Examples and counterexamples to a conjecture in the theory of differentiation of integrals
Let \(\phi_ 1,\phi_ 2,...,\phi_ n\) be n positive real functions of k variables, increasing separately on each variable and assuming arbitrarily small values. Consider the basis \({\mathcal B}\) of all intervals in \({\mathbb{R}}^ n\), whose side length in the ith direction is determined by the function \(\phi_ i\). It was conjectured that the differentiability properties of such a basis would only depend upon the number of parameters, degrees of freedom in some sense, involved. In particular, it should behave not worse than the basis of all intervals in \({\mathbb{R}}^ k\); that is, it should differentiate the Orlicz class \(L(\log^+L)^{k-1}\), independently of the choice of functions \(\phi_ 1,\phi_ 2,...,\phi_ n.\) In this paper we show that such a wild conjecture is false in general. In fact, we give a counterexample in the simplest possible case, when \(n=3\) and \(k=2\), since the cases \(n\leq 2\), for any k, and \(n\geq 3\), for \(k=1\) or \(k\geq n\), follow by general principles such as Vitali's covering lemma or the classical Jessen-Marcinkiewicz-Zygmund theorem. We also explore with some positive results, the possibility of some aspects of the conjecture remaining true. In all these examples we are forced to assume extra conditions on the \(\phi_ i's\).
scientific article; zbMATH DE number 3953496