Fine differentiability and the Hardy–Littlewood maximal function

For an extended real-valued function w defined on Rⁿ, we introduce a relaxed version of the pointwise infinitesimal Lipschitz number Lipw, by using the 1-fine topology. We show that a Stepanoff-type theorem holds, namely that almost everywhere in the set where Lipw ᶠⁱⁿᵉ<∞, the function w is 1-finely differentiable. We then show that the Hardy-Littlewood maximal function of a function of bounded variation is 1-finely differentiable almost everywhere.