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The stability on the Caffarelli-Kohn-Nirenberg and Hardy-type inequalities and beyond

This paper establishes gradient stability and functional properties for Caffareli-Kohn-Nirenberg inequalities, derives refined Sobolev-type embeddings, and investigates weighted versions, extremal functions, and stability constants for a broad class of classical Hardy-type inequalities.

Original authors: Yuxuan Zhou, Wenming Zou

Published 2026-01-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yuxuan Zhou, Wenming Zou

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 the mathematical world as a vast landscape where shapes, distances, and forces interact. In this landscape, there are "rules of the road" called inequalities. These rules tell us that no matter how you stretch or twist a function (a mathematical object representing a shape or a wave), certain limits always hold true. For example, a rule might say, "The energy required to create a shape is always at least X times the size of that shape."

This paper, written by researchers from Tsinghua University, is about sharpening these rules and understanding how close you can get to breaking them without actually breaking the laws of physics (or mathematics).

Here is a breakdown of their work using simple analogies:

1. The Main Goal: Tightening the Safety Net

The authors focus on two famous families of rules:

The Caffarelli-Kohn-Nirenberg (CKN) Inequality: Think of this as a complex safety net that catches falling objects (functions) in a space with uneven gravity (weights). The net has a specific "tightness" (a constant) that determines how well it catches things.
Hardy-type Inequalities: These are similar safety nets, but they deal with objects that are very close to a "singularity" (like a black hole or a sharp point at the center of a room).

The Problem: For a long time, mathematicians knew the exact tightness of these nets. But they didn't fully understand what happens when a function is almost too big to be caught. Does it just barely slip through, or does the net stretch in a predictable way?

The Solution: The authors developed a new set of "magnifying glasses" (mathematical transformations) to look closer at these rules. They didn't just find the tightness; they measured the gap between a function and the perfect shape that fits the rule exactly.

2. Part One: The "Gradient Stability" (How much does it wobble?)

Imagine you have a perfect, smooth balloon (the "extremal function") that fits perfectly inside a mold. This balloon represents the ideal solution where the inequality is an exact equality.

Now, imagine you take a slightly lumpy, misshapen balloon and try to force it into the same mold.

The Question: If the lumpy balloon is almost the right shape, how much "extra energy" (or gap) does it create?
The Discovery: The authors proved that this "extra energy" is directly linked to how far the lumpy balloon is from the perfect shape.
- If the balloon is slightly off, the gap is small.
- If the balloon is very lumpy, the gap grows significantly.
- They calculated the exact formula for this relationship. It's like having a ruler that tells you exactly how much "wobble" you have based on how far you are from perfection.

They also found that this rule works even in tricky situations where the "gravity" of the space changes (weighted spaces), and they figured out how this stability constant behaves when you tweak the parameters of the problem.

3. Part Two: The "Hardy" Variations (New Rules for Different Terrains)

The second half of the paper takes these ideas and applies them to many different, specialized versions of the rules. Think of this as taking the same "magnifying glass" and using it on different terrains:

Logarithmic Sobolev Inequalities: These deal with how information spreads out (like heat or smoke). The authors found the perfect "smoke shapes" (Gaussian functions) that fit these rules and showed how to adjust the rules for different environments.
Hardy-Morrey Inequalities: These rules are for spaces where things can be very rough or jagged. The authors added "smoothness" checks to these rules, ensuring that even jagged shapes obey the law.
Interpolated Inequalities: These are hybrid rules that mix different types of constraints. The authors showed how to build a bridge between the "Hardy" rules and the "Sobolev" rules, creating a new, more flexible safety net.

4. The Secret Weapon: The "Magic Transformation"

The most important tool the authors used is a transformation.

The Analogy: Imagine you have a complicated, twisted knot (a difficult inequality with heavy weights). Trying to untie it directly is a nightmare.
The Trick: The authors found a way to "unwind" the knot into a simple, straight string (a much simpler inequality) without losing any of its length or properties.
The Result: Once they solved the problem for the simple string, they could "re-wind" the solution back into the complex knot. This allowed them to solve very hard problems by reducing them to easy ones they already knew how to handle.

5. What They Didn't Do (Important Boundaries)

The paper is purely theoretical mathematics.

They did not apply these rules to engineering, medicine, or climate change.
They did not predict future technologies.
They did not solve physical problems directly; they solved the mathematical laws that might eventually describe physical problems.

Summary

In short, this paper is like a team of master architects who have spent years studying the blueprints of a building's foundation. They didn't just confirm the foundation is strong; they built a new, ultra-sensitive sensor that can detect even the tiniest cracks. They showed exactly how the foundation reacts when you push it, and they provided a universal tool (the transformation) that allows anyone to check the stability of similar foundations, no matter how complex the building looks on the outside.

Technical Summary: Stability on the Caffarelli-Kohn-Nirenberg and Hardy-type Inequalities and Beyond

Problem Statement
This paper addresses the stability and refinement of two fundamental classes of functional inequalities in analysis: the Caffarelli-Kohn-Nirenberg (CKN) inequalities and various Hardy-type inequalities. While the sharp constants and extremal functions for these inequalities are well-established in specific cases (e.g., the classical Sobolev inequality or specific parameter regimes of CKN), the quantitative stability—measuring how close a function is to an extremal when the inequality is nearly saturated—remains a complex problem, particularly for general parameters and non-Hilbertian settings ( $p \neq 2$ ). Furthermore, the paper seeks to extend known refinements of Hardy inequalities (such as those involving weak Lebesgue norms or logarithmic terms) to weighted versions and to derive explicit stability constants for a broad spectrum of related inequalities.

Methodology
The central methodological tool employed throughout the paper is a class of simple yet powerful transformations. Inspired by Horiuchi's work, the authors utilize a scaling transformation of the form $\bar{u}(r\theta) = k^{1/q} u(r^k \theta)$ (with specific $k$ depending on parameters $n, p, a, b$ ). This transformation serves two primary purposes:

Reduction: It reduces weighted inequalities with complicated weights to simpler, unweighted forms (often mapping the CKN inequality to the classical Sobolev inequality or a standard Hardy inequality).
Transfer: It allows the transfer of known stability results, extremal classifications, and sharp constants from the simpler cases to the weighted cases.

For the stability of the CKN inequality in the functional setting, the authors combine these transformations with spectral gap estimates and compactness arguments. In the critical point setting, they employ careful expansion techniques and spectral gap inequalities for the linearized operator around Talenti bubbles. For the Hardy-type inequalities, the transformation method is used to derive weighted versions of known improved inequalities (e.g., Hardy-Sobolev, logarithmic Hardy, Hardy-Morrey) and to establish explicit stability constants.

Key Contributions and Results

Part 1: Stability of the Caffarelli-Kohn-Nirenberg Inequality
The authors establish gradient stability for the CKN inequality in the full parameter region defined by $1 < p < n$ , $0 \le a < \frac{n-p}{p}$ , $a \le b < a+1$ , and the critical exponent relation.

Functional Stability: They prove that the gap between the left and right sides of the CKN inequality controls the distance of the function to the manifold of extremals ( $M_{p,a,b}$ $M_{p, a, b}$ ).
- For $0 < a < b$ , they establish an estimate with the optimal exponent $\max\{2, p\}$ .
- For $0 < a = b$ , they establish an estimate with exponent $\max\{4, 2p\}$ for general functions, noting that the exponent improves to $\max\{2, p\}$ if the function is centrally symmetric. They conjecture that $\max\{2, p\}$ holds generally for this case.
Improved Sobolev Embeddings: Leveraging the gradient stability, they derive refined Sobolev-type embeddings involving weak Lebesgue norms ( $L^p_w$ ) for functions supported on general domains (both finite and infinite measure). These results provide explicit dependence of the stability constants on domain geometry (e.g., via the principal $p$ -Laplacian eigenvalue).
Stability Constant Properties: The paper investigates the functional properties of the optimal stability constant $K(p, a, b)$ . They prove upper semi-continuity with respect to parameters and establish monotonicity relations between constants for different parameter sets.
Critical Point Stability: For $p > 2$ , the authors provide a stability result in the critical point setting. They show that if a function is close to a sum of Talenti bubbles, the residual of the Euler-Lagrange equation controls the distance to the manifold of positive solutions, yielding alternative-type estimates.

Part 2: Improved Hardy-type Inequalities
The second part of the paper focuses on deriving weighted versions and refinements for various classical Hardy-type inequalities.

Hardy-Sobolev Inequalities: They extend improved Hardy inequalities on bounded domains (originally by Brézis-Vázquez and Gazzola et al.) to weighted versions, providing explicit stability constants and identifying extremizers.
Logarithmic Sobolev and Hardy Inequalities: The authors derive weighted $L^p$ -logarithmic Sobolev inequalities and logarithmic Hardy inequalities. They provide complete classifications of extremizers (Gaussian-type functions) and establish stability estimates, particularly for centrally symmetric functions.
Hardy-Morrey and Interpolation Inequalities: They obtain weighted Hardy-Sobolev-Morrey inequalities with Hölder refinements and series expansions. Additionally, they extend Hardy-Sobolev interpolation inequalities to weighted settings, establishing local improvements under specific assumptions on the ratio of the gradient and potential terms.
Interpolated CKN Inequalities: The paper recovers and extends stability results for interpolated CKN inequalities, providing explicit (though potentially non-sharp) stability constants for general parameters by reducing them to the known case of the Heisenberg-Pauli-Weyl inequality.

Significance and Claims
The paper claims to provide a unified framework for analyzing the stability of a broad class of weighted inequalities. By utilizing the transformation method, the authors demonstrate that complex weighted problems can often be reduced to simpler, well-understood cases.

Explicit Constants: A significant contribution is the derivation of explicit stability constants for various inequalities, which were previously unknown or only known implicitly in the weighted settings.
Generalization: The results generalize previous stability findings (which were often restricted to $p=2$ or specific parameter ranges) to the full parameter range $1 < p < n$ .
New Insights: The paper offers new insights into the symmetry breaking phenomena of extremizers for logarithmic Hardy inequalities and provides the first stability results for the CKN inequality in the critical point setting for $p \neq 2$ .
Modesty: The authors acknowledge that while their derived stability constants are explicit, they may not be sharp in all cases (particularly for the interpolated CKN inequalities with $b > 0$ ). They also note that the stability exponent for the case $a=b>0$ in the functional setting might be improvable to $\max\{2, p\}$ for general functions, a question they leave open (though they note subsequent work by Deng and Tian has resolved this for $a=b>0$ ).

In summary, the paper establishes a robust methodology for deriving stability and refinements for weighted functional inequalities, offering explicit constants and extending the scope of known results to general parameters and domains.