In the present paper we consider boundary value problems for elliptic pseudodifferential operators ($\Psi$DOs) in Besov and Bessel-potential spaces. The most part of the paper is devoted to $\Psi$DOs not possessing the transmission property. In particular we investigate a special case: the case of boundary value problems on two-dimensional manifolds.
Mathematics Subject Classification: 35S15, 35J55
Key words and phrases: Boundary value problems for elliptic pseudo-differential operators, pseudodifferential operators with and without transmission property, anisotropic elliptic pseudodifferential operators, boundary value problems on two-dimensional manifolds for improper elliptic operators, factorization of elliptic symbols, Besov and Bessel-potential spaces.