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Remainder terms and sharp quantitative stability for a nonlocal Sobolev inequality on the Heisenberg group

This paper investigates the nonlocal Sobolev inequality on the Heisenberg group by deriving a gradient-type remainder term for specific dimensions and parameters, and establishing the quantitative stability of critical points for the associated Euler-Lagrange equation in the multi-bubble case.

Original authors: Wenjing Chen, Zexi Wang

Published 2026-02-10

📖 3 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 a professional architect trying to build the most stable, perfectly balanced skyscraper possible. In the world of mathematics, this paper is about finding the "perfect blueprints" for certain complex shapes and then figuring out exactly how much a building will wobble if it isn't built perfectly.

Here is the breakdown of the paper using everyday analogies.

1. The "Perfect Shape" (The Sobolev Inequality)

In math, there are certain "ideal" functions (shapes) that represent the most efficient way to distribute energy. Think of these like a perfectly smooth soap bubble. A soap bubble naturally finds the shape that uses the least amount of surface tension to hold its volume.

The paper deals with a very specific, high-level version of this called the Nonlocal Sobolev Inequality on the Heisenberg Group.

The Heisenberg Group is like a strange, warped version of space where moving "up" is different from moving "sideways" in a way that depends on where you are. It’s like walking on a surface that is constantly twisting as you move.
"Nonlocal" means that what happens at one point in space isn't just affected by its immediate neighbors, but by everything happening far away—like how a ripple in a pond is felt across the entire surface, not just where the stone hit.

2. The "Wobble Factor" (Remainder Terms)

The researchers aren't just interested in the perfect soap bubble; they want to know about the imperfect ones.

If you build a skyscraper that is almost perfect, how much will it lean? If you make a tiny mistake in the blueprint, does the building stay mostly upright, or does it collapse instantly?

The authors prove something called a Remainder Term. In our analogy, this is a mathematical formula that says: "The difference between your imperfect building and the perfect blueprint is directly related to how much the building wobbles." They provide a way to measure that "wobble" (the distance from perfection) so that engineers (mathematicians) can predict stability.

3. The "Multi-Bubble" Problem (Quantitative Stability)

This is the most advanced part of the paper. Imagine instead of one giant soap bubble, you have a cluster of several smaller bubbles all touching and interacting.

When you have multiple "bubbles" (mathematical solutions) interacting, they can get very messy. They might be close to each other, or they might be drifting apart. The researchers established "Quantitative Stability" for these multi-bubble cases.

Think of this like a group of dancers on a stage:

If the dancers are performing a perfectly synchronized routine, they are in a "stable" state.
If one dancer trips slightly, the researchers have proven that the "error" in the dance (how much they are out of sync) can be mathematically bounded. Even if there are many dancers (multi-bubbles), you can still calculate exactly how far the group has drifted from the perfect choreography.

Summary: Why does this matter?

While this is "pure mathematics" (meaning it's about the fundamental laws of logic and shape), this kind of work is the bedrock for:

Physics: Understanding how particles interact in complex, non-linear environments.
Image Processing: Helping computers understand how to smooth out "noise" in a photo without losing the actual edges of the objects.
Engineering: Providing the deep theory used to ensure that complex systems remain stable even when they are pushed to their limits.

In short: The authors have provided a high-precision "ruler" to measure how far a complex, twisting, interacting system has drifted from its most perfect, efficient state.

This paper, authored by Wenjing Chen and Zexi Wang, investigates the nonlocal Sobolev inequality on the Heisenberg group $\mathbb{H}^n$ , specifically focusing on the existence of remainder terms and the quantitative stability of critical points for the associated Euler-Lagrange equation.

1. The Problem

The authors study a nonlocal version of the Folland-Stein-Sobolev inequality on the Heisenberg group $\mathbb{H}^n$ . The inequality, denoted as $SHL(Q, \mu)$ , relates the $L^2$ -norm of the horizontal gradient $\nabla_H u$ to a nonlocal integral term involving the Hardy-Littlewood-Sobolev (HLS) kernel:

$\left( \int_{\mathbb{H}^n} \int_{\mathbb{H}^n} \frac{|u(\xi)|^{Q^*_\mu} |u(\eta)|^{Q^*_\mu}}{|\eta^{-1}\xi|^\mu} d\xi d\eta \right)^{\frac{1}{Q^*_\mu}} \leq C \int_{\mathbb{H}^n} |\nabla_H u|^2 d\xi$

where $Q = 2n+2$ is the homogeneous dimension and $Q^*_\mu$ is the upper critical exponent.

The research addresses two fundamental questions in geometric analysis:

Remainder Terms: If a function $u$ is not an extremal (a "bubble"), how much larger is the left side of the inequality than the right side? Can this difference be bounded by a distance to the manifold of extremal functions?
Quantitative Stability: If a function $u$ "almost" solves the Euler-Lagrange equation (i.e., the equation equals a very small error in a dual norm), must $u$ be quantitatively close to a sum of weakly interacting bubbles?

2. Methodology

The authors employ several advanced techniques from sub-Riemannian geometry and nonlinear analysis:

Concentration-Compactness Principle: Used to handle the loss of compactness inherent in critical exponent problems on unbounded domains.
Spectral Analysis and Non-degeneracy: They utilize the known non-degeneracy of the Jerison-Lee bubbles and the eigenvalues of the linearized operator to control the behavior of functions near the manifold of extremals $M$ .
Asymptotic Analysis: To prove stability in the multi-bubble case, they perform a refined asymptotic expansion of the nonlocal terms, specifically analyzing how bubbles interact when they are "weakly interacting" (separated in scale or location).
Contradiction Arguments: The proofs for the remainder terms and stability results are largely constructed via contradiction, assuming the existence of a sequence that violates the bounds and showing this leads to a violation of the known properties of the bubbles.

3. Key Contributions and Results

A. Remainder Terms (Theorems 1.1 & 1.2)

The authors establish the existence of a gradient-type remainder term.

Theorem 1.1: For $Q \geq 4$ and $\mu \in (0, 4]$ , there exist constants $A_1, A_2 > 0$ such that the deficit in the inequality is bounded from above and below by the square of the distance to the manifold of bubbles $M$ :
$A_2 \text{dist}(u, M)^2 \geq \text{Deficit}(u) \geq A_1 \text{dist}(u, M)^2$
Theorem 1.2: They extend this to bounded domains $\Omega \subset \mathbb{H}^n$ , proving a second-type remainder term involving the weak $L^q$ -norm (specifically $L^{\frac{Q}{Q-2}}_w$ ), which is a sharper result for functions vanishing on the boundary.

B. Quantitative Stability of Critical Points (Theorem 1.4)

The most technically demanding part of the paper is the stability of the multi-bubble case.

Theorem 1.4: For the specific case $Q=4$ and $\mu \in (2, 4)$ , they prove that if $u$ is a sum of $m$ weakly interacting bubbles plus a small error $\Upsilon(u)$ in the dual space, then the distance from $u$ to the manifold of $m$ -bubble sums is linearly bounded by that error:
$\delta(u) \leq C \Upsilon(u)$
This result is sharp, meaning the power of the error term cannot be improved.

4. Significance

This work is significant for several reasons:

Extension to Non-Euclidean Geometry: While these stability problems have been extensively studied in $\mathbb{R}^N$ , this paper provides rigorous proofs in the Heisenberg group, which is the model space for sub-Riemannian geometry.
Nonlocal Complexity: The presence of the nonlocal HLS kernel adds significant complexity compared to the local Sobolev inequality, as the interaction between different parts of the domain must be accounted for globally.
Completing the Theory: By addressing the multi-bubble case for $Q=4$ , the authors bridge a gap in the stability theory for nonlocal equations on the Heisenberg group, providing a complete picture of how "almost-solutions" behave.