Harmonic extension technique for non-symmetric operators with completely monotone kernels

Abstract

We identify a class of non-local integro-differential operators K in mathbb {R} with Dirichlet-to-Neumann maps in the half-plane mathbb {R}times (0, infty ) for appropriate elliptic operators L. More precisely, we prove a bijective correspondence between Lévy operators K with non-local kernels of the form nu (y - x), where nu (x) and nu (-x) are completely monotone functions on (0, infty ), and elliptic operators L= a(y) partial _{xx} + 2 b(y) partial _{x y} + partial _{yy}. This extends a number of previous results in the area, where symmetric operators have been studied: the classical identification of the Dirichlet-to-Neumann operator for the Laplace operator in mathbb {R}times (0, infty ) with -sqrt{-partial _{xx}}, the square root of one-dimensional Laplace operator; the Caffarelli–Silvestre identification of the Dirichlet-to-Neumann operator for nabla cdot (y^{1 - alpha } nabla ) with (-partial _{xx})^{alpha /2} for alpha in (0, 2); and the identification of Dirichlet-to-Neumann maps for operators a(y) partial _{xx} + partial _{yy} with complete Bernstein functions of -partial _{xx} due to Mucha and the author. Our results rely on recent extension of Krein’s spectral theory of strings by Eckhardt and Kostenko.