Logarithmic Laplacian Bochner Representation and Integral Kernels on Complete Riemannian Manifolds

Overview

This study introduces a Bochner representation of the logarithmic Laplacian, a novel development for general complete Riemannian manifolds. The representation is formulated using spectral calculus and the heat semigroup. The research focuses on defining the logarithmic operator to vanish on the kernel of the negative Laplacian, $\ker(-\Delta)$, and employs the orthogonal projection $P_0=E(\{0\})$ onto $\ker(-\Delta)$, with $Q=I-P_0$. The work establishes a specific integral formula for $\log(-\Delta)f$. The investigation also examines the logarithmic Laplacian's behavior under conditions such as lower Ricci curvature bounds and large-time heat-kernel integrability, leading to a pointwise integral representation. Furthermore, it explicitly compares spectral and heat-kernel definitions of the operator, detailing their difference in terms of heat mass loss. Specific computations for fractional and logarithmic kernels, including their asymptotics and integrability properties, are provided for $\mathbb{H}^n$ with $n\ge2$. This paper is an updated version, replacing arXiv:2506.19311v1.

Approach

The research introduces the logarithmic Laplacian on general complete Riemannian manifolds via spectral calculus and the heat semigroup. The methodological steps are as follows:

• The logarithmic operator is defined to vanish on $\ker(-\Delta)$.
• Orthogonal projection $P_0=E(\{0\})$ onto $\ker(-\Delta)$ and $Q=I-P_0$ are utilized.
• A specific integral representation for the logarithmic Laplacian is derived: \[ \log(-\Delta)f = \int_0^\infty \frac{e^{-t}Qf-e^{t\Delta}Qf}{t}\,dt. \] This integral is demonstrated to converge in the strong improper sense within $L^2$.
• The derivation is shown to recover the classical pointwise logarithmic Laplacian on $\mathbb{R}^n$.
• For complete noncompact manifolds, a pointwise integral representation is derived under specific conditions: a lower Ricci curvature bound and an explicit large-time heat-kernel integrability condition.
• The spectral and heat-kernel definitions of the logarithmic Laplacian are compared, with their differences explicitly identified in terms of the loss of heat mass.
• On $\mathbb{H}^n$, for $n\ge2$, the study involves computing fractional and logarithmic kernels.
• The asymptotics of these kernels are determined at zero and infinity.
• Absolute convergence, continuity, and $L^p$ integrability of these kernels are derived under Dini and weighted integrability assumptions.

Findings

• A Bochner representation for the logarithmic Laplacian on general complete Riemannian manifolds was introduced and established. This representation leverages spectral calculus and the heat semigroup.
• The logarithmic operator was defined to vanish on $\ker(-\Delta)$, and the representation is given by: \[ \log(-\Delta)f = \int_0^\infty \frac{e^{-t}Qf-e^{t\Delta}Qf}{t}\,dt, \] where $P_0=E(\{0\})$ is the orthogonal projection onto $\ker(-\Delta)$ and $Q=I-P_0$. This integral converges in the strong improper sense in $L^2$.
• The derived integral representation recovers the classical pointwise logarithmic Laplacian on $\mathbb{R}^n$.
• A pointwise integral representation was derived on complete noncompact manifolds, contingent on a lower Ricci curvature bound and an explicit large-time heat-kernel integrability condition.
• A comparison between the spectral and heat-kernel definitions of the logarithmic Laplacian explicitly identified their difference as being related to the loss of heat mass.
• On $\mathbb{H}^n$, for $n\ge2$, fractional and logarithmic kernels were computed.
• The asymptotics of these kernels were determined at both zero and infinity for $\mathbb{H}^n$.
• Absolute convergence, continuity, and $L^p$ integrability properties of these kernels were derived for $\mathbb{H}^n$ under Dini and weighted integrability assumptions.

Why This Matters

This research provides a foundational mathematical tool by introducing a new Bochner representation for the logarithmic Laplacian on complete Riemannian manifolds. This formalization integrates spectral theory with the heat semigroup, offering a unified framework. The derivation of pointwise integral representations and explicit comparisons between different definitions contribute to a deeper theoretical understanding of this operator's behavior in complex geometric settings.