Michele Villa: Quantitative affine approximation on uniformly rectifiable sets

A basic fact of Lipschitz functions is that they are differentiable almost everywhere. This is Rademacher's theorem. It says a lot about the asymptotic behaviour of a Lipschitz function f at small scales (where they look affine) but not much at any definite scale. How long do we need to wait for the smoothness of f to kick in and make it look like an affine map? Dorronsoro's theorem is a quantification of Rademacher's which tells us: not long - and indeed, a Lipschitz function looks approximately affine at most scales (in some precise sense). Rademacher's theorem holds for functions on rectifiable sets by a theorem of Federer. In this talk, I will discuss (quantitative) affine approximation results for Lipschitz and Sobolev functions on (quantitatively) rectifiable sets.
Based on a joint work with Jonas Azzam and Mihalis Mougoglou.