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Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group

This paper establishes the optimal Bianchi-Egnell-type quantitative stability constant for the critical nonlocal Sobolev inequality on the Heisenberg group, proves that the sharp universal upper bound for the deficit-to-distance comparison is 1, and derives a strict single-bubble upper bound for the residual quotient associated with the Euler-Lagrange equation.

Original authors: Wenjing Chen, Zexi Wang

Published 2026-08-11

📖 8 min read🧠 Deep dive

CC BY 4.0

Original authors: Wenjing Chen, Zexi Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to find the most efficient way to pack a suitcase. You have a fixed amount of space (the suitcase) and a pile of clothes (the energy of a system). In the world of mathematics and physics, there are "rules of the road" called inequalities that tell you the absolute best you can do. One famous rule, the Sobolev inequality, says that if you know how much a shape wiggles (its gradient), you can predict how big it can get. But life is rarely perfect. Sometimes, you have a "deficit"—a little extra wiggle or a little less space than the perfect rule allows.

The big question mathematicians love to ask is: "If I'm not perfect, how far off am I?" This is called quantitative stability. It's like asking, "If my suitcase is 1% too full, how much of my shirt is sticking out?" In simple, flat worlds (like a flat sheet of paper), we know the answer: the "sticking out" is directly proportional to the "overfilling." But what happens when the world isn't flat? What if the space you're packing is a twisted, non-flat universe where directions don't behave the way you expect? This is the world of the Heisenberg group, a mathematical playground where moving forward and then turning left doesn't get you to the same spot as turning left and then moving forward. In this weird, non-commutative space, the old rules of packing get complicated, especially when you add a "long-distance" rule where every piece of clothing feels the pull of every other piece, no matter how far apart they are. This paper dives into that messy, twisted suitcase to see if we can still find a perfect packing rule.

The Twisty Suitcase and the Long-Distance Pull

The authors of this paper, Wenjing Chen and Zexi Wang, are tackling a very specific puzzle in a strange mathematical universe called the Heisenberg group. Think of this group not as a flat floor, but as a space where the rules of movement are "twisted." If you walk forward and then spin, you end up in a different spot than if you spin and then walk. This "non-commutative" nature makes calculating things much harder than in our everyday flat world.

On top of this twisty geometry, they are looking at a nonlocal problem. In normal physics, things usually interact with their immediate neighbors. But here, the "energy" of a point depends on a conversation with every other point in the entire universe, weighted by a specific distance rule. It's like if your mood depended on the mood of every person on Earth, not just your friends next door.

The paper focuses on a critical inequality—a mathematical law that sets a hard limit on how much "wiggle" (energy) a function can have compared to its size. The authors are investigating the stability of this law. They want to know: if a function is almost perfect (satisfying the law almost perfectly), how close is it to being a "perfect" solution? And more importantly, is there a single, perfect "best constant" that describes this closeness for all cases?

The Two Barriers to Perfection

To find the answer, the authors had to look for "loopholes" where the system might break down. They identified two specific ways a sequence of functions could try to evade the system and fail to settle on a perfect solution:

The Single-Bubble Trap: Imagine a perfect solution is a single, smooth hill (a "bubble"). A sequence of functions might try to get closer and closer to this hill, but in doing so, they might wiggle in a very specific, high-frequency way that prevents them from ever quite landing. The authors calculated a "spectral threshold"—a speed limit for this wiggling. They found that if the parameters of the problem (the dimension of the space and the strength of the long-distance pull) fall within certain ranges, this wiggling is strictly controlled. The system cannot evade by just wobbling near the perfect hill.
The Two-Peak Split: Alternatively, a sequence might try to evade by splitting into two separate hills that drift infinitely far apart. It's like trying to pack your suitcase by suddenly having two suitcases appear, one at the top and one at the bottom, so the "average" looks perfect but the individual parts are messy. The authors calculated a "two-peaks threshold" to see if this split is possible. They proved that for specific conditions, the energy cost of splitting is too high, so the system is forced to stay as one single, coherent shape.

The Main Discovery: A New, Stricter Limit

By proving that these two "evading methods" (the wiggling and the splitting) are blocked under certain conditions, the authors achieved something significant. They showed that in these specific scenarios, a minimizer exists. In plain English, this means there is a real, concrete function that sits exactly at the bottom of the energy valley. It's not just a theoretical limit; you can actually point to the "best" function.

But here is the twist: This best function reveals a surprising truth about the stability constant. The authors compared their new, nonlocal stability constant (let's call it $H_{NS}$ ) with the old, well-known stability constant for the flat, local world (let's call it $H_{BE}$ ).

They proved that $H_{NS} > H_{BE}$ .

This is a big deal. In the language of these inequalities, the stability constant acts as a "safety margin" or a minimum guarantee. The inequality states that the "deficit" (how far you are from perfect) must be at least this constant times the square of the distance to the perfect solution. Because $H_{NS}$ is larger than $H_{BE}$ , it means the nonlocal system has a stricter (tighter) lower bound. For the same amount of "wiggle" (deficit), the system is forced to be closer to the perfect solution in this twisted, long-distance universe than it would be in the simple, flat case. The "long-distance pull" changes the geometry of the problem such that the system is more rigid against deviations than the local version would suggest. The safety margin for how far you can stray from perfection is actually narrower here, not wider.

The Universal Ceiling and the Critical Point

The paper doesn't stop at the lower bound (how stable it is). It also looks at the upper bound (how loose the rule can get). They proved that the "worst-case" scenario, where the deficit is as large as possible relative to the distance from perfection, has a universal ceiling of 1. This means the deficit can never be more than the square of the distance. They also showed that this ceiling of 1 is "sharp," meaning it is the best possible number, but equality (reaching exactly 1) holds if and only if you are already on the perfect solution (the manifold of bubbles). If you are even slightly off, the rule is strictly tighter, and the ratio is strictly less than 1.

Finally, they looked at the "critical points"—solutions that aren't necessarily the absolute best, but are still stable in a different way (like a ball sitting in a shallow dip). They derived a strict upper bound for the stability of these critical points as well, showing that even these "almost-perfect" solutions obey a strict rule that is different from the main minimizer rule.

Why This Matters

This work is a triumph of mathematical detective work. It takes a problem that combines two very difficult concepts—twisted geometry and long-distance interactions—and proves that, despite the complexity, the system behaves in a predictable and stable way under specific conditions.

The authors didn't just guess; they used a "profile decomposition" (a way of breaking complex shapes into simple pieces) and "spectral analysis" (looking at the frequencies of vibration) to prove that the system cannot escape its bounds. They showed that the interaction between the twisted geometry of the Heisenberg group and the long-range pull creates a unique stability landscape, one that is strictly different from both the flat world and the local twisted world.

In short, they found the "Goldilocks" zone where the math works perfectly, proved that a perfect solution exists, and showed that the universe of this twisted, long-distance space has a stability profile that is distinct from the flat case, specifically enforcing a stricter (tighter) rule on how far you can stray from perfection before the system breaks down. It's a reminder that even in the most abstract, non-intuitive corners of mathematics, there are deep, orderly structures waiting to be discovered.

Technical Summary: Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group

Problem Statement
This paper investigates the quantitative stability of the critical nonlocal Sobolev inequality on the Heisenberg group $\mathbb{H}^n$ . The central inequality is the sharp Hardy-Littlewood-Sobolev (HLS) type inequality coupled with the sub-Laplacian energy:
$SHL(Q, \mu) \left( \int_{\mathbb{H}^n} \int_{\mathbb{H}^n} \frac{|u(\xi)|^{Q^*_\mu}|u(\eta)|^{Q^*_\mu}}{|\eta^{-1}\xi|^\mu} d\xi d\eta \right)^{\frac{1}{Q^*_\mu}} \leq \int_{\mathbb{H}^n} |\nabla_H u|^2 d\xi,$
where $Q = 2n+2$ is the homogeneous dimension, $0 < \mu < Q$ , and $Q^*_\mu = (2Q-\mu)/(Q-2)$ . The extremal functions (bubbles) for this inequality are the Jerison-Lee bubbles, forming a manifold $\mathcal{M}$ .

The authors seek to determine the optimal stability constant $H_{NS}$ , defined as the infimum of the ratio between the Sobolev deficit and the squared distance to the manifold $\mathcal{M}$ :
$H_{NS} := \inf_{u \in S^{1,2}(\mathbb{H}^n) \setminus \mathcal{M}} \frac{\int_{\mathbb{H}^n} |\nabla_H u|^2 d\xi - SHL(Q, \mu) (\text{HLS term})}{\text{dist}(u, \mathcal{M})^2}.$
The study addresses three specific challenges arising from the interaction between the nonlocal HLS convolution and the noncommutative conformal geometry of $\mathbb{H}^n$ :

The linearization of the nonlocal term generates two distinct coupled bilinear forms, unlike the single form in local problems.
Establishing strict upper bounds for the stability constant requires higher-order expansions dependent on specific parameter ranges $(n, \mu)$ , which cannot be derived by simple dimensional substitution from Euclidean results.
The dichotomy analysis requires a precise expansion of the interaction between Heisenberg translates and dilates through the kernel $|\eta^{-1}\xi|^{-\mu}$ .

Methodology
The authors employ a combination of spectral analysis, variational methods, and asymptotic expansions:

Cayley Transform and Spectral Diagonalization: The problem is mapped from $\mathbb{H}^n$ to the CR sphere $S^{2n+1}$ via the Cayley transform. The second variation of the deficit is analyzed using complex spherical harmonics $\mathcal{H}^{n+1}_{i,j}$ . This allows the diagonalization of two coupled eigenvalue problems associated with the nonlocal linearization.
Perturbation Analysis: To establish strict upper bounds, the authors construct test functions.
- Spectral Threshold: A single bubble perturbed by the lowest normal spectral mode (specifically the $(2,0)$ mode) is used to detect the local loss of compactness.
- Two-Peaks Threshold: A superposition of two asymptotically separated bubbles is used to detect the loss of compactness due to dichotomy (splitting of energy).
Profile Decomposition: A Heisenberg-group profile decomposition combined with a nonlocal Brezis-Lieb splitting is utilized to handle minimizing sequences. This framework separates the sequence into a weak limit and a remainder, allowing the authors to rule out non-compactness scenarios.
Critical Point Stability: For the Euler-Lagrange equation, the authors analyze the residual quotient in the dual space $(S^{1,2})^{-1}$ , requiring a separate expansion distinct from the functional deficit analysis.

Key Contributions and Results

Identification of Strict Compactness Thresholds (Theorem 3.1):
The paper identifies two strict upper bounds for $H_{NS}$ that serve as barriers to compactness:
- Spectral Upper Bound ( $H^{\text{spec}}_{NS}$ ): Derived from the coupled eigenvalue problems on the sphere. The explicit constant is:
  $H^{\text{spec}}_{NS} = \frac{(4n + 8 - 2\mu)(\mu + 4)}{2(n + 4)(4n + 8 - \mu)}.$
  Strictness ( $H_{NS} < H^{\text{spec}}_{NS}$ ) is proven under specific conditions on $(n, \mu)$ , including $\mu \leq 4$ or specific ranges for $\mu > 4$ .
- Two-Peaks Upper Bound ( $H^{\text{2-peak}}_{NS}$ ): Derived from the interaction of two bubbles. The bound is:
  $H^{\text{2-peak}}_{NS} = 2 - 2^{\frac{2n}{4n+4-\mu}}.$
  These bounds are strictly larger than the corresponding thresholds for the local Folland-Stein-Sobolev inequality, indicating that the nonlocal term alters the compactness structure.
Existence of Minimizers (Theorem 3.2):
By combining the strict thresholds with the profile decomposition, the authors prove that the infimum $H_{NS}$ is attained (i.e., a minimizer exists) in the parameter ranges where the strict bounds hold (specifically $\mu \leq 4$ or $\mu > 4$ with $n \geq 5$ and specific constraints on $\mu$ ).
Strict Comparison with Local Stability (Theorem 3.2 & Remark 3.1):
The attainment of a minimizer $u_0 \notin \mathcal{M}$ allows for a strict comparison between the nonlocal and local stability constants. The authors prove:
$H_{NS} > H_{BE},$
where $H_{BE}$ is the optimal stability constant for the local Folland-Stein-Sobolev inequality. This demonstrates that the nonlocal interaction strictly increases the stability constant.
Sharp Universal Upper Bound (Theorem 3.3):
The paper establishes that the sharp universal upper bound for the deficit-to-distance ratio is exactly 1:
$\text{Deficit} \leq 1 \cdot \text{dist}(u, \mathcal{M})^2.$
Equality holds if and only if $u \in \mathcal{M}$ . This result mirrors the local case, showing that while the lower bound (stability constant) changes due to nonlocality, the upper bound remains universal.
Critical Point Stability (Theorem 3.4):
For the associated Euler-Lagrange equation, the authors derive a strict upper bound for the critical-point stability constant $H_{CP}(1)$ in the single-bubble case. This bound is distinct from the functional attainment result and relies on the same coupled spectral mode but a different residual expansion.

Significance and Claims
The paper claims that while the abstract strategy of minimizing stability quotients is not new, the mixed nonlocal-Heisenberg problem possesses a distinct quantitative structure compared to both the Euclidean nonlocal problem and the local Heisenberg problem.

Coupled Spectral Problem: The primary novelty is the identification of a coupled CR-HLS spectral problem. The stability threshold is determined by the simultaneous diagonalization of two bilinear forms, yielding a constant that cannot be obtained from either the Euclidean HLS spectrum or the local CR spectrum alone.
Parameter-Dependent Nonlinearity: The paper highlights that strictness of the bounds is not automatic; it requires a delicate analysis of higher-order terms that depends explicitly on the parameters $n$ and $\mu$ .
Compactness and Comparison: The work provides the first rigorous proof of the attainment of the optimal stability constant for this specific nonlocal inequality on the Heisenberg group and establishes the strict inequality $H_{NS} > H_{BE}$ .

The authors do not claim to have solved the general existence of minimizers for all parameter ranges but provide explicit conditions under which attainment is guaranteed. Similarly, the critical point stability result is presented as a separate derivation that does not follow directly from the attainment of the functional quotient.

The Twisty Suitcase and the Long-Distance Pull

The Two Barriers to Perfection

The Main Discovery: A New, Stricter Limit

The Universal Ceiling and the Critical Point

Why This Matters

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