September 28, 2006
ON A. ZYGMUND DIFFERENTIATION CONJECTURE
Abstract. Consider v a Lipschitz unit vector field on Rn
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 Lp
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 vS-1
satisfy Zygmund's conjecture for all functions in Lp
) and for each p, 1 p < .
Lebesgue differentiation theorem states that given a function f L1(R) the averages