A note on measure-geometric Laplacians

TL;DR

This paper explores measure-geometric Laplacians associated with atomless probability measures, demonstrating their spectral properties, including explicit eigenfunctions and eigenvalues, with applications to fractal measures like Salem and Cantor measures.

Contribution

It provides a detailed analysis of the spectral theory of measure-geometric Laplacians, linking them to classical Laplacians and explicitly computing their eigenfunctions and eigenvalues.

Findings

Eigenvalues and eigenfunctions of $bla^{}$ are explicitly calculated.

Existence of measure-geometric Laplacians with Chebyshev polynomial eigenfunctions.

Applications to fractal measures such as Salem and Cantor measures.

Abstract

We consider the measure-geometric Laplacians $\Delta^{\mu}$ with respect to atomless compactly supported Borel probability measures $\mu$ as introduced by Freiberg and Z\"ahle in 2002 and show that the harmonic calculus of $\Delta^{\mu}$ can be deduced from the classical (weak) Laplacian. We explicitly calculate the eigenvalues and eigenfunctions of $\Delta^{\mu}$. Further, it is shown that there exists a measure-geometric Laplacian whose eigenfunctions are the Chebyshev polynomials and illustrate our results through specific examples of fractal measures, namely Salem and inhomogeneous self-similar Cantor measures.