On A. Zygmund differentiation conjecture

Abstract

Consider $v$ a Lipschitz unit vector field on $R^n$ and $K$ its Lipschitz constant. We show that the maps $S_s:S_s(X)=X+sv(X)$ are invertible for $0\le |s|<1/K$ and define nonsingular point transformations. We use these properties to prove first the differentiation in L^p norm for $1\le p<\infty .$ Then we show the existence of a universal set of values $s\in [-1/2K,1/2K]$ of measure 1/K for which the Lipschitz unit vector fields $v\circ S_s^{-1}$ satisfy Zygmund's conjecture for all functions in $L^p({\mathbb{R}}^n)$ and for each p, $1\le p<\infty .$

Preliminary version