On differentiation of integrals with respect to bases of convex sets.

Abstract

Differentiation of integrals of functions from the class $Lip(1,1)(I^2)$ with respect to the basis of convex sets is established. An estimate of the rate of differentiation is given. It is also shown that there exist functions in $Lip(1,1)(I^N)$, N ≥ 3, and $H^{ω}_{1}(I^2)$ with ω(δ)/δ → ∞ as δ → +0 whose integrals are not differentiated with respect to the bases of convex sets in the corresponding dimension