#### Quantitative $C$1-estimates by Bismut formulae by Li-Juan Cheng, Anton Thalmaier and James Thompson

Abstract
For a $C2$ function $u$ and an elliptic operator $L$, we prove a quantitative estimate for the derivative $du$ in terms of local bounds on $u$ and $L$. An integral version of this estimate is then used to derive a condition for the zero-mean value property of $\Delta u$. An extension to differential forms is also given. Our approach is probabilistic and could easily be adapted to other settings.

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