Integrability of weak mixed first-order derivatives and convergence rates of scrambled digital nets

We consider the $L^p$ integrability of weak mixed first-order derivatives of the integrand and study convergence rates of scrambled digital nets. We show that the generalized Vitali variation with parameter $\alpha \in [\frac{1}{2}, 1]$ from [Dick and Pillichshammer, 2010] is bounded above by the $L^p$ norm of the weak mixed first-order derivative, where $p = \frac{2}{3-2\alpha}$. Consequently, when the weak mixed first-order derivative belongs to $L^p$ for $1 \leq p \leq 2$, the variance of the scrambled digital nets estimator convergences at a rate of $\mathcal{O}(N^{-4+\frac{2}{p}} \log^{s-1} N)$. Numerical experiments further validate the theoretical results.