Admissible Differential Bases in Banach Spaces

In the present work, a new method for constructing differential bases is presented. The bases constructed by this method allow us to distinguish symmetric spaces with different behaviour of fundamental functions at zero, as in the case of Hayes and Stokolos, and even allow us to distinguish Lorentz and Marzinkiewicz spaces or Lebesgue and Marzinkiewicz spaces whose fundamental functions are the same. 1. Introduction We recall certain definitions connected with differential bases [1]. By a differential basis at a point is meant a family of bounded measurable sets with positive measure which contain and are such that there is at least one sequence satisfying the condition . A union of such families is called a differential basis in . We note that the bulk of the problems and assertions in the theory of differentiation of integrals in consists of testing the validity almost everywhere of some fact. Therefore, it is natural to consider differential bases which are defined, not for all points in , but only almost everywhere. In what follows, we will make use of these observations. The classical examples of differential bases are the bases in , usually denoted by ？？ and consisting of all rectangular parallelepipeds of the form which satisfy the condition for and , . A basis made up of open sets is called a Busemann-Feller basis (BF-basis) if it follows from the conditions and such that . The significance of the introduction of -bases lies in the fact that questions arising in the theory of differentiation with respect to bases can be easily resolved for bases. We now define the upper and lower derivatives of the integral of a locally integrable function at a point with respect to a basis by means of the identities We note that the upper and lower derivatives are variants of the functionals and which are the subject of investigation in several sections of the chapter 1 in [2]. Following [1], we say that a basis ？？differentiates the integral of if the identities hold almost everywhere. If differentiates the integral of any function in the space , then we say that the basis ？？differentiates the space . If the basis differentiates , then it is said to be a density basis [1]. One of the fundamental problems of the theory of differentiation of integrals has the following form: given two function spaces , which are different in some sense, is it possible to distinguish these two spaces with the help of differential bases? In other words, does there exist a differential basis, which differentiates all integrals obtained from functions in , but a function can be found