In this talk, we discuss functional inequalities and gradient bounds for the heat kernel on the Vicsek set. The Vicsek set has both fractal and tree structure, whereas neither analogue of curvature nor obvious differential structure exists. We introduce Sobolev spaces in that setting and prove several characterizations based on a metric, a discretization or a weak gradient approach. We also obtain $L^{p}$ Poincaré inequalities and pointwise gradient bounds for the heat kernel. These properties have important applications in harmonic analysis like Sobolev inequalities and the Riesz transform. Moreover, several of our techniques and results apply to more general fractals and trees.[-]

In this talk I will present the approach that using time evolution of Husimi measure of the N particle wave function to get the convergence of Schrödinger to Vlasov equation in the mean field and semiclassical regime. By a reformulation of the many particle Schrödinger equation, one can get a Vlasov ‘like' kinetic equation for Husimi measure. Then the convergence will be obtained by doing appropriate error estimates in comparing these two dynamics. In this first stage result, the estimates have been obtained for regular solutions. This is a joint work with Jinyeop Lee and Matthew Liew.[-]