Riesz Potentials and Liouville Operators on Fractals

An analogue to the theory of Riesz potentials and Liouville operators in R for arbitrary fractal d-sets is developed. Corresponding function spaces agree with traces of euclidean Besov spaces on fractals. By means of associated quadratic forms we construct strongly continuous semigroups with Liouville operators as infinitesimal generator. The case of Dirichlet forms is discussed separately. As an example of related pseudodi#erential equations the fractional heat-type equation is solved. Mathematics Subject Classification. Primary 28A80, Secondary 47B07, 35P20 Keywords. fractal d-set, Riesz potential, pseudodi#erential operator, fractal Besov space, Dirichlet form