Quantitative $C^1$-estimates by Bismut formulae

Li-Juan Cheng; Anton Thalmaier; James Thompson

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arXiv · 1707.07121

Quantitative $C^1$-estimates by Bismut formulae

Li-Juan Cheng·Anton Thalmaier·James Thompson

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Abstract

For a $C^2$ function $u$ and an elliptic operator $L$, we prove a quantitative estimate for the derivative $du$ in terms of local bounds on $u$ and $Lu$. 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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Li-Juan Cheng, Anton Thalmaier, James Thompson. 2017-07-22. Quantitative $C^1$-estimates by Bismut formulae. https://arxiv.org/abs/1707.07121

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