Boundary Hyperbolicity Assumptions

• Boundary hyperbolicity assumptions are a set of conditions—including Gromov hyperbolicity, rough starlikeness, and coercivity—that define the geometry of manifolds and their boundary behavior.
• These assumptions enable analogues of classical Fatou and Calderón–Stein theorems by ensuring non-tangential convergence and establishing equivalent energy criteria for harmonic functions.
• They facilitate unified harmonic analysis on Gromov hyperbolic manifolds, linking analytic properties with geometric boundaries and supporting probabilistic and potential-theoretic applications.

Boundary hyperbolicity assumptions govern the interplay between the geometry of a space (manifold or metric space), the analytic or dynamical properties of functions defined on or near its boundary, and the types of estimates or convergence phenomena that can be deduced. In the context of harmonic functions on non-compact manifolds, these assumptions critically determine the validity of analogues to classical results such as Fatou and Calderón–Stein theorems. The synthesis below presents the rigorous framework, formulations, and implications of boundary hyperbolicity assumptions as developed for harmonic analysis on Gromov hyperbolic manifolds (Petit, 2013).

1. Geometric Framework and Assumptions

Boundary hyperbolicity assumptions in this context refer to a specific triad of properties imposed on a complete simply connected Riemannian manifold $M$:

• Gromov Hyperbolicity: The manifold is geodesic and $\delta$–hyperbolic (for $\delta$ an integer $\geq 3$). The Gromov product

$$(x, y)_o = \frac{1}{2} \left[ d(o, x) + d(o, y) - d(x, y) \right]$$

satisfies the inequality

$$(x, z)_o \geq \min \{ (x, y)_o, (y, z)_o \} - \delta$$

for all $x, y, z \in M$ and any basepoint $o$. This metric property ensures "thin" geodesic triangles and supports the construction of a geometric boundary at infinity via equivalence classes of sequences with diverging distance from $o$.

• Rough Starlikeness: For some basepoint $o$ and constant $K \geq 0$, every point $x \in M$ is within $K$ of some geodesic ray from $o$:

$\forall x \in M\ \exists \gamma~\text{from $o$ s.t.}~d(x, \gamma) \leq K$

This ensures global "visibility" and uniform control approaching the boundary.

• Coercivity: The manifold possesses bounded local geometry and positive spectral gap $\lambda_1(M) > 0$ where

$$\lambda_1(M) := \inf_\varphi \frac{\int_M \|\nabla\varphi\|^2}{\int_M \varphi^2}$$

and every ball $B(x, r)$ is uniformly bi-Lipschitz to a Euclidean ball. Coercivity allows for classical potential-theoretic techniques even in infinite-volume settings.

Implication: For all manifolds satisfying these three conditions (labeled as condition $(\clubsuit)$), the geometric and Martin boundaries coincide up to homeomorphism — a result established by Ancona — so the boundary at infinity has a robust analytic and geometric meaning.

2. Non-Tangential Behavior and Local Fatou Theorem

Within this geometric setup, a central result is a non-tangential boundary convergence theorem for harmonic functions—akin to the classical Fatou theorem. For an open set $U \subset M$ and a nonnegative harmonic function $u$ on $U$, one has:

• For $\mu$–almost every boundary point $\theta$ tangential for $U$ (meaning, for every $c > 0$, the non-tangential cone $\Gamma_c^\theta$ meets $U$ on an unbounded set), the function $u$ admits a non-tangential boundary limit at $\theta$.

The non-tangential cones generalize the classical Euclidean cones and are defined as

$$\Gamma_c^\theta = \{ z \in M\ |\ \exists~\text{geodesic ray}~\gamma~\text{from}~o~\text{to}~\theta~\text{with}~d(z, \gamma) < c \}$$

3. Energy Criteria and Calderón–Stein-Type Equivalence

A principal analytic tool is the density of energy associated to the harmonic function $u$ in the tubular non-tangential approach region:

$$D_c^r(\theta) := -\frac{1}{2} \int_{\Gamma_c^\theta} \Delta|u - r|(dx)$$

for $r \in \mathbb{R}$. The total non-tangential energy is

$$J_c^\theta = \int_{\Gamma_c^\theta} |\nabla u|^2 d\nu_M = \int_{\mathbb{R}} D_c^r(\theta) dr$$

Key equivalence (Calderón–Stein type): For $\mu$–almost every boundary point $\theta$ (with $\mu$ the harmonic measure), the following are equivalent for any $c > 0$:

• $u$ converges non-tangentially at $\theta$,
• The supremum over $r$ of $D_c^r(\theta)$ is finite,
• $D_c^0(\theta)$ is finite,
• $J_c^\theta$ is finite.

Thus, non-tangential convergence, non-tangential boundedness, and finiteness of non-tangential energy are indistinguishable at almost every boundary point, extending the classical result of Calderón and Stein for the Euclidean half-space.

4. Analytic Formulations and Potentials

Key analytic objects and representations in this setting include:

• The Poisson kernel:

$$K(x, \theta) = \lim_{y \to \theta} \frac{G(x, y)}{G(o, y)}$$

where $G(x, y)$ is the Green function on $M$

• Harmonic functions admit Poisson-type representation via

$u(x) = \int_{\partial M} f(\theta) d\mu_x(\theta)$

• The machinery of Harnack inequalities and exponential decay estimates for the Green function is available due to the boundary hyperbolicity assumptions, providing uniform control near the boundary.

5. Applications and Broader Theorems

By leveraging these geometric-analytic conditions, the following generalizations and results are obtained:

• Local Fatou and Calderón–Stein Equivalence: Classical properties regarding non-tangential convergence and boundedness (originally known for harmonic functions in discs or half-spaces) are extended to the non-Euclidean, negatively curved (Gromov hyperbolic) setting.
• Martin Boundary Identification: The analytic and geometric boundaries agree, so results about boundary values, representations, and convergence have a precise topological target.
• Probabilistic Connections: The fine structure of the boundary and non-tangential approach region is intimately tied to the behavior of Brownian motion on $M$ conditioned to exit at a fixed boundary point via Doob's h-transform, allowing stochastic interpretations of boundary limits and the density of energy.

6. Significance and Analytical Consequences

The combination of Gromov hyperbolicity, rough starlikeness, and coercivity is critical in ensuring that the geometric boundary precisely supports robust harmonic analysis. The equivalence of non-tangential convergence, boundedness, and finite energy demonstrates that, even far from the classical Euclidean settings, geometric negative curvature enforces rigorous and unified boundary regularity for harmonic functions. These insights have immediate consequences for the extension of other classical potential-theoretic results—such as uniqueness of boundary value problems, Poisson representations, and Fatou-type theorems—to a broad class of non-compact, infinite-volume manifolds with well-understood boundary behavior.

Summary Table of Core Conditions and Equivalences

Condition on Harmonic Function at Boundary	Analytic Quantity	Description
Non-tangential convergence at θ	$\sup_r D_c^r(\theta) < \infty$	Density of energy along non-tangential cones
Non-tangential boundedness (a.e.)	$D_c^0(\theta) < \infty$	Energy density at a fixed level set
Finite non-tangential energy (a.e.)	$J_c^\theta < \infty$	Integrated energy over tubular neighborhood

These are $\mu$–almost everywhere equivalent in the geometric boundary, as guaranteed by the boundary hyperbolicity assumptions.

7. Outlook

Boundary hyperbolicity assumptions enable the full apparatus of boundary regularity theory in negative curvature—exemplified through the homeomorphism between Martin and geometric boundaries, the validity of non-tangential convergence criteria, and the extension of strong analytic equivalences. Their role is foundational for the analysis of harmonic functions on non-positively curved spaces with rich, non-trivial boundaries, and for developing further generalizations in geometric analysis and probability.