Lipschitz continuity of Cheeger-harmonic functions in metric measure spaces

Nonsmooth calculus

“…References include [164], [165], [107], [108], [24], [25], [117], [110]. Note that these studies contain as special cases potential theories on Carnot-Carathéodory spaces or on Euclidean spaces with weights (on which a large older literature exists).…”

A remark on Poincaré inequalities on metric measure spaces

“…For now we note that, roughly speaking, the above theorem claims that a Poincaré inequality holds for all functions on a metric measure space, with the gradient replaced by an infinitesimal measurement of oscillation, if a discretized version of the Poincaré inequality holds for all functions and at all scales, with bounds independent of the scale of the discretization. The former condition has been widely studied and is rich in application; see [1], [2], [3], [4], [8]. The latter condition is often easier to verify, especially when the metric measure space is studied using discrete approximations.…”

Differentiability of p-Harmonic Functions on Metric Measure Spaces

“…A new technique is therefore needed. We know of two such approaches: Koskela, Rajala, and Shanmugalingam [KRS03] proved that if the space supports a 2-Poincaré inequality and a certain heat kernel estimate, then 2-harmonic functions are locally Lipschitz continuous. Petrunin [Pet03] proved that 2-harmonic functions on Alexandrov spaces are also locally Lipschitz continuous.…”

“…An example provided in [KRS03,p. 150] shows a space that supports a 2-Poincaré inequality and a 2-harmonic function which fails to be Lipschitz continuous at one point.…”