Derivatives 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 David Preiss 1 into a Banach space Y the Gâteaux derivative at a point x 0 ∈ 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 for every u ∈ X , “Fréchet differentiability of Lipschitz functions and porous sets in Banach spaces” f ( x 0 + tu ) − f ( x 0 ) written jointly with lim = Tu t t → 0 Joram Lindenstrauss 2 Jaroslav Tišer 3 and The operator T is called the Fréchet derivative of f at x 0 if it is a with some results coming from a joint work with Gâteaux derivative of f at x 0 and the limit above holds uniformly Giovanni Alberti 4 Marianna Csörnyei 5 and in u in the unit ball (or unit sphere) in X . So T is the Fréchet derivative of f at x 0 if 1 Warwick 2 Jerusalem 3 Prague 4 Pisa 5 London f ( x 0 + u ) = f ( x 0 ) + Tu + o ( � u � ) as � u � → 0 . 1 / 24 2 / 24 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of Lebesgue’s result The first continuous nowhere differentiable f : R → R was constructed by Bolzano about 1820 (unpublished), who Lebesgue’s result is sharp in the sense that for every A ⊂ R of however did not give a full proof. Around 1850, Riemann measure zero there is a Lipschitz (and monotone) function mentioned such an example, which was later found slightly f : R → R which fails to have a derivative at any point of A . incorrect. The first published example with a valid proof is by A more precise result was proved by Zahorski in 1946. Weierstrass in 1875. Theorem. A set A ⊂ R is a G δσ set of Lebesgue measure zero The first general result on existence of derivatives for functions if and only if there is a Lipschitz function f : R − → R which is f : R − → R was found by Lebesgue (around 1900). He proved differentiable exactly at points of R \ A. that a monotone function f : R − → R is differentiable almost 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 G δ if there are open sets G i so that A = � ∞ i = 1 G i . G δσ if there are G δ sets G i so that A = � ∞ i = 1 G i . | f ( s ) − f ( t ) | ≤ C | s − t | 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 : R n → R by Rademacher in 1919 who showed that in this 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 R n , 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 : R 2 → R (Preiss 1990). Doré and Maleva (in preparation) found a compact set in R 2 of This gives rise to the study of derivatives of Lipschitz functions between Banach spaces X and Y . Hausdorff dimension one that contains points of differentiability of all Lipschitz functions f : R 2 → 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 R 2 to R 2 . there is a big difference between Gâteaux and Fréchet differentiability even for simple Lipschitz functions. Many higher dimensional results are also known, but the question of sharpness of Rademacher’s theorem for maps from R 3 to R 3 is still open. 5 / 24 6 / 24 Almost everywhere Further obstacles It is easy to find nowhere differentiable Lipschitz maps f : R → Y , Y a Banach space. For example, for Y = c 0 (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 ( 2 t ) , sin ( 3 t ) � 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 ℓ 1 is nowhere Fréchet differentiable. Spaces for which questions. 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: Theorem (Preiss 1990). Every real-valued Lipschitz function Mankiewicz, Christensen, Aronszajn, Phelps,. . . on an Asplund space has points of Fréchet differentiability. Theorem. Every Lipschitz map from a separable Banach However, no “almost everywhere” result is known. space X into a space Y with the RNP is Gâteaux differentiable In other words, we do not whether every countable collection of almost everywhere. real-valued Lipschitz functions on an Asplund space has a The situation concerning the existence of Gâteaux derivatives common point of Fréchet differentiability. is generally deemed to be quite satisfactory. However, once one We do not even know whether three Lipschitz functions on a goes a bit deeper, fundamental questions remain unanswered. Hilbert space have a common point of Fréchet differentiability. 9 / 24 10 / 24 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 ℓ 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 C 1 surfaces. Theorem (Lindenstrauss, Preiss, Tišer). Every pair of The differentiability result with these sets is rather curious. real-valued Lipschitz function on a Hilbert space has a common Theorem. Every real-valued Lipschitz function on an Asplund point of Fréchet differentiability. space X is Fréchet differentiable Γ -almost everywhere if and Theorem (Lindenstrauss, Preiss, Tišer). Every collection only if every set porous in X is Γ -null. of n real-valued Lipschitz functions on ℓ p , 1 < p < ∞ , p ≥ n Porous sets are special sets of Fréchet nondifferentiability: a has a common point of Fréchet differentiability. set E ⊂ X is porous if an only if the function x → dist ( x , E ) is 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 o ( t n log n − 1 ( 1 / t )) then every collection of n real-valued Lipschitz Originally only asymptotically c 0 spaces were known to satisfy 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 ( t n ) for any n . 11 / 24 12 / 24
Recommend
More recommend
