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

2025/06/25 by Lee, David
#42B15 #42B37 #46F10 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.20121

Abstract

Existence of the fundamental solution of the logarithmic Laplacian (in dimensions d ≥ 3) was established by Huyuan Chen and Laurent Véron (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 Véron regarding the behavior of solutions in dimensions 1 and 2.

Citations

• Convergence rates of the fractional to the local Dirichlet problem
• The Cauchy problem associated to the logarithmic Laplacian with an application to the fundamental solution
• Characterization of solutions of a generalized Helmholtz problem
• The Dirichlet Problem for the Logarithmic Laplacian
• Liouville theorems for a general class of nonlocal operators
• Entire s-harmonic functions are affine

Related

• An extension problem for the logarithmic Laplacian
• The Cauchy problem associated to the logarithmic Laplacian with an application to the fundamental solution
• Fundamental solutions in the Colombeau framework: applications to solvability and regularity theory
• The Dirichlet Problem for the Logarithmic Laplacian
• Isolated singularities for the n-Liouville equation