We prove a complex and a real interpolation theorems on Besov spaces and Triebel-Lizorkin spaces associated with a selfadjoint operator L, without assuming the gradient estimate for its spectral kernel. The result applies to the cases where L is a uniformly elliptic operator or a Schrödinger operator with electromagnetic potential.

Our aim in this paper is to define principal and characteristic directions at points on a smooth 2-dimensional surface in the Euclidean space R in such a way that their equations together with that of the asymptotic directions behave in the same way as the triple formed by their counterpart on smooth surfaces in the Euclidean space R. The definitions we propose are derived from a more general a...