Let $(X,d,mu)$ be a complete metric measure space, with $mu$ a locallydoubling measure, that supports a local weak $L^2$-Poincar'e inequality. Byassuming a heat semigroup type curvature condition, we prove thatCheeger-harmonic functions are Lipschitz continuous on $(X,d,mu)$. Gradientestimates for Cheeger-harmonic functions and solutions to a class of non-linearPoisson type equations are presented.
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机译：令$（X，d， mu）$为一个完整的度量度量空间，其中$ mu $为局部加倍度量，它支持局部弱的$ L ^ 2 $ -Poincar 'e不等式。通过假设热半群型曲率条件，我们证明了Cheeger调和函数在$（X，d， mu）$上是Lipschitz连续的。给出了Cheeger调和函数的梯度估计和一类非线性Poisson型方程的解。
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