Fractional Sobolev-type embedding on CR sphere and Heisenberg group

Abstract: This paper studies critical fractional Sobolev inequalities with lower-order terms on the standard CR sphere $\mathbb S^{2n+1}$. Let $Q=2n+2$, let $s\in(0,1)$, let $1<p<Q$, and let $p_s^*=\frac{Qp}{Q-sp}$. For the inequality $|u|{L^{p_s^*}(\mathbb S^{2n+1})}\le A[u]{s,p}+B|u|{L^p(\mathbb S^{2n+1})}$, we prove that the admissible lower-order coefficients are exactly $\left[|\mathbb S^{2n+1}|^{-s/Q},\infty\right)$. For the power-type inequality $|u|{L^{p_s^*}(\mathbb S^{2n+1})}^p\le A[u]{s,p}^p+B|u|{L^p(\mathbb S^{2n+1})}^p$, we show that the admissible set is $\left[|\mathbb S^{2n+1}|^{-sp/Q},\infty\right)$ when $1<p\le 2$, and $\left(|\mathbb S^{2n+1}|^{-sp/Q},\infty\right)$ when $2<p<Q$. Via the Cayley transform, we derive the exact weighted counterpart on the Heisenberg group and prove that the corresponding admissible sets coincide with those on the sphere. We also show that nonlinear first-moment constraints do not improve the optimal lower-order coefficient, whereas finite-codimensional linear constraints excluding nonzero constants yield coercive inequalities.

• The paper establishes sharp admissible ranges for lower-order coefficients in fractional Sobolev inequalities on the CR sphere.
• It translates results to the Heisenberg group via the Cayley transform, verifying precise weighted critical inequalities.
• The study highlights the impact of constraints, showing how nonlinear and linear conditions affect the necessity of remainder terms.

Fractional Sobolev-type Embedding on the CR Sphere and Heisenberg Group

Mathematical Framework and Objectives

This paper establishes a complete characterization of admissible lower-order coefficients in critical fractional Sobolev-type inequalities on the standard CR sphere $S^{2n+1}$ and the Heisenberg group $H$. The authors focus on inequalities of the form

$$\|u\|_{L^{p^*}(S^{2n+1})} \leq A [u]_{s,p} + B \|u\|_{L^p(S^{2n+1})}$$

and

$$\|u\|_{L^{p^*}(S^{2n+1})}^p \leq A [u]_{s,p}^p + B \|u\|_{L^p(S^{2n+1})}^p,$$

where $[u]_{s,p}$ denotes the CR-invariant fractional seminorm, $p^* = Qp / (Q - sp)$ is the critical exponent, and $Q = 2n + 2$ is the homogeneous dimension. The main goal is to explicitly determine the minimal admissible values for $B$ as a function of $n$, $p$, and $H$0, and to transfer these results to the Heisenberg group via the Cayley transform. The analysis includes both linear and power-type inequalities, covers endpoint sharpness, and explores how constraint classes affect the admissibility of lower-order terms.

Critical Inequalities on the CR Sphere

The compactness of the CR sphere $H$1 fundamentally influences the form of Sobolev-type inequalities. In particular, the presence of constant functions in the energy space imposes a sharp lower bound on the coefficient $H$2 in the lower-order terms, which cannot be eliminated for the unrestricted space. The determination of this lower bound exploits the following mechanism: for constant functions, the fractional seminorm vanishes, so the inequality reduces to a relationship between $H$3 and $H$4 norms, leading directly to the condition that $H$5 for the linear form and $H$6 for the power-type form.

The admissible sets for $H$7 are rigorously proven as follows:

• Linear form: The set of admissible $H$8 is $H$9, and the left endpoint is attainable.
• Power form: For $$\|u\|_{L^{p^*}(S^{2n+1})} \leq A [u]_{s,p} + B \|u\|_{L^p(S^{2n+1})}$$0, the admissible set is $$\|u\|_{L^{p^*}(S^{2n+1})} \leq A [u]_{s,p} + B \|u\|_{L^p(S^{2n+1})}$$1, and the endpoint is attained. For $$\|u\|_{L^{p^*}(S^{2n+1})} \leq A [u]_{s,p} + B \|u\|_{L^p(S^{2n+1})}$$2, the admissible set is $$\|u\|_{L^{p^*}(S^{2n+1})} \leq A [u]_{s,p} + B \|u\|_{L^p(S^{2n+1})}$$3, i.e., the threshold is sharp but not admissible.

The endpoint analysis utilizes expansions around constant functions and precise estimates on fractional norms, showing that perturbations around constants can violate the power-type inequality at the threshold for $$\|u\|_{L^{p^*}(S^{2n+1})} \leq A [u]_{s,p} + B \|u\|_{L^p(S^{2n+1})}$$4.

Weighted Formulation on the Heisenberg Group

Via the Cayley transform, which is a CR diffeomorphism between $$\|u\|_{L^{p^*}(S^{2n+1})} \leq A [u]_{s,p} + B \|u\|_{L^p(S^{2n+1})}$$5 and $$\|u\|_{L^{p^*}(S^{2n+1})} \leq A [u]_{s,p} + B \|u\|_{L^p(S^{2n+1})}$$6, the authors translate the critical inequalities to the Heisenberg group. The transformation introduces a canonical weight on both the Lebesgue norms and the fractional kernel, dictated by the Cayley Jacobian. Thus, the inequalities on $$\|u\|_{L^{p^*}(S^{2n+1})} \leq A [u]_{s,p} + B \|u\|_{L^p(S^{2n+1})}$$7 are not standard homogeneous Sobolev inequalities but are instead weighted, capturing the compactification of $$\|u\|_{L^{p^*}(S^{2n+1})} \leq A [u]_{s,p} + B \|u\|_{L^p(S^{2n+1})}$$8 by a boundary point. Explicitly, for a function $$\|u\|_{L^{p^*}(S^{2n+1})} \leq A [u]_{s,p} + B \|u\|_{L^p(S^{2n+1})}$$9 on $$\|u\|_{L^{p^*}(S^{2n+1})}^p \leq A [u]_{s,p}^p + B \|u\|_{L^p(S^{2n+1})}^p,$$0,

$$\|u\|_{L^{p^*}(S^{2n+1})}^p \leq A [u]_{s,p}^p + B \|u\|_{L^p(S^{2n+1})}^p,$$1

and the weighted seminorm is

$$\|u\|_{L^{p^*}(S^{2n+1})}^p \leq A [u]_{s,p}^p + B \|u\|_{L^p(S^{2n+1})}^p,$$2

where $$\|u\|_{L^{p^*}(S^{2n+1})}^p \leq A [u]_{s,p}^p + B \|u\|_{L^p(S^{2n+1})}^p,$$3 and $$\|u\|_{L^{p^*}(S^{2n+1})}^p \leq A [u]_{s,p}^p + B \|u\|_{L^p(S^{2n+1})}^p,$$4 are determined by the Cayley transform and the CR distance.

The core result is that the admissible sets for lower-order coefficients in these weighted inequalities on $$\|u\|_{L^{p^*}(S^{2n+1})}^p \leq A [u]_{s,p}^p + B \|u\|_{L^p(S^{2n+1})}^p,$$5 coincide exactly with those on the sphere.

Impact of Constraints and Structural Dichotomy

The study also investigates how imposing constraints affects the lower-order coefficient. The findings reveal:

• Nonlinear first-moment constraints, such as vanishing moments against first spherical harmonics or coordinate functions, do not improve the optimal lower bound because constant functions remain admissible. Therefore, the sharp $$\|u\|_{L^{p^*}(S^{2n+1})}^p \leq A [u]_{s,p}^p + B \|u\|_{L^p(S^{2n+1})}^p,$$6 coefficient persists for these classes.
• Finite-codimensional linear constraints that exclude constants (e.g. zero-average spaces or orthogonality to certain harmonics) recover coercivity, leading to pure seminorm inequalities for all $$\|u\|_{L^{p^*}(S^{2n+1})}^p \leq A [u]_{s,p}^p + B \|u\|_{L^p(S^{2n+1})}^p,$$7.

This dichotomy is formalized: the necessity of the lower-order coefficient is equivalent to the class containing non-zero constants.

Subcritical and Further Consequences

By interpolation, a family of subcritical inequalities is established for exponents $$\|u\|_{L^{p^*}(S^{2n+1})}^p \leq A [u]_{s,p}^p + B \|u\|_{L^p(S^{2n+1})}^p,$$8, with arbitrarily small leading seminorm coefficients. These results transfer precisely to the Heisenberg group. The endpoint analysis also motivates future exploration regarding optimal leading coefficients and extensions to more general pseudohermitian manifolds.

Numerical Results and Endpoint Sharpness

The paper provides strong and explicit numerical thresholds for admissible lower-order coefficients:

• Linear threshold: $$\|u\|_{L^{p^*}(S^{2n+1})}^p \leq A [u]_{s,p}^p + B \|u\|_{L^p(S^{2n+1})}^p,$$9
• Power threshold: $[u]_{s,p}$0

For the nonlinear power-type inequality with $[u]_{s,p}$1, the endpoint is not admissible, marking a sharp transition in the qualitative behavior.

Implications and Future Directions

The findings clarify the qualitative and quantitative structure of fractional Sobolev-type inequalities in subelliptic CR geometry and on stratified groups. The explicit description of admissible lower-order terms, together with the weighted Heisenberg formulation, offers tools for geometric analysis, variational PDEs, and potential theory in both compact and noncompact settings. The equivalence under the Cayley transform underscores the deep connection between compact CR manifolds and Carnot groups. Practically, this affects existence and regularity results for critical nonlinear subelliptic equations and informs constraint-based variational methods.

Future developments include identifying optimal leading coefficients, examining broader classes of compact manifolds, and characterizing endpoint admissibility for nonlinear constraint classes. Extensions to spectral fractional operators and their geometric invariants are also anticipated.

Conclusion

This paper rigorously determines the sharp admissible ranges for lower-order coefficients in critical fractional Sobolev-type embeddings on the CR sphere and Heisenberg group. The results are precise and closed for the linear inequality, and exhibit a critical threshold behavior for the power-type inequality. The noncompact (weighted) formulation via the Cayley transform preserves these thresholds exactly. The structural analysis of constraint classes provides a complete dichotomy between coercive and non-coercive regimes. The mathematical implications extend to the geometric analysis of CR manifolds, Carnot groups, and subelliptic PDEs.