Integral representation for fractional Laplace-Beltrami operators

Abstract: In this paper we provide an integral representation of the fractional Laplace-Beltrami operator for general riemannian manifolds which has several interesting applications. We give two different proofs, in two different scenarios, of essentially the same result. One of them deals with compact manifolds with or without boundary, while the other approach treats the case of riemannian manifolds without boundary whose Ricci curvature is uniformly bounded below.

• The paper provides explicit integral representations for the fractional Laplace-Beltrami operator on Riemannian manifolds using singular integral kernels with rigorously controlled error terms.
• It employs techniques such as spectral theory, the Hadamard parametrix, and heat kernel analysis to bridge nonlocal PDE results from Euclidean spaces to curved geometries.
• The findings facilitate sharp analytic inequalities and regularity proofs, notably advancing the understanding of fractional Sobolev embeddings and nonlinear evolution equations on manifolds.

Integral Representation for Fractional Laplace-Beltrami Operators on Riemannian Manifolds

Introduction and Motivation

The fractional Laplacian and, more generally, fractional powers of Laplace-Beltrami operators play a central role in the theory of nonlocal PDEs, with applications to regularity theory, geometric analysis, and fluid dynamics. While explicit integral representations for the fractional Laplacian on Euclidean spaces and tori are classical, their extension to general Riemannian manifolds—especially those with boundaries or with varying curvature—poses significant technical barriers. The absence of such expressions has hindered the development of pointwise estimates and critical inequalities for nonlocal operators in curved geometries.

This paper presents two comprehensive approaches yielding principal value integral representations (plus explicit error terms) for the fractional Laplace-Beltrami operator $(-\Delta_g)^s$ ($0 < s < 1$) on smooth, compact Riemannian manifolds, as well as in certain classes of complete, non-compact manifolds with Ricci curvature uniformly bounded below. Beyond the foundational representation, the work establishes tools for extending sharp analytic inequalities to the manifold setting, clarifying the interplay between geometry, spectral theory, and singular integral operators.

Main Results

The central achievement is providing, for a smooth function $f$ on a compact $n$-dimensional Riemannian manifold $(M, g)$ without boundary, a singular integral representation: $$(-\Delta_g)^s f(x) = \mathrm{P.V.} \int_M \frac{f(x) - f(y)}{d(x, y)^{n+2s}} \left(C_{n,s} \chi u_0 + k_N \right)(x, y) d\mathrm{vol}_g(y) + O\big(\|f\|_{H^{-N}(M)}\big)$$ where $\chi$ is a diagonal cutoff, $u_0$ is a smooth function with $u_0(x,x)=1$, $k_N(x, y)$ is smooth and $O(d(x, y))$, and $N$ is arbitrary. The error is controlled in negative Sobolev norms. The precise structure enables explicit comparisons with the Euclidean case and underpins applications to regularity and maximum principles.

For non-compact manifolds with Ricci curvature and injectivity radius bounded below, the construction relies on the heat kernel parametrix, leading to a similar representation involving the fundamental solution of the heat operator, with geometric constants entering via sharp heat kernel bounds.

Two distinct but related proofs are provided:

• On compact manifolds (possibly with boundary, using appropriate eigenfunction expansions and local parametrices),
• On complete, non-compact manifolds with Ricci lower bounds and positive injectivity radius, using heat kernel analysis.

Analytical Framework and Proof Techniques

The technical foundation draws on spectral theory, the Hadamard parametrix, and heat kernel analysis.

On compact manifolds, the authors exploit the functional calculus for $-\Delta_g$, constructing a right parametrix using Bessel potential theory, leading to explicit singular integral kernels akin to the Euclidean case but adapted to the metric structure. The Hadamard parametrix provides local expansions for the resolvent kernel, with explicit control of error terms via the recursive structure of Bessel functions and the smooth cutoff function.

On non-compact manifolds, the heat semigroup representation is central: $$(-\Delta_g)^s f(x) = \frac{1}{\Gamma(-s)} \int_0^\infty \left(e^{-t\Delta_g}f(x) - f(x)\right) \frac{dt}{t^{1+s}}$$ This formula, combined with Li-Yau heat kernel estimates and short-time asymptotics of the heat kernel, yields the singular integral representation with explicit kernel structure and quantifiable lower-order remainders.

The analysis in both cases carefully disentangles the leading singularity—governed by the manifold dimension and the fractional order—from geometric corrections and error terms, yielding a robust formula amenable to manifold-dependent applications.

Implications and Applications

One explicit application is a direct and transparent proof of fractional Sobolev embeddings on compact manifolds: $$\|f\|_{L^p(M)} \leq C \left( \|f\|_{L^2} + \|(-\Delta_g)^{s}f\|_{L^2} \right), \qquad p = \frac{2n}{n-2s},~ 0 < s < 1.$$ Unlike approaches relying purely on abstract functional analysis or complex interpolation, the singular integral formula allows the use of the Hardy-Littlewood-Sobolev inequality, connecting directly with pointwise control and elliptic regularity.

Crucially, the integral formula allows for pointwise nonlinear lower bounds for the nonlocal operator, such as

$$(-\Delta_g)^s |f(x)| \geq c\,|f(x)|^{1+\alpha} + \text{lower order},$$

sharpening classical maximum principles and facilitating regularity proofs for critical equations. In particular, these results underpin recent advances in the global regularity theory for the critical surface quasigeostrophic (SQG) equation on the standard sphere, extending the pioneering analysis previously limited to flat or boundaryless domains. The framework can, in principle, be extended to other active scalar equations with fractional dissipation on manifolds.

The explicit error control with respect to the geometry also implies that the method is robust under perturbations of the metric, anisotropy, and up to the presence of boundary, modulo degeneration of constants near the boundary.

Theoretical and Practical Implications

From a theoretical standpoint, this work provides a bridge between the classical singular integral theory of nonlocal elliptic operators and spectral-geometric analysis on manifolds. The explicit kernels reveal the precise effect of curvature and local geometry on the nonlocal behavior, illuminating the influence of metric structure on analytic estimates and functional inequalities.

Practically, the formulae are directly applicable in the study of regularity of weak solutions to nonlocal drift-diffusion equations, well-posedness for active scalar equations at critical dissipation, and the derivation of nonlinear bounds within geometric contexts that historically have lacked such sharp estimates.

Future Directions

Several directions emerge from this work:

• Extension to higher fractional orders $s>1$ and endpoint cases where singularities are more intricate (e.g., logarithmic terms in even dimension).
• Adaptation to noncompact manifolds with degenerate or unbounded geometry, possibly replacing Ricci lower bounds with other geometric controls.
• Application of these explicit formulas to probabilistic representations (e.g., Lévy processes on manifolds) and to quantifying the geometry-induced corrections in stochastic processes.
• Exploitation of the framework for anisotropic or metric-dependent nonlocal operators, relevant to differential geometry and mathematical physics.

Conclusion

This work delivers precise, utilitarian representations for fractional Laplace-Beltrami operators on general classes of Riemannian manifolds, accompanied by detailed error analysis and broad potential for applications in geometric analysis and nonlocal PDE theory. The explicit singular kernel structures bridge the analytic gap between Euclidean and manifold settings, offering concrete tools for both functional inequalities and nonlinear evolution equations involving nonlocal geometric operators (1704.06126).