A SURVEY ON MEASURE GEOMETRIC LAPLACIANS ON CANTOR LIKE SETS

Generalized second order differential operators of the form ddµ d dν are considered, where µ and ν are finite atomless Borel measures with compact supports on the real line. If the Hausdorff dimension of supp µ is less than one, such an operator allows an interpretation as a measure geometric Laplacian for this fractal, which has similar analytical properties to the Euclidean Lapla-cian. In the special case of self-similar measures — Hausdorff measures or, more general, self-similar measures with arbitrary weights —, spectral asymptotics are presented. We apply these results to the special case where ν is the Lebesgue measure, i.e. to the well-known Sturm–Liouville operator ddµ d dx. Key Words: differential operators; Hausdorff dimension; measure geometric Laplacian; fractal; Sturm–Liouville operator