A classical pseudodifferential operator P on Rn satisfies the
μ-transmission condition relative to a smooth open subset Ω, when
the symbol terms have a certain twisted parity on the normal to ∂Ω. 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 dμ−k,
d(x)=dist(x,∂Ω). Examples include fractional
Laplacians (−Δ)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 Rn∖Ω
reduce to problems supported on Ωˉ, and how the so-called "large"
solutions arise. Moreover, the results are extended to general function spaces
Fp,qs and Bp,qs, including H\"older-Zygmund spaces B∞,∞s. This leads to optimal H\"older estimates, e.g. for Dirichlet
solutions of (−Δ)au=f∈L∞(Ω), u∈daCa(Ωˉ)
when 0<a<1, a=1/2 (in daCa−ϵ(Ωˉ) when a=1/2).Comment: Title slightly changed, 34 page