Some remarks on Rademacher's theorem in infinite dimensions

Journal of Mathematical Sciences, 1997

Among the most notable events in the nonlinear functional analysis in the past two decades, one can mention the development of the theory of differentiable measures and the creation of the Malliavin calculus. These two theories can be regarded as infinite-dimensional analogs of such classical fields as geometric measure theory, the theory of Sobolev spaces, and the theory of generalized functions.

Journal of Mathematical Analysis and Applications, 1973

Introduction. Let J be a <r-field of subsets of an abstract set M and let m(e) be a non-negative measure function defined on J. The classical Radon-Nikodym theorem [17, p. 36](1) states that, if M is the union of a countable number of sets of finite measure, then a necessary and sufficient condition for a completely additive real function R(e), defined over J, to be a Lebesgue integral (with respect to m(e)) is that R(e) be absolutely continuous relative to m(e). Our purpose is to extend this theorem to functions with values in an arbitrary Banach space and apply the resulting theorem to obtain an integral representation for the general bounded linear transformation on the space of summable functions to an arbitrary Banach space. A number of writers [4, 6, 7, 8, 11, 12, 13, 14] have obtained similar extensions; however they have all imposed restrictions either on the Banach space or on the completely additive functions considered. The theorem proved here is free of all such restrictions. It is evident that any such generalization of the Radon-Nikodym theorem will involve a corresponding generalization of the Lebesgue integral, of which there are many. A variation of an integral studied in detail by B. J. Pettis(2) will be used here. A point function x(p) defined on ¥ to a Banach space X is said to be Pettis integrable [12] provided there exists a function X(e) on J to Ï such that, for each element x of the space 3-adjoint to ï and each element e of J, the function x(x(p)) is Lebesgue integrable on the set e to the value x(X(e)). Whenever X(e) exists, it is completely additive and absolutely continuous relative to m(e). On the other hand, Pettis [12, p. 303] gave an example of a completely additive function which is absolutely continuous but is not an integral in his sense. This shows that the ordinary Pettis integral cannot appear in a general Radon-Nikodym theorem. However, without changing essentially the definition or general properties of the integral, we can enlarge the class of functions admissable for integration (so that it contains certain functions other than point functions) and thus obtain an integral which will serve our purposes. The class of functions which we will admit for integration consists of all multivalued set functions x(e) defined for elements of J having finite, nonzero Except for §5, the contents of this paper were presented to the Society, September 12, 1943. The results in §5 were presented February 27, 1944, under the title Representation of linear transformations on summable functions.

Nagoya Mathematical Journal, 1972

In a recent paper, Sato [6] has shown that for every Gaussian measure n on a real separable or reflexive Banach space (X, ‖ • ‖) there exists a separable closed sub-space X〵 of X such that and is the σ-extension of the canonical Gaussian cylinder measure of a real separable Hilbert space such that the norm is contiunous on and is dense in The main purpose of this note is to prove that ‖ • ‖ x〵 is measurable (and not merely continuous) on .

Pacific Journal of Mathematics, 1975

Lecture Notes in Mathematics, 1991

Recently Talagrand [T] estimated the deviation of a function on {0, 1} n from its median in terms of the Lipschitz constant of a convex extension of f to ℓ n 2 ; namely, he proved that

Bulletin of the Australian Mathematical Society, 2000

We prove that a Banach space X has the Radon-Nikodym property if, and only if, every weak*-lower semicontinuous convex continuous function f of X* is Gâteaux differentiable at some point of its domain with derivative in the predual space x.

Lecture Notes in Mathematics, 1978

The study of differential equations for functions of infinitely many variables leads naturally to distribution theory on infinite dimensional spaces. Several kinds of test functions and the corresponding distributions have been introduced by Alvarez [1],Berezanskii and Samoilenko [4], Dudin [7, 8], Elson [i0], Fomin [12], Kr~e [30] and Kuo [18]. In the infinite dimensional case there is no natural way to regard bounded measurable functions as distributions because the Lebesgue measure does not exist. Thus one cannot expect to represent certain distributions~ e.g. harmonic distributions, by smooth functions. However, finite Borel measures can be regarded as distributions in the natural way. It is then desirable to develop differential calculus for measures so that, in particular, harmonic distributions can be represented by smooth measures. The notion of differentiable measures was first introduced by Fomin [ii, 13] and studied in details by Averbuh, Smoljanov and Fomin [2, 3]. Differentiable measures and differential equations for them have also been studied by Daleckii and Fomin [6], Kuo [16, 21] and Uglanov . Formally, the derivative of a Borel measure ~ on a topological vector space V can be defined by considering the limit lim ~I{~A + ev) -~(A)] for v in V and a ¢40 Borel subset A of V. Denote the limit by ~'(A)(v) if it exists. Thus ~' is a V*valued Borel measure. If V =~n then ~ is differentiable Iff ~ is absolutely continuous with respect to Lebesgue measure and its density is a.e. differentiable.

Mathematical Notes, 2015

In this paper, we study various generalizations of the classical Wiener algebra on a Banach space and prove analogs of Wiener's theorem on the invertibility of elements of such algebras.

Journal of Geometric Analysis, 2004

We extend Cheeger's theorem on differentiability of Lipschitz functions in metric measure spaces to the class of functions satisfying Stepanov's condition. As a consequence, we obtain the analogue of Calderon' s differentiability theorem of Sobolev functions in metric measure spaces satisfying a Poincard inequality. Recent years have seen an intense ongoing research activity in extending classical results of analysis in Euclidean spaces to the setting of general metric-measure spaces. We do not intend to present a full list of achievements in this area, but we mention the works [12, 15, 21, 11, 9], and [10] for results on Sobolev spaces; and works of [14] and [19] for results on quasiconformal mappings in this general framework. We refer to the monograph of Heinonen [13] for an overview of this development. A major advance in this area of research was marked by the work of Cheeger [5] who extended Rademacher's differentiability theorem to the fairly large class of metric-measure spaces which Math Subject Classifications. 28A 15, 26B05.

Publicacions Matemàtiques, 2005

In the setting of R d with an n−dimensional measure µ, we give several characterizations of Lipschitz spaces in terms of mean oscillations involving µ. We also show that Lipschitz spaces are preserved by those Calderón-Zygmund operators T associated to the measure µ for which T (1) is the Lipschitz class 0.

Journal of Mathematical Analysis and Applications, 2006

Let μ be a scalar measure of bounded variation on a compact metrizable abelian group G. Suppose that μ has the property that for any measure σ whose Fourier-Stieltjes transformσ vanishes at ∞, the measure μ * σ has Radon-Nikodým derivative with respect to λ, the Haar measure on G. Then L. Pigno and S. Saeki showed that μ itself has Radon-Nikodým derivative. Such property is not shared by vector measures in general. We say that a Banach space X has the near differentiability property if every X-valued measure of bounded variation shares the above property. We prove that Banach spaces with the Radon-Nikodým property have the near differentiability property, while Banach spaces with the near differentiability property enjoy the near Radon-Nikodým property. We also show that the Banach spaces L 1 [0, 1] and L 1 /H 1 0 have the near differentiability property. Lastly, we show that Banach spaces with the near differentiability property have type II-Λ-Radon-Nikodým property, whenever Λ is a Riesz subset of type 0 of G.

Proceedings of the Edinburgh Mathematical Society, 2017

It is well known that the standard Lipschitz space in Euclidean space, with exponent α ∈ (0, 1), can be characterized by means of the inequality , where is the Poisson integral of the function f. There are two cases: one can either assume that the functions in the space are bounded, or one can not make such an assumption. In the setting of the Ornstein–Uhlenbeck semigroup in ℝ n , Gatto and Urbina defined a Lipschitz space by means of a similar inequality for the Ornstein–Uhlenbeck Poisson integral, considering bounded functions. In a preceding paper, the authors characterized that space by means of a Lipschitz-type continuity condition. The present paper defines a Lipschitz space in the same setting in a similar way, but now without the boundedness condition. Our main result says that this space can also be described by a continuity condition. The functions in this space turn out to have at most logarithmic growth at infinity.

We study differentiability properties of the Riesz potential, with kernel of ho-mogeneity 2 − d in R d , d ≥ 3, of a finite Borel measure. In the plane we consider the logarithmic potential of a finite Borel measure. We introduce a notion of differentiability in the capacity sense, where capacity is Newtonian capacity in dimension d ≥ 3 and Wiener capacity in the plane. We require that the first order remainder at a point is small when measured by means of a normalized weak capacity " norm " in balls of small radii centered at the point. This implies L p dif-ferentiability in the Calderón–Zygmund sense for 1 ≤ p < d/d − 2. If d ≥ 3, we show that the Riesz potential of a finite Borel measure is differentiable in the capacity sense except for a set of zero C 1-harmonic capacity. The result is sharp and depends on deep results in non-doubling Calderón–Zygmund theory. In the plane the situation is different. Surprisingly there are two distinct notions of differentia-bility in the capacity sense. For each of them we obtain the best possible result on the size of the exceptional set in terms of Hausdorff measures. Finally we obtain, for d ≥ 3, results on Peano second order differentiability in the sense of capacity with exceptional sets of zero Lebesgue measure.

Proyecciones (Antofagasta), 2001

In this paper we define the absolutely continuous relation between nonarchimedean scalar measures and then we give and prove a version of the Radon-Nykodym Theorem in this setting. We also define the nonarchimedean vector measure and prove some results in order to prepare a version of this Theorem in a vector case.

Physica D: Nonlinear Phenomena, 2010

Functions of bounded variation in an abstract Wiener space, i.e., an infinite dimensional Banach space endowed with a Gaussian measure and a related differentiable structure, have been introduced by M. Fukushima and M. Hino using Dirichlet forms, and their properties have been studied with tools from stochastics. In this paper we reformulate, with purely analytical tools, the definition and the main properties of BV functions, and start investigating further properties.