Gaussian upper heat kernel bounds and Sobolev inequalities on graphs with unbounded geometry

07.06.2023, 13:00  –  Haus 9, Raum 0.17 und Zoom
Forschungsseminar Diskrete Spektraltheorie

Dr. Christian Rose (Potsdam)

Abstract: On Riemannian manifolds the conjunction of Gaussian upper heat kernel bounds and the volume doubling property of balls are equivalent to Sobolev inequalities in arbitrarily small balls. The small-time behaviours of heat kernels of manifolds and graphs differ significantly and results in a big difference of the Gaussian behaviour of the heat kernels at time zero. Hence, a characterization like the above cannot hold for arbitrarily small times and balls. In contrast, the long-time behaviours of the heat kernels are similar. In this talk I will present a new characterization of Gaussian upper bounds for large times in terms of Sobolev inequalities and volume doubling properties on large balls.
This is an ongoing joint work with Matthias Keller.