Differentiability problems in Banach spaces For vector valued functions there are two main version of derivatives: Gâteaux (or weak) derivatives and Fréchet (or strong) derivatives. For a function f from a Banach space X 1 David Preiss into a Banach space Y the Gâteaux derivative at a point x0 ∈ X is by definition a bounded linear operator T : X −→ Y so that Expanded notes of a talk based on a nearly finished research monograph “Fréchet differentiability of Lipschitz functions and porous sets in Banach for every u ∈ X, spaces” f (x + tu) − f (x ) written jointly with lim 0 0 = Tu t→0 t Joram Lindenstrauss2 and Jaroslav Tišer3 The operator T is called the Fréchet derivative of f at x if it is a with some results coming from a joint work with 0 4 5 Gâteaux derivative of f at x0 and the limit above holds uniformly Giovanni Alberti and Marianna Csörnyei in u in the unit ball (or unit sphere) in X. So T is the Fréchet derivative of f at x0 if 1Warwick 2Jerusalem 3Prague 4Pisa 5London
f (x0 + u) = f (x0) + Tu + o(kuk) as kuk → 0.
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Existence of derivatives Sharpness of Lebesgue’s result
The first continuous nowhere differentiable f : R → R was constructed by Bolzano about 1820 (unpublished), who Lebesgue’s result is sharp in the sense that for every A ⊂ of however did not give a full proof. Around 1850, Riemann R measure zero there is a Lipschitz (and monotone) function mentioned such an example, which was later found slightly f : → which fails to have a derivative at any point of A. incorrect. The first published example with a valid proof is by R R Weierstrass in 1875. A more precise result was proved by Zahorski in 1946.
The first general result on existence of derivatives for functions Theorem. A set A ⊂ R is a Gδσ set of Lebesgue measure zero f : R −→ R was found by Lebesgue (around 1900). He proved if and only if there is a Lipschitz function f : R −→ R which is that a monotone function f : R −→ R is differentiable almost differentiable exactly at points of R \ A. everywhere. As a consequence it follows that every Lipschitz Explanation. A set A ⊂ R is function f : R −→ R i.e., a function which satisfies T∞ Gδ if there are open sets Gi so that A = i=1 Gi . G if there are G sets G so that A = S∞ G . |f (s) − f (t)| ≤ C|s − t| δσ δ i i=1 i
for some constant C and every s, t ∈ R, has a derivative a.e.
3 / 24 4 / 24 Rademacher’s Theorem Infinitely many dimensions
Lebesgue’s theorem was extended to Lipschitz functions f : n → by Rademacher in 1919 who showed that in this R R The notion of a Lipschitz function makes sense for functions case f is also differentiable a.e. f : M → N between metric spaces, However, this result is not as sharp as Lebesgue’s: in Rn, n ≥ 2 there are sets of measure zero containing points of dist(f (x), f (y)) ≤ const · dist(x, y). differentiability of all Lipschitz f : R2 → R (Preiss 1990). Doré and Maleva (in preparation) found a compact set in R2 of This gives rise to the study of derivatives of Lipschitz functions Hausdorff dimension one that contains points of differentiability between Banach spaces X and Y . 2 of all Lipschitz functions f : R → R. If dim(X) < ∞ and f is Lipschitz, the two notions of derivative With Alberti and Csörnyei (in preparation) we proved that coincide. However, if dim X = ∞ easy examples show that Rademacher’s theorem is sharp for maps from R2 to R2. there is a big difference between Gâteaux and Fréchet Many higher dimensional results are also known, but the differentiability even for simple Lipschitz functions. question of sharpness of Rademacher’s theorem for maps from R3 to R3 is still open.
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Almost everywhere Further obstacles
It is easy to find nowhere differentiable Lipschitz maps f : R → Y , Y a Banach space. For example, for Y = c0 (the Recall that in infinite dimensional spaces there is no Lebesgue space of sequences converging to zero with maximum norm), measure. If we wish to extend Lebesgue’s theorem to infinite dimensional setting, we have to extend the notion of a.e. sin(t) sin(2t) sin(3t) f (t) = , , ,... (almost everywhere, except for a null set) to such spaces. 1 2 3 So we have to define in a reasonable way a family of negligible Spaces Y for which this pathology does not happen were sets on such spaces. They should form a proper σ-ideal of characterized in various ways by many authors (including subsets of the given space X, i.e., be closed under subsets and Walter Schachermayer). They are called spaces with the RNP countable unions, and should not contain all subsets of X. (Radon-Nikodým property) and include all reflexive spaces. It turns out that there are infinitely many non-equivalent natural There are more obstacles to Fréchet differentiability; e.g. the σ-ideals, some of which are suitable for some differentiability norm on ` is nowhere Fréchet differentiable. Spaces for which questions. 1 this behaviour does not happen are called Asplund spaces. Among separable spaces they are precisely those with separable dual.
7 / 24 8 / 24 Gâteaux differentiability Fréchet differentiability
The unsolved questions about Gâteaux differentiability pale in Starting from about 1970, the theorem of Lebesgue has been comparison with those concerning Fréchet differentiability. extended to Gâteaux differentiability, with various notions of The only general positive result is negligible sets, independently by a number of authors: Mankiewicz, Christensen, Aronszajn, Phelps,. . . Theorem (Preiss 1990). Every real-valued Lipschitz function on an Asplund space has points of Fréchet differentiability. Theorem. Every Lipschitz map from a separable Banach space X into a space Y with the RNP is Gâteaux differentiable However, no “almost everywhere” result is known. almost everywhere. In other words, we do not whether every countable collection of The situation concerning the existence of Gâteaux derivatives real-valued Lipschitz functions on an Asplund space has a is generally deemed to be quite satisfactory. However, once one common point of Fréchet differentiability. goes a bit deeper, fundamental questions remain unanswered. We do not even know whether three Lipschitz functions on a Hilbert space have a common point of Fréchet differentiability.
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Γ-null sets What about Hilbert spaces Lindenstrauss and Preiss (2003) defined a new σ-ideal of In Hilbert spaces (as well as in all `p, 1 < p < ∞) there are negligible sets: A set N ⊂ X is γ-null if it is null on residually porous sets that are not Γ-null. The best we can do is many infinite dimensional C1 surfaces. The differentiability result with these sets is rather curious. Theorem (Lindenstrauss, Preiss, Tišer). Every pair of real-valued Lipschitz function on a Hilbert space has a common Every real-valued Lipschitz function on an Asplund Theorem. point of Fréchet differentiability. space X is Fréchet differentiable Γ-almost everywhere if and only if every set porous in X is Γ-null. Theorem (Lindenstrauss, Preiss, Tišer). Every collection Porous sets are special sets of Fréchet nondifferentiability: a of n real-valued Lipschitz functions on `p, 1 < p < ∞, p ≥ n set E ⊂ X is porous if an only if the function x → dist(x, E) is has a common point of Fréchet differentiability. Fréchet nondifferentiable at any point of E. These results are special cases of a more general result. Notice that the class of porous sets is considerably smaller than Theorem (Lindenstrauss, Preiss, Tišer). If a Banach the class of Fréchet nondifferentiability sets, already on . R space X has modulus of asymptotic smoothness − Originally only asymptotically c0 spaces were known to satisfy o(tn logn 1(1/t)) then every collection of n real-valued Lipschitz the assumption of this Theorem. We now know that it holds for functions on X has a common point of Fréchet differentiability. spaces asymptotically smooth with modulus o(tn) for any n.
11 / 24 12 / 24 Why are 2 and 3 so different? Asymptotic smoothness
The modulus of asymptotic uniform smoothness of a Banach Theorem (Preiss, Tišer). There are three real-valued space X is defined by Lipschitz functions f1, f2, f3 on a Hilbert space so that ρ¯X (t) = sup inf sup kx + yk − 1, t > 0. k k= ( / )<∞ y∈Y ∂f1(x) ∂f2(x) ∂f2(x) x 1 dim X Y + + = 0 kyk≤t ∂x1 ∂x2 ∂x3 The moduli of asymptotic uniform smoothness of ` , at every point at which they are all Fréchet differentiable but not p 1 ≤ p < ∞, and c are at every point at which they are all Gâteaux differentiable. 0 p 1/p We call this phenomenon a (rather strong) failure of the ρ¯`p (t) = (1 + t ) − 1
multi-dimensional mean value estimate. In our positive results ρ¯c0 (t) = max(t − 1, 0) we can actually prove that the multi-dimensional mean value estimate holds. So the above is impossible with two functions. Unlike the usual moduli of uniform smoothness which cannot go to zero faster that t2, there is no limit on behaviour of these moduli as t & 0.
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ε-differentiability Mean value estimates
So called ε-Fréchet differentiability results for Lipschitz functions in asymptotically uniformly smooth spaces say that for Mean value estimates are inequalities controlling increment of every Lipschitz map f of such a space X into a finite the function with the help of its derivative. For Gâteaux dimensional space Y and for every ε > 0 there are x ∈ X and a derivatives they follow from any of the existing results. bounded linear operator T : X → Y so that Theorem. Let f be a real-valued locally Lipschitz function on an open subset G of a separable Banach space X and let kf (y) − f (x) − T (y − x)k lim sup < ε. a, b ∈ G be such that the straight segment [a, b] is contained y→x ky − xk in G. Then for every ε > 0 there is a point x ∈ G at which f is Gâteaux differentiable and For uniformly smooth spaces this was proved by Lindenstrauss and Preiss in 1996. To prove the general case we were joined f 0(x)(b − a) > f (b) − f (a) − ε. by Bates, Johnson and Schechtman (1999). This result is highly exceptional. It is the only general Theorem. On Asplund spaces this holds also for Fréchet differentiability result for Lipschitz function proved in a situation derivatives. when the multi-dimensional mean value estimates may fail.
15 / 24 16 / 24 Multi-dimensional mean value estimates Variational arguments and differentiability
The multi-dimensional mean value estimates for vector-valued The following statement illustrates the role of perturbational functions come from the Gauss-Green (divergence) theorem. variational principles in proving differentiability. Again, they hold for Gâteaux derivatives. Observation. Suppose that Θ: X → is Fréchet Theorem (Lindenstrauss, Preiss, Tišer). If X has R differentiable, ψ : X → is continuous and f : X → is modulus of asymptotic smoothness o(tn logn−1(1/t)) these R R Lipschitz and everywhere Gâteaux differentiable. Suppose estimates hold for Fréchet derivatives of maps X → Rn. further that at some (x0, u0) the function h: E × X → R. Theorem (Lindenstrauss, Preiss, Tišer). These n h(x, u) = f 0(x)(u) + Θ(u) + ψ(x) estimates fail for Fréchet derivatives of maps X → R provided that for some q > n/(n − 1),X ∗ contains a normalized attains its minimum. Then f is Fréchet differentiable at x0. sequence such that that for any real ci and natural k, More delicate versions of the variational principle of Ekeland’s k k X X 1/q type are used to handle the situation when f is not everywhere c x ≤ const · |c |q . i i i Gâteaux differentiable. We describe these principles as infinite i=1 i=1 two player games, which allows some unusual applications.
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The perturbation game The perturbation game — precise rules
The game is played by two players, (A) and (B). Set up. The players are given They are given a metric space (M, d), a function f : M −→ R I a metric space (M, d), and a number ε0 > 0. I a function f : M −→ R bounded from below, and I The game begins by (A) choosing a positive number η0. I a number 0 < ε0 < ∞.
I Then (B) chooses a function F0 : M −→ R, a point x0 ∈ M, Move k of (A). Player (A) chooses ηk > 0 and denotes and positive numbers r , ε . 0 1 X ( ) = ( ) + ( ). I Then (A) chooses a positive number η1. hk x f x Fj x 0≤j
19 / 24 20 / 24 Porosity Γn-null sets A set E in a Banach space X is called porous if there is 0 < c < 1 such that for every x ∈ E and ε > 0 there is y ∈ X so A set N ⊂ X is Γn-null if it is null on residually many 1 that 0 < kyk < ε and n-dimensional C surfaces. The Γ1 null sets are reasonably well understood. B(x + y, ckyk) ∩ E = ∅. Theorem. A separable Banach space X has separable dual if and only if every set porous in X is Γ1-null. In our work, σ-porous sets (countable unions of porous sets) appear as sets of points with special type of Fréchet Much less is known for Γn sets, n > 2. Also the connection non-differentiability, namely that f is differentiable but not between Γn sets for finite n and Γ-null sets is not completely regularly differentiable in some direction u. clear. But we have A map f : X → Y is regularly differentiable at x in the direction Theorem. Gδσ sets that are Γn null for infinitely many n are of u ∈ X if it differentiable at x in the direction some u and for Γ-null. every ε > 0 there is δ > 0 such that Since Fréchet non-differentiability sets are Gδσ, this statement can be used, and has been used, to improve existing result on 0 kf (x + z + tu) − f (x + z) − tf (x; u)k ≤ ε(|t| + kzk) Fréchet differentiability Γ-almost everywhere.
whenever t ∈ R, z ∈ X and |t| + kzk < δ.
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A very optimistic differentiability conjecture One of many problems
Conjecture. In every Asplund space X there is a nontrivial Problem. Can a separable Hilbert space be written as notion of negligible sets such that for every locally Lipschitz ∞ ∞ map f of an open subset G ⊂ X into a Banach space Y having [ [ H = Pi ∪ Gi the RNP: i=1 i=1 I f is Gâteaux differentiable almost everywhere in G. where Pi are porous and Gi are Gâteaux nondifferentiability I If S ⊂ G is a set with negligible complement such that f is sets of Lipschitz maps into RNP spaces (or of real-valued Gâteaux differentiable at every point of S, then Lipschitz functions)? Lip(f ) = sup kf 0(x)k. x∈S More generally, which spaces can be decomposed in this way? I If the set of Gâteaux derivatives of f attained on some set E ⊂ G is norm separable, then f is Fréchet differentiable at The only known results are almost every point of E. I `1 can be written in this way, In spite of having only little evidence for validity of this I spaces admitting a norm with modulus of asymptotic n conjecture, we seem to be very far from disproving it and uniform smoothness o(t ) for every n cannot be written in equally far from proving it. this way.
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