Harmonic Calculus on Fractals – a Measure Geometric Approach Ii

Riesz potentials of fractal measures μ in metric spaces and their inverses are introduced . They define self–adjoint operators in the Hilbert space L2(μ) and the former are shown to be compact. In the Euclidean case the corresponding spectral asymptotics are derived with Besov space methods. The inverses of the Riesz potentials are fractal pseudodifferential operators. For the order two operator the spectral dimension coincides with the Hausdorff dimension of the underlying fractal. Introduction In part I of the paper the Laplace operator ∆μ with respect to an atomless finite Borel measure μ on an interval (a, b) is introduced by