Fundamental solutions of the logarithmic Laplacian: An approach via the division problem
Abstract
Existence of the fundamental solution of the logarithmic Laplacian (in dimensions d ≥ 3 d \geq 3 ) was established by Huyuan Chen and Laurent Véron [J. Funct. Anal. 287 (2024), p. 72]. In this note, we present an alternative approach, based on a modification of the classical division problem. This is inspired by the theory of fundamental solutions by Malgrange [Ann. Inst. Fourier (Grenoble) 6 (1955/56), pp. 27–355] and Ehrenpreis [Amer. J. Math. 76 (1954), pp. 88–903; Amer. J. Math. 77 (1955), pp. 28–292]. 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.
Fields of Study & Subjects
Nonlinear Partial Differential Equations
Advanced Differential Equations and Dynamical Systems
Nonlinear Differential Equations Analysis
Publication details
- Journal / source
- Proceedings of the American Mathematical Society
- Volume
- 154
- Issue
- 8
- Pages
- 3479–3494
- Publication date
- 8 April 2026
- Language
- EN