The logarithmic Dirichlet Laplacian on Ahlfors regular spaces

TL;DR

This paper introduces a logarithmic analogue of the Laplace-Beltrami operator on Ahlfors regular spaces, revealing spectral and functional properties similar to classical elliptic operators on manifolds.

Contribution

It defines a new operator with spectral properties akin to elliptic pseudo-differential operators, extending analysis to Ahlfors regular metric-measure spaces.

Findings

The heat semigroup of the operator is trace-class after a critical time.

The domain forms a Banach module over Dini continuous functions.

Hölder continuous functions are smooth vectors for the operator.

Abstract

We introduce the logarithmic analogue of the Laplace-Beltrami operator on Ahlfors regular metric-measure spaces. This operator is intrinsically defined with spectral properties analogous to those of elliptic pseudo-differential operators on Riemannian manifolds. Specifically, its heat semigroup consists of compact operators which are trace-class after some critical point in time. Moreover, its domain is a Banach module over the Dini continuous functions and every H\"older continuous function is a smooth vector. Finally, the operator is compatible, in the sense of noncommutative geometry, with the action of a large class of non-isometric homeomorphisms.