Integration by parts for nonsymmetric fractional-order operators on a halfspace

Abstract For a strongly elliptic pseudodifferential operator L of order 2a ( 0 a 1 ) with real kernel, we show an integration-by-parts formula for solutions of the homogeneous Dirichlet problem, in the model case where the operator is x-independent with homogeneous symbol, considered on the halfspace R + n . The new aspect compared to ( − Δ ) a is that L is nonsymmetric, having both an even and an odd part. Hence it satisfies a μ-transmission condition where generally μ ≠ a . We present a complex method, relying on a factorization in factors holomorphic in ξ n in the lower or upper complex halfplane, using order-reducing operators combined with a decomposition principle originating from Wiener and Hopf. This is in contrast to a real, computational method presented very recently by Dipierro, Ros-Oton, Serra and Valdinoci. Our method allows μ in a larger range than they consider. Another new contribution is the (model) study of “large” solutions of nonhomogeneous Dirichlet problems when μ > 0 . Here we deduce a “halfways Green's formula” for L: ∫ R + n L u v ¯ d x − ∫ R + n u L ⁎ v ‾ d x = c ∫ R n − 1 γ 0 ( u / x n μ − 1 ) γ 0 ( v ¯ / x n μ ⁎ ) d x ′ , when u solves a nonhomogeneous Dirichlet problem for L, and v solves a homogeneous Dirichlet problem for L ⁎ ; μ ⁎ = 2 a − μ . Finally, we show a full Green's formula, when both u and v solve nonhomogeneous Dirichlet problems; here both Dirichlet and Neumann traces of u and v enter, as well as a first-order pseudodifferential operator over the boundary.