The conformal logarithmic Laplacian on the sphere: Yamabe-type problems and Sobolev spaces

Abstract

We study the conformal logarithmic Laplacian on the sphere, an explicit singular integral operator that arises as the derivative (with respect to the order parameter) of the conformal fractional Laplacian at zero. Our analysis provides a detailed investigation of its spectral properties, its conformal invariance, and the associated \(Q\)-curvature problem. Furthermore, we establish a precise connection between this operator on the sphere and the logarithmic Laplacian in \(\mathbb{R}^N\) via stereographic projection. This correspondence bridges classification results for two Yamabe-type problems previously studied in the literature, extending one of them to the weak setting. To this end, we introduce a Hilbert space that serves as the logarithmic counterpart of the homogeneous fractional Sobolev space, offering a natural functional framework for the variational study of logarithmic-type equations in unbounded domains.

Revised version. In particular, a miscalculation in Lemma 3.15 has been corrected; this does not affect any of the statements