On the finiteness of strong maximal functions associated to functions whose integrals are strongly differentiable

Besicovitch proved that if f is an integrable function on R2 whose associated strong maximal function MSf is finite a.e., then the integral off is strongly differentiable. On the other hand, Papoulis proved the existence of an integrable function on R2 (taking on both positive and negative values) whose integral is strongly differentiable but whose associated strong maximal function is infinite on a set of positive measure. In this paper, we prove that if n >= 2 and if f is a measurable nonnegative function on Rn whose integral is strongly differentiable and moreover such that f (1 +log+ f )n-2 is integrable, then MSf is finite a.e. We also show this result is sharp by proving that, if phi is a continuous increasing function on [0, infinity) such that phi(0) = 0 and with phi(u) = o(u(1 + log+ u)n-2) (u -> infinity), then there exists a nonnegative measurable function f on Rn such that phi(f) is integrable on Rn and the integral off is strongly differentiable, although MSf is infinite almost everywhere.(c) 2023 Elsevier Inc. All rights reserved.

Automated summary

Besicovitch proved that if f is an integrable function on R2 with finite strong maximal function MSf a.e., then the integral off is strongly differentiable. On the other hand, Papoulis proved the existence of an integrable function on R2 whose integral is strongly differentiable but with infinite associated strong maximal function on a set of positive measure. In this paper, we prove that if n >= 2 and if f is a measurable nonnegative function on Rn with strongly differentiable integral and f(1 + log+ f)n-2 integrable, then MSf is finite a.e. We also show that this result is sharp by constructing a function f satisfying certain conditions.