TY - JOUR AU - Baudoin, Fabrice AU - Chen, Li PY - 2022/10/06 Y2 - 2024/03/28 TI - Sobolev spaces and Poincaré inequalities on the Vicsek fractal JF - Annales Fennici Mathematici JA - Ann. Fenn. Math. VL - 48 IS - 1 SE - Articles DO - 10.54330/afm.122168 UR - https://afm.journal.fi/article/view/122168 SP - 3-26 AB - <pre>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 \(p&gt;1\). 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 \(p \ge 1\).</pre> ER -