On Fréchet differentiability of Lipschitzian functions on spaces with gaussian measures

Abstract

We construct two counter-examples related to Frechet differentiability in infinite dimensions. The first one gives a convex Lipschitzian function on a Banach space such that its convolution with a given measure is Frechet differentiable only on a measure zero set. The second one gives a Borel function on a space with a Gaussian measure such that it is Lipschitzian along the Cameron–Martin subspace, but is Frechet differentiable along this subspace only on a measure zero set. This answers a long standing open question. AMS Subject Classification: 28C20, 49J50 The problem of Frechet differentiability of Lipschitzian functions has attracted a considerable attention in the last decades. One of the major achievements in this area is Preiss’s theorem [1] according to which every Lipschitzian function on a Hilbert space is Frechet differentiable on a dense set. However, this set may be small in many respects, in particular, it may have measure zero with respect to every nondegenerate Gaussian measure. The consideration of Gaussian measures on infinite dimensional spaces in relation with Frechet differentiability brings new problems that are specifically infinite dimensional. Every Radon Gaussian measure γ on a space X (which is a Banach space or, more generally, a locally convex space) possesses the so called Cameron–Martin space H (called also the reproducing kernel), which is a separable Hilbert space with some norm | · |H and is compactly embedded into X. If X is infinite dimensional, then H is much smaller than X, although it may be dense in X. For many reasons, it is natural to consider functions on X that are Frechet differentiable along H. A function f on X is called Frechet differentiable along H (or Frechet H-differentiable) at a point x ∈ X if there is a vector DHf(x) ∈ H such that f(x+ h)− f(x)− (DHf(x), h)H = o(h), h ∈ H, where lim |h|→0 |h|−1 H |o(h)|H = 0. It turns out that this weaker property is much more flexible and that many natural functions are Frechet differentiable along H not even being continuous on X. For example, the convolution