The Conformal Fractional--Logarithmic Laplacian on the Sphere: Yamabe Problems and Sharp Inequalities

In this paper, we introduce the conformal fractional--logarithmic Laplacian on the unit sphere, defined as the derivative of the conformal fractional Laplacian with respect to the order parameter $s\in(0,1)$. We investigate its fundamental analytic and spectral properties, including its relation to the conformal logarithmic Laplacian, its spectral representation, and the explicit form of its eigenvalues and eigenfunctions. We further establish its conformal covariance law and derive the associated Yamabe-type equation, proving its equivalence to the corresponding conformal equation in $\mathbb R^N$ through stereographic projection. Finally, we apply this framework to sharp Sobolev-type inequalities, recovering the sharp logarithmic Sobolev inequality, revealing the failure of a naive fractional--logarithmic analogue, and establishing new sharp fractional--logarithmic inequalities.