Liminf sufficiency for pointwise-difference Sobolev characterizations

Determine whether finiteness of a liminf weak-type threshold functional, in place of the corresponding supremum condition, suffices for the derivative-free characterizations of Sobolev spaces based on pointwise differences developed by Brezis, Seeger, Van Schaftingen, and Yung.

Background

The paper contrasts its oscillation-based characterization, involving the function m_f on balls, with other derivative-free characterizations of Sobolev spaces based on pointwise differences. In most of those works, the relevant criterion uses a supremum over threshold levels rather than a liminf as the threshold tends to zero.

The authors note that the sufficiency of replacing such a supremum condition by a liminf condition was explicitly posed as an open problem in the work of Brezis, Seeger, Van Schaftingen, and Yung. The present paper resolves an analogous question for the ball-oscillation functional m_f, but does not claim to resolve the pointwise-difference formulation identified here.