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

TL;DR

This paper investigates the spectral and conformal properties of the conformal logarithmic Laplacian on the sphere, connecting it to the Euclidean logarithmic Laplacian and extending Yamabe-type problem classifications.

Contribution

It introduces a new functional framework for logarithmic equations, analyzes the operator's spectral and conformal invariance, and links spherical and Euclidean settings for Yamabe problems.

Findings

Spectral analysis of the conformal logarithmic Laplacian.

Establishment of conformal invariance and Q-curvature properties.

Connection between spherical and Euclidean logarithmic Laplacians.

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.