  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 $a=1/2$).
