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

## TL;DR

This paper establishes a correspondence between certain non-local operators with completely monotone kernels and elliptic operators, extending previous symmetric operator results to a broader class using spectral theory.

## Contribution

It introduces a bijective link between non-symmetric Lévy operators with completely monotone kernels and specific elliptic operators, broadening the scope of Dirichlet-to-Neumann map identifications.

## Key findings

- Identified a class of non-local operators with completely monotone kernels.
- Established a bijective correspondence with elliptic operators.
- Extended previous symmetric operator results to non-symmetric cases.

## 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 L\'evy 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.