Fundamental Solutions of the Logarithmic Laplacian: An Approach via the Division Problem

TL;DR

This paper offers a new method to construct fundamental solutions for the logarithmic Laplacian in higher dimensions, using a division problem approach inspired by classical theories, and explores related Liouville theorems and conjectures.

Contribution

It introduces an alternative approach to fundamental solutions of the logarithmic Laplacian, extending classical division problem techniques and clarifying solution behaviors in lower dimensions.

Findings

Established existence of fundamental solutions in dimensions d ≥ 3

Provided a variant of the Liouville theorem for the logarithmic Laplacian

Clarified conjecture on solution behavior in dimensions 1 and 2

Abstract

Existence of the fundamental solution of the logarithmic Laplacian (in dimensions $d \geq 3$) was established by Huyuan Chen and Laurent V\'eron (2024). In this note, we present an alternative approach, based on a modification on the classical division problem. This is inspired by the theory of fundamental solutions by Malgrange and Ehrenpreis. Moreover, we give a variant of the Liouville theorem for the logarithmic Laplacian and give some further clarification regarding a conjecture posed by Chen and V\'eron regarding the behavior of solutions in dimensions 1 and 2.