Chen–Véron conjecture on decaying fundamental solutions of the logarithmic Laplacian

Let $d\in\{1,2\}$. A fundamental solution of the logarithmic Laplacian is a distribution $E$ solving the fundamental-solution equation for the logarithmic Laplacian. The conjecture concerns a radial, locally integrable representative $E$ and the asymptotic bound

$$|E(x)|\lesssim |x|^{-\left(\frac{d-1}{2}\right)},\qquad |x|\geq 2.$$

Chen–Véron conjecture. In dimensions $1$ and $2$, such a fundamental solution exists, is radial and locally integrable, and satisfies the displayed bound.

This is a modified version of a conjecture attributed to Chen and Véron concerning a fundamental solution with decay properties analogous to a Sommerfeld radiation condition. The preceding classification describes all fundamental solutions, while the existence of one with these additional properties remains open in the supplied source.