On a new Kato class and positive solutions of Dirichlet problems for the fractional Laplacian in bounded domains

The purpose of this paper is to extend some results of the potential theory of an elliptic operator to the fractional Laplacian (−Δ)^α/2, 0<α<2, in a bounded C^1,1 domain D in Rⁿ. In particular, we introduce a new Kato class K_α(D) and we exploit the properties of this class to study the existence of positive solutions of some Dirichlet problems for the fractional Laplacian.