Existing negative results on invalidity of analogues of classical Density and Differentiation Theorems in infinite-dimensional spaces are considerably strengthened by a construction of a Gaussian measure gamma on a separable Hilbert space H for which the Density Theorem fails uniformly, i.e., there is a set M subset of H of positive gamma-measure such that lim(r SE arrow 0 )sup(x is an element of H) gamma(B(x,r) boolean AND M)/gamma B(x,r) = 0.

By constructing a map of a separable Hilbert space into itself that is continuous, expanding, nonsurjective, and equal to the identity on the unit ball we answer a problem stated by Louis Nirenberg.

We provide sufficient conditions for a set E ⊂ ℝn to be a non-universal differentiability set, i.e., to be contained in the set of points of non-differentiability of a real-valued Lipschitz function. These conditions are motivated by a description of the ideal generated by sets of non-differentiability of Lipschitz self-maps of ℝn given by Alberti, Csörnyei and Preiss, which eventually led to the result of Jones and Csörnyei that for every Lebesgue null set E in ℝn there is a Lipschitz map f: ℝn → ℝn not differentiable at any point of E, even though for n > 1 and for Lipschitz functions from ℝn to ℝ there exist Lebesgue null universal differentiability sets. Among other results, we show that the new class of Lebesgue null sets introduced here contains all uniformly purely unrectifiable sets and gives a quantified version of the result about non-differentiability in directions outside the decomposability bundle with respect to a Radon measure.

The monotonically controlled integral defined by Bendová and Malý, which is equivalent to the Denjoy-Perron integral, admits a natural parameter α>0 thereby leading to the whole scale of integrals called α-monotonically controlled integrals. While the power of these integrals is easily seen to increase with increasing α, our main results show that their exact dependence on α is rather curious. For α<1 they do not even contain the Lebesgue integral, for 1≤α≤ 2 they coincide with the Denjoy-Perron integral, and for α>2 they are mutually different and not even contained in the Denjoy-Khintchine integral.

Motivated by an attempt to find a general chain rule formula for differentiating the composition f circle g of Lipschitz functions f and g that would be as close as possible to the standard formula (f circle g)' (x) = f' (g(x)) circle g'(x), we show that this formula holds without any artificial assumptions provided derivatives are replaced by complete derivative assignments. The idea behind these assignments is that the derivative of f at y is understood as defined only in the direction of a suitable "tangent space" U(f, y) (and so it exists at every point), but these tangent spaces are chosen in such a way that for any g they contain the range of g' (x) for almost every x. Showing the existence of such assignments leads us to a detailed study of derived sets and the ways in which they describe pointwise behavior of Lipschitz functions.

We show a new game characterizing various types of σ-porosity for Souslin sets in terms of winning strategies. We use the game to prove and reprove some new and older inscribing theorems for σ-ideals of σ-porous type in locally compact metric spaces.

Joram's PhD dealt with extensions of linear operators between Banach spaces, leading also to the study of preduals of L 1 spaces.Some of his other groundbreaking research results include a study with A. Pełczyński of Grothendieck's work in Banach space theory and applications thereof,

We show that if n>1 then there exists a Lebesgue null set in R^n containing a point of differentiability of each Lipschitz function mapping from R^n to R^(n-1); in combination with the work of others, this completes the investigation of when the classical Rademacher theorem admits a converse. Avoidance of sigma-porous sets, arising as irregular points of Lipschitz functions, plays a key role in the proof.

We give a geometric description of the smallest σ-ideal Open image in new window of subsets of a separable Banach space with respect to which cone-monotone functions are Gâteaux differentiable almost everywhere. We also show that the usual generalizations of Rademacher’s and Stepanov’s theorems for metric and weak*-differentiability, as well as for Gâteaux and Hadamard differentiability of functions with values in spaces with the Radon-Nikodym property, hold with the exceptional sets belonging to Open image in new window .

It is well known that most continuous functions are nowhere differentiable. Furthermore, in terms of Dini derivatives, most continuous functions are nondifferentiable in the strongest possible sense except in a small set of points. In this paper, we completely characterise families S of sets of points for which most continuous functions have the property that such small set of points belongs to S. The proof uses a topological zero-one law and the Banach-Mazur game.

This chapter shows that if a Banach space with a Fréchet smooth norm is asymptotically smooth with modulus o(tⁿ logⁿ⁻¹(1/t)) then every Lipschitz map of X to a space of dimension not exceeding n has many points of Fréchet differentiability. In particular, it proves that two real-valued Lipschitz functions on a Hilbert space have a common point of Fréchet differentiability. The chapter first presents the theorem whose assumptions hold for any space X with separable dual, includes the result that real-valued Lipschitz functions on such spaces have points of Fréchet differentiability, and takes into account the corresponding mean value estimate. The chapter then gives the estimate for a “regularity parameter” and reduces the theorem to a special case. Finally, it discusses simplifications of the arguments of the proof of the main result in some special situations.

This chapter shows how spaces with separable dual admit a Fréchet smooth norm. It first considers a criterion of the differentiability of continuous convex functions on Banach spaces before discussing Fréchet smooth and nonsmooth renormings and Fréchet differentiability of convex functions. It then describes the connection between porous sets and Fréchet differentiability, along with the set of points of Fréchet differentiability of maps between Banach spaces. It also examines the concept of separable determination, the relevance of the σ‎-porous sets for differentiability and proves the existence of a Fréchet smooth equivalent norm on a Banach space with separable dual. The chapter concludes by explaining how one can show that many differentiability type results hold in nonseparable spaces provided they hold in separable ones.

This chapter describes the modulus of smoothness of a function in the direction of a family of subspaces and the much simpler notion of upper Fréchet differentiability. It also considers the notion of spaces admitting bump functions smooth in the direction of a family of subspaces with modulus controlled by ω‎(t). It shows that this notion is related to asymptotic uniform smoothness, and that very smooth bumps, and very asymptotically uniformly smooth norms, exist in all asymptotically c₀ spaces. This allows a new approach to results on Γ‎-almost everywhere Frechet differentiability of Lipschitz functions. The chapter concludes by explaining an immediate consequence for renorming of spaces containing an asymptotically c₀ family of subspaces.

This chapter introduces the notions of Γ‎-null and Γ‎ₙ-null sets, which are σ‎-ideals of subsets of a Banach space X. Γ‎-null set is key for the strongest known general Fréchet differentiability results in Banach spaces, whereas Γ‎ₙ-null set presents a new, more refined concept. The reason for these notions comes from an (imprecise) observation that differentiability problems are governed by measure in finite dimension, but by Baire category when it comes to behavior at infinity. The chapter first relates Γ‎-null and Γ‎ₙ-null sets to Gâteaux differentiability before discussing their basic properties. It then describes Γ‎-null and Γ‎ₙ-null sets of low Borel classes and presents equivalent definitions of Γ‎ₙ-null sets. Finally, it considers the separable determination of Γ‎-nullness for Borel sets.

This chapter presents a number of results and notions that will be used in subsequent chapters. In particular, it considers the concept of regular differentiability and the lemma on deformation of n-dimensional surfaces. The idea is to deform a flat surface passing through a point x (along which we imagine that a certain function f is almost affine) to a surface passing through a point witnessing that f is not Fréchet differentiable at x. This is done in such a way that certain “energy” associated to surfaces increases less than the “energy” of the functionf along the surface. The chapter also discusses linear operators and tensor products, various notions and notation related to Fréchet differentiability, and deformation of surfaces controlled by ω‎ⁿ. Finally, it proves some integral estimates of derivatives of Lipschitz maps between Euclidean spaces (not necessarily of the same dimension).

This chapter presents the main results on Gâteaux differentiability of Lipschitz functions by recalling the notions of the Radon-Nikodým property (RNP) and null sets. The discussion focuses not only on the mere existence of points of Fréchet differentiability, but also, and often more important, on the validity of the mean value estimates. After considering the RNP of a Banach space, the chapter examines Haar and Aronszajn-Gauss null sets. It then analyzes the existence result for Gâteaux derivatives as well as the meaning of multidimensional mean value estimates. It also explains how, for locally Lipschitz maps of separable Banach spaces to spaces with the RNP, the condition for the validity of the multidimensional mean value estimate may be simplified.

This chapter gives an account of the known genuinely infinite dimensional results proving Fréchet differentiability almost everywhere except for Γ‎-null sets. Γ‎-null sets provide the only notion of negligible sets with which a Fréchet differentiability result is known. Porous sets appear as sets at which Gâteaux derivatives can behave irregularly, and they turn out to be the only obstacle to validity of a Fréchet differentiability result Γ‎-almost everywhere. Furthermore, geometry of the space may (or may not) guarantee that porous sets are Γ‎-null. The chapter also shows that on some infinite dimensional Banach spaces countable collections of real-valued Lipschitz functions, and even of fairly general Lipschitz maps to infinite dimensional spaces, have a common point of Fréchet differentiability.

This chapter introduces the notion of porosity “at infinity” (formally defined as porosity with respect to a family of subspaces) and discusses the main result, which shows that sets porous with respect to a family of subspaces are Γ‎ₙ-null provided X admits a continuous bump function whose modulus of smoothness (in the direction of this family) is controlled by tⁿ logⁿ⁻¹ (1/t). The first of these results characterizes Asplund spaces: it is shown that a separable space has separable dual if and only if all its porous sets are Γ‎₁-null. The chapter first describes porous and σ‎-porous sets as well as a criterion of Γ‎ₙ-nullness of porous sets. It then considers the link between directional porosity and Γ‎ₙ-nullness. Finally, it tackles the question in which spaces, and for what values of n, porous sets are Γ‎ₙ-null.