The fractional $p$-Laplacian on hyperbolic spaces

Technical Summary: The Fractional $p$-Laplacian on Hyperbolic Spaces

Problem Statement
The paper addresses the definition and analysis of the fractional $p$-Laplacian, denoted $(−\Delta_{\mathbb{H}^n})^s_p$, on hyperbolic spaces $\mathbb{H}^n$ for $n \in \mathbb{N}$, $0 < s < 1$, and $p > 1$. While fractional operators are well-studied in Euclidean spaces $\mathbb{R}^n$ and various other geometric settings (Riemannian manifolds, metric measure spaces, etc.), the lack of a Fourier transform in nonlinear settings and the specific geometric structure of hyperbolic space prevent the direct application of standard definitions. Specifically, the authors note that definitions relying on the Fourier transform are unavailable for nonlinear operators, and not all definitions of fractional Laplacians (e.g., regional vs. spectral) are equivalent in general Riemannian manifolds. The primary challenge is to establish a rigorous, equivalent set of definitions for the nonlinear fractional $p$-Laplacian on $\mathbb{H}^n$ with explicit normalizing constants that allow for the study of pointwise convergence to the standard $p$-Laplacian as $s \to 1^-$.

Methodology
The authors employ a multi-faceted approach combining hyperbolic geometry, heat kernel analysis, and asymptotic properties of special functions:

1. Geometric Framework: The study utilizes the hyperboloid model of $\mathbb{H}^n$ equipped with the Lorentzian metric. The distance function $d(x, \xi)$ and the volume element $\sinh^{n-1} r \, dr \, d\omega$ are central to the integral formulations.
2. Heat Semigroup Representation: The authors relate the fractional operator to the heat semigroup $\{e^{t\Delta_{\mathbb{H}^n}}\}_{t \geq 0}$ generated by the Laplacian on $\mathbb{H}^n$. They utilize explicit formulas for the heat kernel $h(t, \rho)$ on hyperbolic spaces, distinguishing between odd and even dimensions, which involve derivatives of the heat kernel with respect to the distance $\rho$.
3. Modified Bessel Functions: The kernels of the fractional operators are expressed using modified Bessel functions of the second kind, $K_\nu$. The authors leverage integral representations and asymptotic behaviors of $K_\nu$ (as $\rho \to 0^+$ and $\rho \to \infty$) to analyze the singular kernels.
4. Caffarelli–Silvestre Extension: The paper investigates the extension problem on $\mathbb{H}^n \times (0, \infty)$, relating the fractional operator to a Dirichlet-to-Neumann map. This involves constructing a Poisson kernel $P(\rho, y)$ and an extension operator $E_{s,p}$.
5. Asymptotic Analysis for Convergence: To prove the convergence of $(−\Delta_{\mathbb{H}^n})^s_p$ to the $p$-Laplacian as $s \to 1^-$, the authors perform detailed asymptotic analysis of integrals involving the normalizing constants $c_{n,s,p}$ and the kernels $K_{n,s,p}$. They use Taylor expansions and the dominated convergence theorem, carefully handling the volume element $\sinh^{n-1} \rho$ which differs from the Euclidean $\rho^{n-1}$.

Key Contributions and Results

1. Three Equivalent Definitions: The paper establishes three equivalent definitions of the fractional $p$-Laplacian on $\mathbb{H}^n$, providing explicit normalizing constants for each:

   • Singular Integral Definition (Definition 1.1): A pointwise integral representation involving a singular kernel $K_{n,s,p}(\rho)$. The kernel is defined explicitly using modified Bessel functions and derivatives with respect to the distance, with distinct formulas for odd and even dimensions.
   • Heat Semigroup Definition (Theorem 1.3): The operator is represented as an integral over time of the heat semigroup applied to a nonlinear function $\Phi_p(u(x) - u(\cdot))$. The constant $C_2$ is explicitly derived.
   • Extension Problem Definition (Theorem 1.4): The operator is realized as a limit of a weighted normal derivative of a solution to a degenerate elliptic extension problem in the upper half-space $\mathbb{H}^n \times (0, \infty)$. The constant $C_3$ is explicitly provided.

2. Explicit Normalizing Constants: A critical contribution is the derivation of the precise normalizing constants ($c_{n,s,p}, C_1, C_2, C_3, C_4$) involving Gamma functions and powers of 2 and $\pi$. These constants are chosen specifically to ensure the correct limiting behavior as $s \to 1^-$.

3. Kernel Asymptotics (Proposition 1.2 & Theorem 1.2): The authors characterize the behavior of the kernel $K_{n,s,p}(\rho)$. They show that near the origin ($\rho \to 0^+$), the kernel behaves like the Euclidean singular kernel $\rho^{-n-sp}$, while at infinity ($\rho \to \infty$), it exhibits exponential decay due to the hyperbolic geometry ($\sim \rho^{-1-sp/2} e^{-(n-1)\rho}$). This confirms the well-definedness of the operator for bounded $C^2$ functions.

4. Pointwise Convergence (Theorem 1.5): The main convergence result states that for $u \in C^2_b(\mathbb{H}^n)$ and $x \in \mathbb{H}^n$ where $\nabla u(x) \neq 0$,
   $$\lim_{s \to 1^-} (-\Delta_{\mathbb{H}^n})^s_p u(x) = (-\Delta_{\mathbb{H}^n})_p u(x)$$
   where $(-\Delta_{\mathbb{H}^n})_p$ is the standard $p$-Laplacian on hyperbolic space. The proof relies on the explicit constants and the asymptotic analysis of the integrals of the kernel against the volume element, showing that the non-Euclidean volume factor does not prevent convergence to the local operator.

Significance
The paper claims significance primarily in extending the representation formulas for fractional operators from the linear case ($p=2$) established in previous works (e.g., [2]) to the nonlinear regime ($p > 1$) on hyperbolic spaces. By providing explicit normalizing constants, the authors enable the rigorous study of the limit $s \to 1^-$, a property essential for connecting fractional models to local models in geometric analysis. The work highlights that while Fourier-based methods fail in this nonlinear setting, the rich structure of hyperbolic space (specifically the heat kernel and Poisson kernel representations) allows for a robust definition of the fractional $p$-Laplacian. The explicit constants are noted as crucial for applications in regularity theory. The authors do not propose new experimental applications or future directions beyond the mathematical establishment of these definitions and their convergence properties.