Differentiability and Measures in Banach Spaces

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1 Differentiability and Measures in Banach Spaces David Preiss Department of Mathematics, University College London, London WC1E 6BT, UK The purpose of this contribution is to give information about new results concerning natural questions about differentiability and measures in real Banach spaces (of infinite but also of finite dimension) and, possibly more importantly, to point out some of the many open problems we are still faced with in this area of research. 1. Differentiability We recall two well known notions. 1. A real valued function / defined on an open subset G of a Banach space E is said to be Fréchet differentiate at a point x G G if there is f(x) e E* such that lim \f(x + u)-f(x)-(f(x),u)\ = 0 «-0 u f(x) is called the Fréchet derivative of/ at x. 2. A real valued function / defined on an open subset G of a Banach space E is said to be Lipschitz on G if there is a constant C such that \f(x) f(y)\ < C\\x y\\ whenever x,y G G. The smallest such constant C is denoted by Lip(/). From the work of Lebesgue (in the one dimensional case) and of Rademacher (in the finite dimensional case) we know that Lipschitz functions on finite dimensional spaces are (Fréchet) differentiate almost everywhere with respect to the Lebesgue measure. Infinite dimensional results of similar nature are known for Gateaux differentiability. (See [1,3,5,6]). These extension are obtained by a linear approximation of the infinite dimensional situation by finite dimensional spaces. However, the question of Fréchet differentiability seems to need a different approach. This might be also seen from many examples of nowhere Fréchet differentiable Lipschitz mappings of a separable Hilbert space into itself, since for such mappings the Gateaux differentiability results mentioned above still hold. Thus our first result answers a natural question. Proceedings of the International Congress of Mathematicians, Kyoto, Japan, 1990

2 924 David Preiss Theorem 1. Every Lipschitz function defined.on a separable Hilbert] space is Fréchet differentiate at least at one point. Hilbert spaces are, of course, not the most general spaces in which one would hope for such a result. Indeed, from the extensive investigations of differentiability questions for continuous convex functions (e.g., [13,14]) we know that the result may hold in all Asplund spaces. (A Banach space is said to be an Asplund space if the dual of every its separable subspace is separable.) This generalization of Theorem 1 is given in the following statement. Theorem 2. Every locally Lipschitz function defined on an open subset of an Asplund space is Fréchet dijferentiable on a dense subset of its domain. The method we use need not be confined to Fréchet differentiability. It applies also to so called & derivatives, in the definition of which we require the uniform convergence on the members of a given family ^ of bounded subsets of the Banach space satisfying some mild additional assumptions. (The details can be found in [10].) This gives the most general form of the above differentiability results. (However, a recent Haydon's example of an Asplund space without equivalent smooth norms shows that the deduction of Theorem 2 from Theorem 3 is not straightforward.) Theorem 3. Let E be a Banach space admitting an equivalent norm which is M dijferentiable away from the origin. Then every locally Lipschitz function defined on an open subset G of E is & differentiable on a dense subset of G. These statements, as given, are not satisfactory from the point of view of possible applications. For example, suppose that a Lipschitz. function / on a separable Hilbert space has derivative zero at every point at which it is Fréchet differentiable. We would like to be able to deduce that / is constant. This can be done, since in all the above results the mean value theorem holds. For example, in case of Theorem 3 we prove that the increment of the function over any segment [u, v] c: G is majorized by the supremum of the derivatives in the direction v u at points at which the function is & differentiable. The proof of the above results requires new information about Lipschitz functions in finite dimensional spaces. Thus, as a byproduct, we get the following curious statement. Theorem 4. There is a piarle set N of Lebesgue measure zero such that every Lipschitz function defined on the plane is differentiable at some point of N. To describe a set having such a property is quite easy: Any Gs plane set of Lebesgue measure zero containing all lines passing through two different points with rational coordinates will do. This particular example also suggests the reasons why our proof of Fréchet differentiability results is not straightforward. It combines in some way two notions of smallness of a set: First category (hence the G s part) and measure zero (hence the lines). It seems to be intuitively clear that a similar mixture is impossible on the line. That this is true has been shown in [2] and [15] : Theorem 4 is false on the line..

3 Differentiability and Measures in Banach Spaces Construction of a Point of Fréchet Differentiability The details of the proof can be found in [10]. Here we just point out the main observations. Because of that we restrict our attention to the proof of Theorem 1 only. We introduce the directional derivatives of/ by f'(x,e) = lim f{x + re) - f{x \ r->0 r and we denote by M the set of all pairs (x,e) e E x E* such that \\e\\ = 1 and f f (x,e) exists. The first basic observation is that if (x,e) e M and f(x,e) = Lip(/) then / is Fréchet differentiable at x. Even though such a pair need not exist, this suggests that we might attempt to use a maximizing procedure. Thus our plan is to construct inductively a sequence (xk,eu) G M so that: 1. The sequence Xk converges to some x. 2. The sequence e/ c converges to some e. 3. The directional derivative f(x,e) exists. 4. For the pair (x, e) some variant of the above observation can be used. To achieve 1, we simply choose x^+i close to x*. This is based on a local form of our observation, namely, that the equality of f'(x,e) to the limit of the Lipschitz constants on balls around x suffices for Fréchet differentiability of / at x. Unfortunately, to get 2 is not so simple. Since requirement 4 forces us to take /'(xfcjß/c) as large as possible, we cannot at the same time prescribe how close should ek be to e^-\. There is also a different objection we should take into account: If our method worked, we would construct not only a point of differentiability, but also a point at which gradient vector exists. This causes no problem in Hilbert spaces, but is impossible in non-reflexive Asplund spaces. (Every liner functional not attaining its maximum on the unit ball gives an example.) Thus an idea suggests itself: We should change the norm (and the change should depend on /), at least in the general case. Recalling that we are constructing a sequence e^ of unit vectors, and observing that a small change of the norm can drastically change the set of pairs considered for the choice of (x/ c+ i,e/ c+ i), we find that 2 can be achieved by constructing, together with the sequence (x/ c, ej ( ), a sequence of norms p/ c, where pu+i is the (e.g., h) sum of pk and of a (small) multiple of the distance to the one dimensional subspace of H generated by ej c. Then the conditions p*+i(e/h-i) = 1 (= p(eu)) and f(xk+uek+i) > /'(^/CJ^/C) already imply that ej<+i is close to ej c. The requirement 3 seems to be the most difficult. To get it, we observe that the problem is essentially one dimensional and requires some method of interchange of limit and derivative. Since we cannot hope to be able to use anything like the uniform convergence of the derivatives, the only possibility seems to be to choose the points at which the increment of the function is approximated by the derivative globally. The following one dimensional lemma says that this can be done.

4 926 David Preiss Lemma 5. Suppose that a < Ç < b, 0 < a < 1/4, and L > 0 are real numbers, h is a Lipschitz function defined on [a,b], Lip(/i) < L, h(a) = h(b) 0, and h(ç) ^ 0. Then there is a measurable set A c (a, b) such that 1. The Lebesgue measure of the set A is at least a\h(t;)\/l, 2. h'(x) > (T\h(Ç)\/(b -v- a) for every % e A, and 3. \h(t) h(n)\ < 4(1 + 2a)^h'(x)L\t T for every % e A and every t e [a,b]. The most important third statement of the lemma says that the approximation of the increment of the function by its derivative at the point % is "globally good" in the whole interval [a,b]. The second statement just says that the derivative at T increased as much as we could hope for. From the first statement we just use that T can be chosen sufficiently far from the end points. This is needed in order to get a bilateral approximation. Because in the first statement of Lemma 5 we do not have to speak about measure, the Lemma can be formulated without the notion of the Lebesgue measure. We can then try to prove it without any use of measure theory. This sounds difficult, since we also claim that h is differentiable at x. But we can also replace the derivatives by lower derivatives and get a version of the lemma that really can be proved without any use of measure theory. Surprisingly enough, this statement then easily implies that Lipschitz functions on the real line have at least one point of differentiability. Though I did not follow this way, since to use the Lebesgue measure and maximal operator technique turned out to be much easier, these remarks suggest that the proof of differentiability discussed here is different from the usual measure theoretic proofs. Having done this, we can already imagine how to construct the sequence (x/c,ßfc) so that 3 holds: We will choose (xk+uck+i) so that the approximation of the the increment of the function by its directional derivative at the point xj^+i in the direction e^i is "globally good" on the whole Une through Xfc+i in the direction ek+\. However, the previous choice implies that our construction will lead to a pair (x,e) for which the equality f'(x,e) = Lip(/) is quite far from being true. Hence to achieve 4 we need to improve upon our main observation. We first reformulate this observation as: A Lipschitz function / on a Hilbert space is Fréchet differentiable at x if there is a unit vector e such that f(x,e) exists and lim sup{/'(x, e); (x, e) e M and x x < 3} < f(x, e). A simple proof of this statement together with Lemma 5 gives the following differentiability criterion, which we formulate in the most general situation. Theorem6. Suppose that E is a Banach space, x y G E, eo e E, \\eo\\ = 1, and that f is a Lipschitz function defined on E such that f'(xo, eo) exists. Let M denote the set of all pairs (x,e) E Ex{e G E;\\e\\ = 1} such that f(x, e) exists, f(x, e) > f(xo, eo), and

5 Differentiability and Measures in Banach Spaces 927 \(f(x + teo) -f(x)) - (/(xo + too) -/(*>))! < 6\tW(f f (x,e)-f'(xo,e 0 ))Liv(f) for every t G R. Then, if the norm is & differentiable at eo, and if \ims SlQ sup{f, (x,e);(x,e) G M and x-x 0 < ö} </'(xo,ßo), / is $ differentiable at xo. Now, the way of constructing the sequences (xk, e^) and Pk is more or less clear. We always pick up the next pair from the set M described in the previous Theorem. The additional requirement is only that /'(x/c+i^+i) is very close to the supremum of the directional derivatives f'(x,e) for (x,e) G M. Then we define the norm pk+i and continue our construction. Though we still have to be quite careful and make some technical estimates, since, for example, the set M from the previous Theorem depends upon the choice of the norm, this construction leads to a sequence satisfying all our requirements. 1.2 Problems From the previous discussion it is clear that the theory of differentiability still abounds with open problems. I would just like to point out the following two. Problem 7. Does every pair of Lipschitz functions on a separable Hilbert space have a common point of differentiability? Problem 8. For which finite Borei measures in separable Banach spaces is it true that every Lipschitz function is differentiable almost everywhere? The second problem is purely finite dimensional since such measures do not exist in infinite dimensional spaces. (See [12].) The answer is not known in the plane (or in any higher dimensional space). In the one dimensional case the required measures are precisely those that are absolutely continuous with respect to the Lebesgue measure. In spite of Theorem 4 I do not know any example that would show that this is not true in all finite dimensional spaces. 2. Measures The question whether measures on separable Banach spaces are determined by their values on balls has been around since R. O. Davies [4] published his beautiful example of two different probability measures on a compact metric space that agree on all balls. Together with J. Tiser [11] we recently answered it by proving: Theorem 9. Whenever two finite Borei measures in a separable Banach space agree on all balls, then they agree.

6 928 David Preiss To prove this statement, we first use the Fourier transform to reduce the problem to showing that the measures agree on all halfspaces. Then, by blowing up balls, we come to the situation when the halfspace contains a nonempty open cone C on every translate of which the measures agree. An approximation argument (or a differentiability result from [7]) reduces the problem further to the case when C n C is a subspace of finite codimension. Thus we can pass to the factor space and we have to solve the corresponding problem in finite dimensional spaces : Do we know the measure of a halfspace provided we know the measure of each translate of a nonempty open cone contained in it? Since this turned out to be true, our approach has been successful. Instead of giving further details, it might be more interesting to point out some examples. The motivation for them comes from the Besicovitch-Morse differentiability theorem, which is a much stronger statement than that measures in finite dimensional normed spaces are determined by their values on balls : For every (locally) finite Borei measure p in a finite dimensional Banach space and for every p integrable function / the limit lim * xx / f(u)dp(u) (1) JK) K) r^p(b(x,r))j B{Xir) ^' exists and equals f(x) for p alrnost every x. As a corollary of this statement one can prove that, if p and v are two finite Borei measures in a finite dimensional Banach space satisfying p(b) > v(b) for every ball B then p>v. Example 1 ([9]). There is a Gaussian measure y\ in fe and a yi integrable function / such that the limit in (1) is infinite uniformly for x G h, i.e., lim inf - / JK) ' r^o x zi 2 y 1 (B(x,r))J B{Xir) f(u) dyau) m = oo. ' Example 2 ([8]). There is a Gaussian measure 72 in h and a bounded y2 measurable function / such that, for p almost every x, the limit in (1) does not exist. Example 3 (J. Tiser). There is a non-degenerated Gaussian measure y3 in fe such that (1) holds for every / G L p (y 3 ), p > 1. Example 4. In a separable Hilbert space the statement "p(b) > v(b) for balls with radius less than one implies p > v" holds if and only if the dimension of the space is finite. Example 5. In a separable Hilbert space the statement "p(b) > v(b) for balls with radius greater than one implies p>v" holds if and only if the dimension of the space is infinite. Example 6. In the Zoo sum of a separable Hilbert space with the line there are measures p and v such that p ^ v but p(b) > v(b) for all balls.

7 Differentiability and Measures in Banach Spaces 929 However, in spite of the above result and examples, the investigation of the behaviour of measures on balls cannot be considered as finished. For example, the following question is still far from being answered. Problem 10. Are finite Borei measures in separable Banach spaces determined by their values on balls with radii less than one? References 1. Aronszajn, N.: Differentiability of Lipschitz functions in Banach spaces. Studia Math. 57 (1976) Choquet, G. : Applications des propriétés descriptives de la fonction contingent à la théorie des fonctions de variable réelle et à la géométrie différentielle des variétés cartésiennes. J. Math. Pures Appi. 26 (1947) Christensen, J. P. R. : Measure theoretic zero sets in infinitely dimensional spaces and application to differentiability of Lipschitz mappings. Actes du Deuxième Colloque danalyse Fonctionele de Bordeaux 2 (1973) Davies, R. O.: Measures not approximable or not specifiable by means of balls. Mathematika 18 (1971) Mankiewicz, P.: On the differentiability of Lipschitz mappings in Fréchet spaces. Studia Math. 45 (1973) Phelps, R. R. : Gaussian null sets and differentiability of Lipschitz maps on Banach spaces. Pacific J. Math. 77 (1978) Preiss, D. : Almost differentiability of convex functions on Banach spaces and determination of measures by their values on balls. In : Proc. Conf. Geometry of Banach Spaces (Strobl 1989) (to appear) 8. Preiss, D.: Gaussian measures and the density theorem. Comment. Math. Univ. Carolinae 22 (1981) Preiss, D. : Differentiation of measures in infinitely dimensional spaces. In : Proc. Conf. Topology and Measure III, pages , Greifswald, Preiss, D.: Differentiability of Lipschitz functions on Banach spaces. J. Functional Anal. 91 (1990) Preiss, D., Tiser, J.: Measures on Banach spaces are determined by their values on balls (to appear) 12. Preiss, D., Zajicek, L.: Fréchet differentiation of convex functions in a Banach space with a separable dual. Proc. Amer. Math. Soc. 91 (1984) Stegall, Ch.: The duality between Asplund spaces and spaces with the Radon- Nikodym property. Israel J. Math. 29 (1978) Stegall, Ch.: The Radon-Nikodym property in conjugate Banach spaces II. Trans. Amer. Math. Soc. 264 (1981) Zahorski, Z.: Sur l'ensemble des points de non-derivabilité d'une fonction continue. Bull. Soc. Math. France 74 (1946)

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