On the comparison of translation invariant convex differentiation bases

Abstract

Abstract It is known that if B and B ′ {B^{\prime}} are translation invariant convex density differentiation bases and the maximal operators associated to them locally majorize each other, then B and B ′ {B^{\prime}} differentiate the integrals of the same class of non-negative functions. We show that under the same conditions it is not possible to assert more about similarity of the differential properties of B and B ′ {B^{\prime}} in view of their positive equivalence.