrary in a certain sense, or null.It is shown moreover that the isomorphic component itself is composed of two other isomorphism...

Journal of Mathematical Analysis and Applications, 1979

In recent papers (see 61) we were concerned with the question of when a p E MR(Zr), extended to a v E MR(L?'a) where, in general, 9i C gS were lattices of subsets of an abstract set X, and MR(S) was the collection of all (totally finite) Z-regular finitely additive measures defined on a(Y), the algebra generated by 9. Analogous questions were raised and answered for o-smooth measures and numerous applications given to both the solution of topological and measure theoretic questions. A more recent generalization of these procedures, (see ), led to a very general measure extension procedure for arbitrary (not necessarily bounded) measures.

Journal of Soviet Mathematics, 1986

Bull. Amer. Math. Soc, 1965

Lecture Notes in Mathematics, 1997

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

2019

We study properties of functions with bounded variation in Carnot-Carathéodory spaces. In Chapter 2 we prove their almost everywhere approximate differentiability and we examine their approximate discontinuity set and the decomposition of their distributional derivatives. Under an additional assumption on the space, called property R, we show that almost all approximate discontinuities are of jump type and we study a representation formula for the jump part of the derivative. In Chapter 3 we prove a rank-one theorem à la G. Alberti for the derivatives of vector-valued maps with bounded variation in a class of Carnot groups that includes all Heisenberg groups H n with n ≥ 2. Some important tools for the proof are properties linking the horizontal derivatives of a real-valued function with bounded variation to its subgraph. In Chapter 4 we prove a compactness result for bounded sequences (u j) of functions with bounded variation in metric spaces (X, d j) where the space X is fixed, but the metric may vary with j. We also provide an application to Carnot-Carathéodory spaces. The results of Chapter 4 are fundamental for the proofs of some facts of Chapter 2. dimension Q = s i=1 i(n i −n i−1) and the metric measure space (R n , d, L n) (where L n denotes the n-dimensional Lebesgue measure) is locally Ahlfors Q-regular (see Theorem 1.2.4), i.e., for every compact set K ⊆ R n there exist C ≥ 1 and R > 0 such that 1 C r Q ≤ L n (B(p, r) ≤ Cr Q , R will be discussed into details later on, together with the definition of rectifiability. Some of the main results about fine properties of BV functions presented in Chapter 2 need some fine blow-up analysis about intrinsic regular hypersurfaces (see Section 1.5). Chapter 2 and Section 1.5 are mostly new and contained in the work of the author and his supervisor Davide Vittone [30]. Part of the analysis of singular points for BV X functions requires some blow-up technique together with the nilpotent approximation of a CC space. Chapter 4 contains a technical but fundamental lemma (contained in [29]) that ensure compactness of equi-The aim of Chapter 2 is to establish "fine" properties of BV functions in CC spaces. A first non-trivial part of this Chapter consists in fixing the appropriate language in a consistent and robust manner. Section 2.1 is therefore devoted to the introduction of approximate notions of continuity, jump point and differentiability point for generic L 1 loc maps in CC spaces. The notion of approximate continuity has been already worked out in the literature (see e.g. [48, Section 2.7]) by the extension of the Lebesgue Theorem The proof of Theorem 1 is based on Lemma 2.2.6, i.e., on a suitable extension to CC spaces of the inequalitŷ B(p,r) |u(q) − u(p)| |q − p| dL n (q) ≤ Cˆ1 0 |Du|(B(p, tr)) t n dt valid for a classical BV function u on R n. Lemma 2.2.6 answers an open problem stated in [8] and it is new even in Carnot groups. Theorem 1 was proved in the setting of Carnot groups in [8] together with the following result, which we also extend to our more general setting. We denote by H Q−1 the Hausdorff measure of dimension Q − 1 and by S u the set of points where a function u does not possess an approximate limit in the sense of Definition 2.1.1. However, for classical BV functions much stronger results than Theorems 1 and 3 are indeed known: some of them are proved in Section 2.2 for BV X functions under the additional assumption that the space (R n , X) satisfies the following condition. Definition 1 (Property R). Let (R n , X) be an equiregular CC space with homogeneous dimension Q. We say that (R n , X) satisfies the property R if, for every open set Ω ⊆ R n and every E ⊆ R n with locally finite X-perimeter in Ω, the essential boundary ∂ * E ∩ Ω of E in Ω is countably X-rectifiable, i.e., there exists a countable family {S i : i ∈ N} of C 1 X hypersurfaces such that H Q−1 (∂ * E ∩ Ω \ ∞ i=0 S i) = 0. D X u to any countably X-rectifiable set R. D s G u |D s G u| (x) has rank one for |D s G u|-a.e. x ∈ Ω. It is worth pointing out that Theorem 7 applies to the n-th Heisenberg group H n , provided n ≥ 2. Heisenberg groups are defined in Example 1.3.24 and they represent some of the most simple non-trivial examples of Carnot groups. Notice also that to the rectifiable set ∂ * E u coincides H Q-almost everywhere with the measure-theoretic horizontal normal to E u. As already pointed out, by Theorem 8, property w-R is weaker than property R but we conjecture they are actually equivalent. Property w-R is a non-trivial technical obstruction one has to face when following the strategy of [67]: the rectifiability of sets with finite G-perimeter in Carnot groups is indeed a major open problem, which has been solved only in step 2 Carnot groups (see [38, 39]) and H Q−2 (Σ) might be either 0 or +∞ (even locally) as shown by A. Kozhevnikov [53]. See also the recent paper [63]. The fact that Theorem 9 does not apply to H 1 (actually, to H 1 × R × R, see the proof of Lemma 3.2.7) prevents us from proving the Rank-One Theorem for G = H 1. This does not follow from [26] either (see Remark 3.4.7) and, thus, it remains a very interesting open problem. BV X j functions with constant independent of j; these two results (Theorems 4.2.4 and 4.2.5, respectively) use in a crucial way some outcomes of the papers [18, 73]. As it is clear by the techniques used in Chapter 2, in the study of fine properties of BV X functions in CC spaces, and in particular of their local properties, one often needs to perform a blow-up procedure around a fixed point p: as explained in Theorem 1.4.5, this produces a sequence of CC metric spaces (R n , X j) that converges to (a quotient of) a Carnot group structure G. In this blow-up, the original BV X function u 0 gives rise to a sequence (u j) of functions in BV X j which, up to subsequences, will converge

p-Adic Numbers, Ultrametric Analysis and Applications, 2019

This work will be centered in commutative Banach subalgebras of the algebra of bounded linear operators defined on free Banach spaces of countable type. The main goal of this work will be to formulate a representation theorem for these operators through integrals defined by spectral measures type. In order to get this objective, we will show that, under special conditions, each one of these algebras is isometrically isomorphic to some space of continuous functions defined over a compact set. Then, we will identify such compact sets developing the Gelfand space theory in the non-Archimedean setting. This fact will allow us to define a measure which is known as spectral measure. As a second goal, we will formulate a matrix representation theorem for this class of operators in which the entries of the matrices will be integrals coming from scalar measures.

European Journal of Mathematics

The notion of a Banach space is one of the most fundamental notions of modern mathematics. Such spaces were named to honour Stefan Banach (1892-1945), one of the founders of Functional Analysis, who lived, worked and died in Lviv (now the largest city in western part of Ukraine). Of course, there are many important Banach spaces: spaces of sequences, functions, operators, etc. Yet, there exists one very concrete Banach space, called the Banach space. It includes numerous historical places in Lviv related to Stefan Banach: the houses where B Taras Banakh

2010

If ϕ is an analytic function bounded by 1 on the bidisk D 2 and τ ∈ ∂(D 2) is a point at which ϕ has an angular gradient ∇ϕ(τ) then ∇ϕ(λ) → ∇ϕ(τ) as λ → τ nontangentially in D 2. This is an analog for the bidisk of a classical theorem of Carathéodory for the disk. For ϕ as above, if τ ∈ ∂(D 2) is such that the lim inf of (1 − |ϕ(λ)|)/(1 − λ) as λ → τ is finite then the directional derivative D −δ ϕ(τ) exists for all appropriate directions δ ∈ C 2. Moreover, one can associate with ϕ and τ an analytic function h in the Pick class such that the value of the directional derivative can be expressed in terms of h.

Topological Methods in Nonlinear Analysis, 2016

Let T be a measurable space, X a Banach space while Y a Banach lattice. We consider the class of "upper separated" set-valued functions F : T × X → 2 Y and investigate the problem of the existence of Carathéodory type selection, that is, measurable in the first variable and order-convex in the second variable.

2020

We introduce Kuelbs-Steadman-type spaces for real-valued functions, with respect to countably additive measures, taking values in Banach spaces. We investigate their main properties and embeddings in L^p-type spaces, considering both the norm associated to norm convergence of the involved integrals and that related to weak convergence of the integrals.