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Exact Green's formula for the fractional Laplacian and perturbations

Let $Ω$ be an open, smooth, bounded subset of $ \Bbb R ^n$. In connection with the fractional Laplacian $(-Δ)^a$ ($a>0$), and more generally for a $2a$-order classical pseudodifferential operator ($ψ$do) $P$ with even symbol, one can define the Dirichlet value $γ_0^{a-1}u$ resp. Neumann value $γ_1^{a-1}u$ of $u(x)$ as the trace resp. normal derivative of $u/d^{a-1}$ on $\partialΩ$, where $d(x)$ is the distance from $x\inΩ$ to $\partialΩ$; they define well-posed boundary value problems for $P$. A Green's formula was shown in a preceding paper, containing a generally nonlocal term $(Bγ_0^{a-1}u,γ_0^{a-1}v)_{\partialΩ}$, where $B$ is a first-order $ψ$do on $\partialΩ$. Presently, we determine $B$ from $L$ in the case $P=L^a$, where $L$ is a strongly elliptic second-order differential operator. A particular result is that $B=0$ when $L=-Δ$, and that $B$ is multiplication by a function (is local) when $L$ equals $-Δ$ plus a first-order term. In cases of more general $L$, $B$ can be nonlocal.