The halo problem in the theory of differentiation of integrals

Let there be given a Lorentz space and an Orlicz space with equal fundamental functions. We construct a differential basis that differentiates the integrals of functions belonging to the Lorentz space, but does not differentiate the integral of some function belonging to the Orlicz space. Such bases enable us to obtain a negative solution of the so-called halo problem for $p\in(1,\infty)$. Morillon [1], Russian p. 186, proved that this problem has a positive solution in the case when $p=1$.