A classical pseudodifferential operator \(P\) on \(R^n\) satisfies the \(\mu\)-transmission condition relative to a smooth open subset \(\Omega \), when the symbol terms have a certain twisted parity on the normal to \(\partial\Omega \). As shown recently by the author, the condition assures solvability of Dirichlet-type boundary problems for elliptic \(P\) in full scales of Sobolev spaces with a singularity \(d^{\mu -k}\), \(d(x)=\operatorname{dist}(x,\partial\Omega)\). Examples include fractional Laplacians \((-\Delta)^a\) and complex powers of strongly elliptic PDE. We now introduce new boundary conditions, of Neumann type or more general nonlocal. It is also shown how problems with data on \(R^n\setminus \Omega \) reduce to problems supported on \(\bar\Omega\), and how the so-called "large" solutions arise. Moreover, the results are extended to general function spaces \(F^s_{p,q}\) and \(B^s_{p,q}\), including H\"older-Zygmund spaces \(B^s_{\infty ,\infty}\). This leads to optimal H\"older estimates, e.g. for Dirichlet solutions of \((-\Delta)^au=f\in L_\infty (\Omega)\), \(u\in d^aC^a(\bar\Omega)\) when \(0<a<1\), \(a\ne 1/2\) (in \(d^aC^{a-\epsilon}(\bar\Omega)\) when \(a=1/2\)).