# Jacob Schach Møller: Weighted estimates for the metric Laplacian on the cubic lattice

**Time: **
Thu 2019-02-28 15.30 - 16.30

**Lecturer: **
Jacob Schach Møller, Aarhus University

**Location: ** Seminar Hall Kuskvillan, Institut Mittag-Leffler

Abstract: In this talk we will study the free metric Laplacian with Kirchoff condition at the vertices of the cubic lattice in dimension d>= 3.

The free metric Laplacian has embedded eigenvalues of infinite multiplicity at energies \((\pi j)^2 , j=1,2,3,...\)

We study bounds of the free Laplacian sandwiched by weights that are locally \(L^4\) and \(L^q\) at infinity with q>=2 (but close to 2). The bounds are uniform up to the spectrum, except for the aforementioned embedded eigenvalues. The limiting resolvents are Holder continuous, and in dimension d>4 they are even Lipschitz. The analysis relies on a certain embedding estimate, a Privalov lemma, and an earlier study of corresponding weighted estimates for the discrete Laplacian on the cubic lattice.

The talk is based on joint work with E. Korotyaev and M. G. Rasmussen.