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 $L^p$-Poincaré inequalities for all values of $p \ge 1$.
Keywords:
Vicsek set, Sobolev spaces, Poincaré inequalities, p-energies, real interpolation
Affiliations des auteurs :
Fabrice Baudoin
1
;
Li Chen
2
1
University of Connecticut, Department of Mathematics
2
Louisiana State University, Department of Mathematics
Fabrice Baudoin; Li Chen. Sobolev spaces and Poincaré inequalities on the Vicsek fractal. Annales Fennici Mathematici, Tome 48 (2023) no. 1, pp. 3-26. doi: 10.54330/afm.122168
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