A note on measure-geometric Laplacians

We consider the measure-geometric Laplacians ${\mathrm{\Delta }}^{\mu }$ 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 ${\mathrm{\Delta }}^{\mu }$ can be deduced from the classical (weak) Laplacian. We explicitly calculate the eigenvalues and eigenfunctions of ${\mathrm{\Delta }}^{\mu }$. 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.