Measure theoretic Laplace operators on fractals (Q2702425)

The author defines the derivative \(f'\) with respect to a finite atomless Borel regular measure \(\mu\) on an interval \([a,b]\) by the fundamental theorem of calculus, that is, \(f(x) = \int_a^x f' d\mu\). The \(\mu\)-Laplacian is then given by the second \(\mu\)-derivative. The concept is extended to \(\mathbb R^n\) and corresponding measures with compact support and finite \((n-1)\)-energy \(\iint|x-y|^{-(n-1)} \mu(dx)\mu(dy)\). According to results of Mattila the measure \(\mu\) can be almost surely disintegrated into section measures on lines. On these lines the previous techniques are applied. Unfortunately, the self-adjointness of the resulting operators on compacts cannot be proved.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00049].