Strong differentiability of Lipschitz functions

Abstract:

Let F be a differentiation basis in ${R^n}$, i.e., a family of measurable sets S contracting to 0 such that ${\left \| {{M_F}f} \right \|_p} \leqslant {A_p}{\left \| f \right \|_p}$, where ${M_F}$ is the Hardy-Littlewood maximal operator. For $f \in \Lambda _\alpha ^{pq}$, we let ${E_F}(f)$ be the complement of the Lebesgue set of f relative to F, and we show that ${E_F}$ has $L_\alpha ^{pq}$-capacity 0, where $L_\alpha ^{pq}$ is a capacity associated with $\Lambda _\alpha ^{pq}$ in much the same way as the Bessel capacity ${B_{\alpha p}}$ is associated with $L_\alpha ^p$.