Sobolev spaces and Poincaré inequalities on the Vicsek fractal
Keywords:Vicsek set, Sobolev spaces, Poincaré inequalities, p-energies, real interpolation
AbstractIn 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 \(L^p\)-Poincaré inequalities for all values of \(p \ge 1\).
How to Cite
Baudoin, F., & Chen, L. (2022). Sobolev spaces and Poincaré inequalities on the Vicsek fractal. Annales Fennici Mathematici, 48(1), 3–26. https://doi.org/10.54330/afm.122168
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