A note on measure-geometric Laplacians

作者：Kesseboehmer M; Samuel T^*; Weyer H

来源：Monatshefte fur Mathematik, 2016, 181(3): 643-655.

DOI：10.1007/s00605-016-0906-0

• Measure-geometric Laplacians
• Spectral asymptotics
• Singular measures
• Chebyshev polynomials
• Salem measures

We consider the measure-geometric Laplacians with respect to atomless compactly supported Borel probability measures as introduced by Freiberg and Zahle (Potential Anal. 16(1):265-277, 2002) and show that the harmonic calculus of can be deduced from the classical (weak) Laplacian. We explicitly calculate the eigenvalues and eigenfunctions of . 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 inhomogeneous self-similar Cantor measures and Salem measures.

• 出版日期2016-11