Harmonic calculus on fractals -- a measure geometric approach. I
Differentiation, \(\frac{df}{d\mu}\), of functions \(f\) w.r.t. finite atomless measures \(\mu\) with compact support on the real line is considered. The derivative is defined via the fundamental theorem of calculus, where the Lebesgue measure is replaced by \(\mu\). The ``Laplacian'' then is \(\frac{d^2f}{d\mu^2}\). Its spectral asymptotics in \(L^2([0,1], \mu)\) has Weyl's classical form when the measure is supported by a Cantor set with arbitrary weights. The main technical point is a scaling property of the resolvent operator.
scientific article; zbMATH DE number 1742919