On the differentiability of Lipschitz functions with respect to measures in the Euclidean space
نویسندگان
چکیده
Rademacher theorem states that every Lipschitz function on the Euclidean space is differentiable almost everywhere, where “almost everywhere” refers to the Lebesgue measure. Our main result is an extension of this theorem where the Lebesgue measure is replaced by an arbitrary measure μ. In particular we show that the differentiability properties of Lipschitz functions at μ-almost every point are related to the decompositions of μ in terms of rectifiable one-dimensional measures. In the process we obtain a differentiability result for Lipschitz functions with respect to (measures associated to) k-dimensional normal currents, which we use to extend certain formulas involving normal currents and maps of class C to Lipschitz maps.
منابع مشابه
Quasicompact and Riesz unital endomorphisms of real Lipschitz algebras of complex-valued functions
We first show that a bounded linear operator $ T $ on a real Banach space $ E $ is quasicompact (Riesz, respectively) if and only if $T': E_{mathbb{C}}longrightarrow E_{mathbb{C}}$ is quasicompact (Riesz, respectively), where the complex Banach space $E_{mathbb{C}}$ is a suitable complexification of $E$ and $T'$ is the complex linear operator on $E_{mathbb{C}}$ associated with $T$. Next, we pr...
متن کاملDifferentiability of Cone-monotone Functions on Separable Banach Space
Motivated by applications to (directionally) Lipschitz functions, we provide a general result on the almost everywhere Gâteaux differentiability of real-valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone with nonempty interior. This seemingly arduous restriction is useful, since it covers the case of directionally Lips...
متن کاملOn Fréchet differentiability of Lipschitz maps between Banach spaces
A well-known open question is whether every countable collection of Lipschitz functions on a Banach space X with separable dual has a common point of Fréchet differentiability. We show that the answer is positive for some infinite-dimensional X. Previously, even for collections consisting of two functions this has been known for finite-dimensional X only (although for one function the answer is...
متن کاملWeak differentiability of solutions to SDEs with semi-monotone drifts
In this work we prove Malliavin differentiability for the solution to an SDE with locally Lipschitz and semi-monotone drift. To prove this formula, we construct a sequence of SDEs with globally Lipschitz drifts and show that the $p$-moments of their Malliavin derivatives are uniformly bounded.
متن کامل