On Faber-Krahn inequality for Dirichlet Fractional-Logarithmic Laplacian

We study the Faber–Krahn problem for the fractional–logarithmic Laplacian $(-\mathrm{\Delta }{)}^{s+\ln }$, $0<s<1$, with Fourier symbol $\mathrm{\mid }\xi {\mathrm{\mid }}^{2s}\ln \mathrm{\mid }\xi {\mathrm{\mid }}^2$. We prove a small-volume and large-volume dichotomy. For sufficiently small volume, balls minimize the first Dirichlet eigenvalue, with rigidity up to translation.