Elliptic Harnack inequality and Poincar\'e inequality for $p$-energies on metric measure spaces

Meng Yang

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arXiv · 2606.16405

Elliptic Harnack inequality and Poincar\'e inequality for $p$-energies on metric measure spaces

Meng Yang

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Abstract

For $p>1$ and a $p$-energy on a volume doubling metric measure space, we prove the Poincar\'e inequality under the elliptic Harnack inequality, two-sided capacity bounds, and additional geometric and analytic assumptions. Combining this with our previous results, we obtain, under the same structural assumptions, the equivalence between the elliptic Harnack inequality together with two-sided capacity bounds and the conjunction of the Poincar\'e inequality and the cutoff Sobolev inequality.

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Meng Yang. 2026-06-15. Elliptic Harnack inequality and Poincar\'e inequality for $p$-energies on metric measure spaces. https://arxiv.org/abs/2606.16405

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