$\mathcal H$-Harmonic Bergman-Besov Spaces on the Real Hyperbolic Ball

Abstract: Using the characterizations in terms of various differential operators including partial, normal, and tangential derivatives, we extend the family of Bergman spaces of $\mathcal H$-harmonic functions on the real hyperbolic ball from $α>-1$ to all $α\in\mathbb R$. We then generalize several properties of Bergman spaces such as projection, duality, and inclusion relations, to this extended family.

• The paper introduces a unified theory of H-harmonic Bergman-Besov spaces on the real hyperbolic ball, extending classical Bergman spaces to all α in ℝ.
• It establishes norm equivalence through partial, normal, and tangential derivatives along with coefficient multiplier operators aligned with reproducing kernel structures.
• It constructs robust reproducing kernels, projection operators, and duality mappings that facilitate applications in PDEs and hyperbolic harmonic analysis.

$\mathcal H$-Harmonic Bergman-Besov Spaces on the Real Hyperbolic Ball

Introduction and Fundamental Definitions

The paper rigorously develops the theory of $\mathcal H$-harmonic Bergman-Besov spaces on the real hyperbolic ball $\mathbb B \subseteq \mathbb R^n$, deeply integrating analysis on symmetric spaces, reproducing kernel techniques, and advanced function space theory. The hyperbolic metric is central, with the associated Laplace-Beltrami operator $\Delta_h$ induced via Möbius transformations on $\mathbb B$ and acting as the generator of hyperbolic harmonicity. Functions annihilated by $\Delta_h$—the $\mathcal H$-harmonic functions—form the foundation for specialized Bergman-type spaces.

Weighted Lebesgue measures $$d\nu_\alpha(x) = \frac{1}{V_\alpha}(1 - |x|^2)^\alpha d\nu(x)$$ are introduced parametrized by $\alpha \in \mathbb R$, extending classical Bergman spaces to the entire real line, including the regime $\alpha \leq -1$, which is essential for Möbius-invariance and reproducing kernel constructions beyond the conventional domain.

The classical $\mathcal H$0-harmonic Bergman spaces $\mathcal H$1 for $\mathcal H$2 are defined by integrability of $\mathcal H$3-harmonic functions with respect to $\mathcal H$4 in $\mathcal H$5. The principal advancement is the systematic extension to $\mathcal H$6, yielding the Bergman-Besov spaces $\mathcal H$7, characterized via differential operator norms, tangential derivatives, and specialized coefficient multipliers.

Structural Characterization via Differential Operators

The central contribution is a series of equivalence theorems that characterize $\mathcal H$8 precisely through various differential operators:

• Partial Derivatives: For $\mathcal H$9, membership in $\mathbb B \subseteq \mathbb R^n$0 is equivalent to the $\mathbb B \subseteq \mathbb R^n$1-integrability of all $\mathbb B \subseteq \mathbb R^n$2-th order partial derivatives.
• Normal Derivatives: Norms involving repeated radial derivatives $\mathbb B \subseteq \mathbb R^n$3 (with $\mathbb B \subseteq \mathbb R^n$4 defined by $\mathbb B \subseteq \mathbb R^n$5) provide an equivalent norm.
• Tangential Derivatives $\mathbb B \subseteq \mathbb R^n$6: Iterated tangential derivatives (infinitesimal rotations $\mathbb B \subseteq \mathbb R^n$7) serve as an alternative norm, crucially preserving $\mathbb B \subseteq \mathbb R^n$8-harmonicity for all $\mathbb B \subseteq \mathbb R^n$9.
• Coefficient Multiplier Operators $\Delta_h$0: The $\Delta_h$1 operators act as general order differential/integral operators on the homogeneous expansion of $\Delta_h$2, setting norms compatible with reproducing kernels and harmonic decomposition.

Explicit norm equivalences are provided, showing the robustness of the space definitions across derivative-based descriptions and kernel-based expansions.

Extension to Besov Regime ($\Delta_h$3)

A major technical achievement is the systematic extension of Bergman spaces to all $\Delta_h$4, including the non-classical Besov zone. The paper demonstrates that tangential derivatives and $\Delta_h$5 operators preserve $\Delta_h$6-harmonicity for arbitrary $\Delta_h$7, allowing for full norm equivalence and definition of $\Delta_h$8 as Banach spaces for any $\Delta_h$9. In contrast, partial and normal derivative characterizations require $\mathbb B$0 to ensure $\mathbb B$1-integrability—a subtlety analyzed in detail.

The space $\mathbb B$2 is identified as Möbius-invariant, echoing previous results that it is the unique Möbius-invariant Hilbert space among $\mathbb B$3-harmonic functions—this regime is particularly significant in hyperbolic harmonic analysis.

Reproducing Kernel and Projection Theory

The paper provides detailed constructions and asymptotic estimates of reproducing kernels $\mathbb B$4 in the extended regime, utilizing gamma function expansions for coefficients. The $\mathbb B$5 operators are shown to interact compatibly with the kernel structure, ensuring the validity of reproducing formulas for all $\mathbb B$6 and enabling bounded projection operators $\mathbb B$7 under precise parameter constraints ($\mathbb B$8).

This facilitates right invertibility and enables duality pairings and Banach isomorphisms between different $\mathbb B$9 regimes. Strong quantitative estimates for derivatives of reproducing kernels and corresponding integral transforms $\Delta_h$0 are established, underpinning the $\Delta_h$1 boundedness arguments.

Duality, Predual, and Inclusion Relations

Leveraging reproducing kernel constructions, the paper generalizes classical duality theory to the extended $\Delta_h$2 spaces for $\Delta_h$3, showing that the dual of $\Delta_h$4 is isomorphic to $\Delta_h$5, using explicit pairings through $\Delta_h$6. For $\Delta_h$7, the dual space is the $\Delta_h$8-harmonic Bloch space, with predual structure given by the little Bloch space.

Inclusion relations between $\Delta_h$9 and $\mathcal H$0 are established with sharp parameter bounds, distinguishing the cases $\mathcal H$1 and $\mathcal H$2 via precise inequalities involving $\mathcal H$3 and $\mathcal H$4, ensuring continuity of inclusion.

Theoretical and Practical Implications

The systematic extension of Bergman spaces to arbitrary $\mathcal H$5 and the robust equivalence of numerous differential norms provides a comprehensive functional analytic framework for $\mathcal H$6-harmonic function theory on the real hyperbolic ball. The results facilitate advanced harmonic analysis, interpolation, inclusion, atomistic decompositions, and Möbius-invariant function space constructions significant for both PDEs and complex analysis on symmetric spaces.

The reproduction of kernel-based projection operators and their boundedness properties are critical for explicit operator constructions, functional calculus, and duality theory in harmonic analysis. The inclusion of Bloch-type spaces solidifies the connection to the boundary behavior, essential for potential theory and complex function theory.

Future research will likely employ these spaces for more general domains, extend Möbius-invariant harmonic function theory, analyze automorphism groups, and pursue analogues in higher rank symmetric spaces.

Conclusion

The paper rigorously establishes a unified theory for $\mathcal H$7-harmonic Bergman-Besov spaces on the real hyperbolic ball, elucidating their structural properties, norm equivalences, kernel-driven projection techniques, and duality theory for all $\mathcal H$8. The results bridge classical harmonic Bergman space theory and modern Besov-type spaces, offering deep insights and robust technical tools for function theoretic analysis in hyperbolic geometries.