Local and nonlocal boundary conditions for μ-transmission and fractional elliptic pseudodifferential operators

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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 Ω, and how the so-called "large" solutions arise. Moreover, the results are extended to general function spaces Fˢₚ,q and Bˢₚ,q, including Hölder-Zygmund spaces Bˢ∞,∞. This leads to optimal Hölder estimates, e.g. for Dirichlet solutions of (-Δ)ᵃu=f∈ L∞ (Ω), u∈ dᵃCᵃ(Ω) when 0<a<1, a≠ 1/2 (in dᵃCᵃ⁻ε(Ω) when a=1/2).