Sharp trace inequalities for conformally invariant fractional powers of the sublaplacian on the Heisenberg group and the CR sphere

Abstract: We establish sharp Sobolev trace inequalities for conformally invariant fractional powers of the sublaplacian on the Heisenberg group and the CR sphere, extending the corresponding Euclidean results of Einav-Loss, Beckner, and Bez-Machihara-Sugimoto to these non-Euclidean settings. In the limiting case, sharp trace Beckner-Onofri inequalities are also established on the CR sphere. The proofs are based on a duality argument due to Bez-Machihara-Sugimoto, together with the Frank-Lieb sharp form of the Hardy-Littlewood-Sobolev inequalities on the Heisenberg group and the CR sphere. The same approach also yields trace Beckner-Onofri inequalities on the standard sphere.

• The paper demonstrates that sharp fractional Sobolev trace inequalities hold on both the Heisenberg group and CR sphere using conformally invariant operators.
• It employs duality arguments and exact Hardy-Littlewood-Sobolev constants to characterize optimal embedding constants and extremal functions.
• The findings extend classical Euclidean inequalities to sub-Riemannian geometries, offering new benchmarks for geometric analysis and PDE research.

Sharp Trace Inequalities for Conformally Invariant Fractional Powers of the Sublaplacian on the Heisenberg Group and the CR Sphere: Summary and Implications

Introduction and Scope

This paper systematically constructs sharp Sobolev trace inequalities for conformally invariant fractional powers of the sublaplacian in two non-Euclidean settings: the Heisenberg group $\mathbb{H}^n$ and the CR sphere $\mathbb{S}^{2n+1}$. These results extend and generalize prior classical and fractional sharp trace inequalities from the Euclidean context—such as those by Einav-Loss, Beckner, and Bez-Machihara-Sugimoto—to sub-Riemannian and CR geometries.

The research covers both the full range $2m < s < Q_n$ for the order $s$ of the operator (with $Q_n = 2n+2$ the homogeneous dimension) and the endpoint, establishing sharp Beckner-Onofri-type trace inequalities. Methodologically, the duality argument of Bez-Machihara-Sugimoto and refined Hardy-Littlewood-Sobolev (HLS) constants of Frank-Lieb for CR and sub-Riemannian structures are central technical tools.

Mathematical Framework

Operators and Spaces

• Heisenberg Group and sublaplacian: The fractional powers $\mathcal{L}_{s,\mathbb{H}^n}$ of the sublaplacian, conformally invariant with respect to non-isotropic dilations and left translations, are defined spectrally and satisfy precise homogeneity and intertwining properties. The associated Sobolev spaces $W^{s,2}(\mathbb{H}^n)$ are natural domains for trace inequalities.
• CR Sphere: The conformally invariant differential operators $\mathcal{A}_{s,\mathbb{S}^{2n+1}}$, diagonalized by CR harmonics, enable formulation of inequalities on $\mathbb{S}^{2n+1}$, with restriction mappings to CR subspheres $\mathbb{S}^{2(n-m)+1}$ permitting trace statements.
• Restriction/Trace Mappings: For subgroups or subspheres, restriction operators $\mathbb{S}^{2n+1}$0, $\mathbb{S}^{2n+1}$1 serve as analytical traces, mapping functions from the ambient to lower-dimensional structures.

Main Inequalities

• Sharp Fractional Trace Inequalities: For $\mathbb{S}^{2n+1}$2,

$\mathbb{S}^{2n+1}$3

with optimal constants $\mathbb{S}^{2n+1}$4 (specified in terms of Gamma functions and geometric invariants). Parallel results hold on the CR sphere.

• Endpoint (Beckner-Onofri) Cases: For $\mathbb{S}^{2n+1}$5, trace inequalities at the endpoint yield sharp logarithmic-type (Beckner-Onofri) inequalities involving determinants and CR-pluriharmonic functions.

Attainability and Extremals

Optimal functions attaining equality are explicitly characterized by their conformal (Möbius-type or dilation-type) covariance and are parameterized by group translations and dilations, verifying rigidity up to the full symmetry group of the setting.

Technical Methodology

• Duality and Intertwining: The duality argument leverages the adjoint relationship between energy and trace spaces, expressing the trace inequality as a dual of a sharp HLS inequality.
• Sharp HLS Constants: Frank-Lieb’s computation of exact HLS constants and extremals on the Heisenberg group and CR sphere underpins both the foundational fractional Sobolev and new sharp trace inequalities.
• Cayley/Streographic Projections: Central to transferring results between model spaces and canonical spheres are explicit formulas for the Jacobians of the Cayley and stereographic transforms, preserving conformal invariance of both the operators and measures.
• Limiting Process for Endpoint Results: The Beckner-Onofri type trace inequalities are derived via careful limits and derivative computations at the critical order, leading to log-determinant-type functionals relevant to sharp logarithmic Sobolev inequalities.

Numerical and Structural Results

• Strong Results: The constants in all inequalities are shown to be exact and, in particular, agree with the minimal constants for the fractional (and, in the limit, logarithmic) trace embedding. The expressions for the constants are given explicitly in terms of Gamma functions and geometric parameters.
• Rigidity: Equality is achieved only for functions that arise from the action of the conformal group, confirming rigidity; no other extremizers exist.

Implications and Future Directions

• Non-Euclidean Analysis: These results reinforce the deep analogy between conformal geometry on spheres and the sub-Riemannian geometry of the Heisenberg group, establishing the Heisenberg-CR geometry as a canonical context for sharp inequalities previously known only in Euclidean settings.
• Sharp Constants and Quantitative Geometry: The methods and constants provide benchmarks for further study in geometric analysis, potential applications to PDEs (e.g., CR Yamabe-type problems), and serve as constraints in bounds on singular integrals and potential theory in sub-Riemannian spaces.
• Fractional and Higher-Order Operators: The explicit extension to the full fractional range (and endpoint) opens avenues for studying non-local phenomena and higher-order conformally covariant operators in CR and other parabolic or subelliptic geometries.
• Further Generalization: The techniques are amenable to application in more general settings (e.g., stratified Lie groups, weighted measures, or other conformally invariant structures), and the methods might extend to even finer multilinear or vector-valued inequalities.

Conclusion

This work achieves a comprehensive characterization of sharp fractional Sobolev trace inequalities in the Heisenberg and CR spheres, culminating in new Beckner-Onofri-type trace inequalities at critical orders. The precise constants, extremal functions, and direct methodologies provide a foundational basis advancing the analysis of conformally invariant inequalities in non-Euclidean and non-elliptic frameworks. Future research may exploit these results to analyze related geometric flows, non-local operators, or conformally invariant nonlinear PDEs in strictly pseudoconvex or sub-Riemannian geometries.