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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 \(p \ge 1\).more » « less
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Abstract We study the Brownian motion on the non-compact Grassmann manifold $$\frac {\textbf {U}(n-k,k)} {\textbf {U}(n-k)\textbf {U}(k)}$$ and some of its functionals. The key point is to realize this Brownian motion as a matrix diffusion process, use matrix stochastic calculus and take advantage of the hyperbolic Stiefel fibration to study a functional that can be understood in that setting as a generalized stochastic area process. In particular, a connection to the generalized Maass Laplacian of the complex hyperbolic space is presented and applications to the study of Brownian windings in the Lie group $$\textbf {U}(n-k,k)$$ are then given.more » « less
