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Stability for Critical Points of the Hardy--Littlewood--Sobolev Inequality and a Dual Stability Framework

This paper establishes the first quantitative stability results for critical points and Palais--Smale sequences of the Hardy--Littlewood--Sobolev inequality in a non-Hilbertian distance by introducing a novel weak-decomposition–strong-stability method and a duality framework that subsequently removes the nonnegativity assumption in stability results for the fractional Sobolev inequality.

Original authors: Lu Chen, Guozhen Lu, Hanli Tang

Published 2026-05-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Lu Chen, Guozhen Lu, Hanli Tang

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

The Big Picture: The "Perfect Shape" and the "Wobbly Neighbor"

Imagine you have a mathematical rule (an inequality) that describes the most efficient, "perfect" way to arrange energy or mass in a space. In the world of math, this perfect arrangement is called a critical point. Think of it like a perfectly round, smooth soap bubble. It's the most stable, efficient shape possible.

For a long time, mathematicians have known exactly what these perfect shapes look like for certain rules (specifically the Sobolev inequality). They also figured out a very important question: If you have a shape that is almost perfect, how close is it to the real perfect one?

This is called stability. If you nudge the perfect bubble slightly, does it stay close to its original shape, or does it collapse into something totally different? For the "Sobolev" rule, mathematicians have already built a detailed map showing exactly how close a wobbly shape is to the perfect one.

The Problem:
There is a cousin to the Sobolev rule called the Hardy–Littlewood–Sobolev (HLS) inequality. It's like a different type of physics game. While we know what the "perfect bubble" looks like for HLS, nobody knew how to measure the "wobble" of a shape that is almost perfect.

Why? Because the ruler mathematicians usually use to measure distance (the "Hilbert space" ruler) doesn't work for this specific game. It's like trying to measure the distance between two cities using a ruler meant for measuring the weight of apples. The tools didn't fit.

The Authors' Solution: A New Toolkit

Chen, Lu, and Tang stepped in to fix this. They developed a brand new way to measure the "wobble" for the HLS inequality.

1. The "Weak-Strong" Strategy

Imagine you are trying to fix a wobbly table.

The Old Way (Failed): You tried to push the legs straight using a tool that only works on square tables. It didn't work because the HLS table has round, curved legs.
The New Way: The authors invented a "Weak-Decomposition–Strong-Stability" method.
- Weak Decomposition: They first break the wobbly shape down into two parts: the "perfect bubble" part and the "messy remainder" part. They do this in a very gentle, flexible way (a "weak" way) that doesn't require the rigid tools that failed before.
- Strong Stability: Once they isolate the "messy remainder," they use a powerful, "strong" mathematical technique to prove that this remainder is actually very small. This proves that if the shape is almost perfect, it must be very close to the perfect bubble.

2. The "Mirror" Trick (Duality)

The paper also introduces a clever "mirror" concept called Duality.

Think of the Sobolev inequality and the HLS inequality as two sides of the same coin, or reflections in a mirror.
For a long time, mathematicians could only prove stability for the "Sobolev side" if the shape was made of positive numbers (like only positive energy). They couldn't prove it if the shape had negative parts mixed in.
The authors realized that if they could prove stability for the HLS side (using their new toolkit), they could look in the mirror and instantly know the answer for the Sobolev side, even for shapes with negative numbers mixed in.
The Result: They successfully used the stability of the HLS inequality to prove a new, stronger stability result for the Sobolev inequality, removing a major restriction (the "non-negativity" rule) that had been in place for decades.

Key Takeaways in Plain English

The Gap: We knew how to measure how close a shape is to perfect for one type of math rule (Sobolev), but we were stuck on the other type (HLS) because our measuring tools didn't fit.
The Breakthrough: The authors built a new measuring tool (the weak-decomposition method) specifically designed for the HLS rule. They proved that if a shape is close to satisfying the HLS rule, it is mathematically guaranteed to be close to the perfect "bubble" shape.
The Bonus: By proving this for HLS, they used a "mirror" effect to also improve the rules for the Sobolev inequality. They showed that you don't need to assume the shapes are "positive" to prove they are stable; the math works even if the shapes are messy or mixed with negative values.
The "Firsts": This is the first time anyone has been able to give a precise, numerical lower bound (a specific "minimum wobble" number) for these types of sequences in this specific, difficult mathematical setting.

Summary Analogy

Imagine you have a perfect, round balloon (the critical point).

Old Math: We knew how to measure how squashed a balloon was if it was made of a specific material (Sobolev).
The Problem: We didn't know how to measure a squashed balloon made of a different, slippery material (HLS) because our tape measure kept slipping off.
This Paper: The authors invented a sticky, custom tape measure that grips the slippery material. They showed that even if the balloon is a bit squashed, it's definitely still a balloon and not a cube. Furthermore, by measuring the slippery balloon, they figured out a better way to measure the first type of balloon too, even if that balloon had weird, jagged edges.

The paper is a pure mathematics achievement. It solves a specific puzzle about how stable certain mathematical shapes are, providing new tools and connections that were previously missing.

Technical Summary: Stability for Critical Points of the Hardy–Littlewood–Sobolev Inequality and a Dual Stability Framework

Problem Statement
While quantitative stability for critical points of the Sobolev and fractional Sobolev inequalities has been extensively studied, the corresponding stability theory for critical points of the Hardy–Littlewood–Sobolev (HLS) inequality remains largely unexplored. A primary obstacle is that the natural stability problem for HLS critical points involves a non-Hilbertian distance (specifically the $L^{\frac{2n}{n+2s}}$ norm). Consequently, classical orthogonal decomposition methods, which are standard in Hilbert-space settings (such as those used for the Sobolev inequality), are not directly applicable. Furthermore, existing stability results for the fractional Sobolev inequality often rely on non-negativity assumptions for the functions involved, limiting their generality.

Methodology
The authors develop a novel "weak-decomposition–strong-stability" method tailored to the specific structure of HLS critical points. The approach involves several key technical components:

Sphere Conformal Mapping: The analysis is frequently transferred to the sphere $S^n$ via stereographic projection to utilize spherical harmonics techniques, which facilitate the derivation of local stability estimates.
Weak Decomposition in $H^{-s}$ : To circumvent the lack of Hilbertian structure in the $L^{\frac{2n}{n+2s}}$ metric, the authors decompose a function $u$ in the dual space $H^{-s}(S^n)$ as $u = \phi + r$ , where $\phi$ is a critical point (bubble) and $r$ satisfies specific orthogonality conditions. They then establish a comparison theorem relating the $L^{\frac{2n}{n+2s}}$ distance to the remainder term $r$ .
Local to Global Stability: The proof strategy establishes global stability by combining local stability results (valid when the distance to the manifold of critical points is small) with a new Struwe-type decomposition.
Duality Framework: A central innovation is the establishment of a duality framework connecting the stability of critical points for the Sobolev inequality with those of the HLS inequality. This allows results derived for one inequality to be transferred to the other. Specifically, the authors use the HLS stability to derive Sobolev stability without non-negativity assumptions.
Sign Testing: To remove the non-negativity assumption for the HLS equation, the authors test the integral equation with the positive and negative parts of the function. This yields energy identities that prove the critical point does not change signs, a technique that fails for the fractional Sobolev equation when $s > 1$ due to the lack of closure of positive/negative parts in $\dot{H}^s$ .

Key Contributions and Results

Theorem 1.1 (HLS Stability): The paper establishes a quantitative stability inequality for critical points of the HLS inequality. For functions $f \in L^{\frac{2n}{n+2s}}(\mathbb{R}^n)$ with energy bounded within a specific range, the paper proves:
$\||f|^{-\frac{4s}{n+2s}}f - (-\Delta)^{-s}f\|_{L^{\frac{2n}{n-2s}}(\mathbb{R}^n)} \geq C_{n,s} d(f, \mathcal{M}_{HLS})$
where $d(f, \mathcal{M}_{HLS})$ is the distance to the manifold of Aubin-Talenti bubbles in the $L^{\frac{2n}{n+2s}}$ norm.
Explicit Lower Bound for Palais–Smale Sequences: The authors provide an explicit lower bound for the stability of Palais–Smale sequences of the HLS integral equation. To the best of their knowledge, this is the first quantitative stability result for Palais–Smale sequences of a variational functional measured in a non-Hilbertian distance.
Theorem 1.2 (Sobolev Stability via Duality): By introducing a duality framework, the authors derive Struwe-type decomposition and stability results for critical points of the fractional Sobolev inequality for general functions (not restricted to non-negative ones). This removes the non-negativity assumption imposed in previous works (e.g., De Nitti and König [26]) for all $s \in (0, n/2)$ .
Struwe-Type Decomposition: The paper proves a Struwe-type decomposition lemma for critical points of both the HLS and Sobolev inequalities without assuming non-negativity, extending previous results that were limited to non-negative functions.

Significance
The paper claims to fill a significant gap in the stability theory of variational inequalities by addressing the HLS inequality, which had been neglected due to the difficulties posed by non-Hilbertian metrics. The development of the "weak-decomposition–strong-stability" method provides a new tool for analyzing stability in non-Hilbertian settings. Furthermore, the duality framework introduced offers a powerful mechanism to transfer stability results between the HLS and Sobolev inequalities, thereby generalizing existing Sobolev stability theorems to functions that may change signs. The explicit lower bounds provided for Palais–Smale sequences represent a novel contribution to the quantitative understanding of variational functionals in non-standard metric spaces.