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

Abstract

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