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 $(-\Delta {)}^a$ is that $L$ is nonsymmetric, having both an even and an odd part. Hence it satisfies a $\mu$-transmission condition where generally $\mu \ne a$. We present a complex method, relying on a factorization in factors holomorphic in ${\xi }_n$ 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 $\mu$ in a larger range than they consider. Another new contribution is the (model) study of "large" solutions of nonhomogeneous Dirichlet problems when $\mu >0$. Here we deduce a "halfways Green's formula" for $L$: $${\int }_{R_{+}^n}Lu\bar{v}dx-{\int }_{R_{+}^n}u\overline{L^{\ast }v}dx=c{\int }_{R^{n-1}}{\gamma }_0(u/x_n^{\mu -1}){\gamma }_0(\bar{v}/x_n^{{\mu }^{\ast }})d{x}^{\prime },$$ when $u$ solves a nonhomogeneous Dirichlet problem for $L$, and $v$ solves a homogeneous Dirichlet problem for $L^{\ast }$; ${\mu }^{\ast }=2a-\mu$. 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.

The results in this paper are valid for operators satisfying the mu-transmission condition. It was overlooked that the main example L does not satisfy that condition in all cases, but only a principal mu-transmission condition. The missing cases are now treated in the subsequent paper arXiv:2104.05581

Cited by in corpus (2)

• The principal transmission condition
• Analytic functional calculus and Gårding inequality on graded Lie groups with applications to diffusion equations