A survey of regularity estimates for the Laplacian and p-Laplacian on metric measure spaces, with a proof overview of the author's second-order regularity theorem in bounded RCD spaces.
Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds
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abstract
We prove that a metric measure space $(X,d,m)$ satisfying finite dimensional lower Ricci curvature bounds and whose Sobolev space $W^{1,2}$ is Hilbert is rectifiable. That is, a $RCD^*(K,N)$-space is rectifiable, and in particular for $m$-a.e. point the tangent cone is unique and euclidean of dimension at most $N$. The proof is based on a maximal function argument combined with an original Almost Splitting Theorem via estimates on the gradient of the excess. To this aim we also show a sharp integral Abresh-Gromoll type inequality on the excess function and an Abresh-Gromoll-type inequality on the gradient of the excess. The argument is new even in the smooth setting.
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An overview of regularity results for the Laplacian and $p$-Laplacian in metric spaces
A survey of regularity estimates for the Laplacian and p-Laplacian on metric measure spaces, with a proof overview of the author's second-order regularity theorem in bounded RCD spaces.