ON A. ZYGMUND DIFFERENTIATION CONJECTURE

Consider v a Lipschitz unit vector field on R n and K its Lipschitz constant. We show that the maps Ss : Ss(X) = X + sv(X) are invertible for 0 ≤ |s| < 1/K and define nonsingular point transformations. We use these properties to prove first the differentiation in L p norm for 1 ≤ p < ∞. Then we show the existence of a universal set of values s ∈ (−1/2K, 1/2K) of measure 1/K for which the Lipschitz unit vector fields v ◦S −1 s satisfy Zygmund's conjecture for all functions in L p (R n ) and for each p, 1 ≤ p < ∞.