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arXiv · 2207.02949

Sobolev spaces and Poincar\'e inequalities on the Vicsek fractal

Fabrice Baudoin·Li Chen
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Abstract

In this paper we prove that several natural approaches to Sobolev spaces coincide on the Vicsek fractal. More precisely, we show that the metric approach of Korevaar-Schoen, the approach by limit of discrete $p$-energies and the approach by limit of Sobolev spaces on cable systems all yield the same functional space with equivalent norms for $p>1$. As a consequence we prove that the Sobolev spaces form a real interpolation scale. We also obtain $L^p$-Poincar\'e inequalities for all values of $p \ge 1$.

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Fabrice Baudoin, Li Chen. 2022-07-06. Sobolev spaces and Poincar\'e inequalities on the Vicsek fractal. https://arxiv.org/abs/2207.02949

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