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

Abstract

Besicovitch proved that if f is an integrable function on R2 whose associated strong maximal function MSf is finite a.e., then the integral of f 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 φ is a continuous increasing function on [0,∞) such that φ(0)=0 and with φ(u)=o(u(1+log+⁡u)n−2)(u→∞), then there exists a nonnegative measurable function f on Rn such that φ(f) is integrable on Rn and the integral of f is strongly differentiable, although MSf is infinite almost everywhere.