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A strong-form stability for a class of $L^p$ Caffarelli-Kohn-Nirenberg interpolation inequality

This paper establishes strong-form stability results for a class of $L^p$ Caffarelli-Kohn-Nirenberg interpolation inequalities by proving that the distance to the manifold of optimizers is controlled by the deficit, while demonstrating that such stability cannot be achieved for the individual norms separately and extending these findings to second-order inequalities for radial functions.

Original authors: Yingfang Zhang, Wenming Zou

Published 2026-07-20

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Yingfang Zhang, 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 universe as a giant, invisible trampoline made of mathematical fabric. In physics and engineering, we often need to predict how things move or settle on this trampoline. To do this, mathematicians use special rules called "inequalities." Think of these not as strict laws that say "this must equal that," but as safety nets. They guarantee that if you pull the fabric in one way (like stretching a spring), it won't snap or behave wildly in another way. One of the most famous of these safety nets is the Caffarelli-Kohn-Nirenberg (CKN) inequality. It's a complex rule that connects how fast a function changes (its slope), how big the function is, and how it behaves near a specific point (like the center of a whirlpool). For decades, mathematicians have known these safety nets exist and have even found the "perfect" shapes that fit them exactly—these are called minimizers. But knowing a net exists isn't enough; we need to know how tight it is. If you pull the fabric just a tiny bit away from the perfect shape, does the net hold you firmly, or do you slip through? This question of "stability" is what this paper tackles.

The authors of this paper, Yingfang Zhang and Wenming Zou, are diving deep into the stability of these CKN safety nets. They are asking a very specific question: If a function is almost a perfect minimizer (meaning it almost satisfies the inequality perfectly), how close is it to being a perfect one? They measure this "almost" using something called a "deficit"—a number that tells you how much the inequality is being violated. The paper proves that for certain types of these inequalities, this deficit acts like a strong magnet, pulling the imperfect function very close to the perfect shape. However, they also discover a surprising twist: you cannot measure this closeness by looking at just one part of the function (like just its slope or just its size) separately. It's like trying to judge how close a spinning top is to standing still by only looking at its height; you need to look at the whole spinning motion together. The paper establishes a "strong-form" stability, proving that the deficit controls the distance to the perfect shape in a very precise way, but only when you look at the right combination of measurements. They also extend these findings to more complex, second-order versions of these rules, but only for functions that look the same in every direction (radial functions), like a perfectly round balloon.

The Main Discovery: A Stronger Grip on Stability

The core achievement of this paper is establishing a "strong-form stability" for a specific class of Caffarelli-Kohn-Nirenberg (CKN) interpolation inequalities. In simpler terms, the authors proved that if you have a function that is slightly "off" from being a perfect minimizer, the size of that "off-ness" (the deficit) directly controls how far away the function is from the set of perfect minimizers.

The paper finds that this relationship works differently depending on the power $p$ used in the math:

When $p = 2$ : The distance to the perfect shape is directly proportional to the deficit. If the deficit is small, the distance is small in a very straightforward, linear way.
When $p > 2$ : The relationship is slightly different; the distance is proportional to the deficit raised to the power of $1/p$ . This means the "grip" of the stability is slightly different for higher powers, but it is still firmly established.

Crucially, the authors prove that this stability is "strong" because it looks at a combined measure of the function's properties (a weighted product of its slope-norm and size-norm) rather than just one in isolation.

What the Paper Rules Out

The paper explicitly argues against the idea that you can establish this kind of stability by looking at the function's slope (the gradient) or its size (the function value) separately.

The authors prove that it is impossible to say that the deficit controls the distance to the minimizers if you only measure the distance in the "slope" norm or only in the "size" norm. They demonstrate this by showing that you can stretch or shrink the function in specific ways (using a scaling transformation) that keeps the deficit exactly the same but makes the distance in either the slope-norm or size-norm grow infinitely large or shrink to zero. This proves that the deficit is "blind" to these individual measurements. You must look at the combined, weighted product of these two norms to see the stability.

How Sure Are They?

The paper provides rigorous mathematical proofs for all its main claims. There are no simulations, guesses, or suggestions here. The authors have constructed logical arguments using calculus, inequalities, and variational methods to prove that:

The stability results they found are true for all functions in the specified spaces.
The separate stability results they ruled out are mathematically impossible.
The "strong-form" results they derived are sharp, meaning the exponents they found (like $t=1$ or $t=1/p$ ) cannot be improved.

They also establish that for the second-order inequalities (involving the Laplacian or "curvature" of the function), these strong stability results hold true, but only for functions that are radial (symmetric in all directions). For non-radial functions in the second-order case, they note that while the weak stability might be possible, the strong-form result is much harder to prove and remains an open challenge, though their methods suggest a path forward if the weak stability can be established.

The Story in a Nutshell

Imagine you are trying to find the perfect shape for a soap bubble. You know the perfect shape is a sphere. The "deficit" is how much extra energy your bubble has compared to a perfect sphere. This paper proves that if your bubble has very little extra energy, it must be very close to a sphere, but you have to measure "closeness" by looking at both the surface tension and the volume together. If you try to measure closeness by just looking at the surface tension or just the volume, you might get fooled—the bubble could look like a sphere in one way but be totally distorted in another, even with the same low energy. The authors have mapped out exactly how these measurements relate, proving that the "perfect sphere" is a very stable target, provided you measure it with the right combination of tools. They also showed that for more complex, "double-layered" bubbles, this stability holds if the bubble is perfectly round, but the rules get trickier if it's lopsided.

Technical Summary: A Strong-Form Stability for a Class of $L^p$ Caffarelli-Kohn-Nirenberg Interpolation Inequality

Problem Statement
This paper investigates the stability of the Caffarelli-Kohn-Nirenberg (CKN) interpolation inequality. The central question is whether the "deficit" of a function $u$ (the difference between the left-hand side and the right-hand side of the inequality) controls the distance of $u$ from the set of minimizers. Specifically, the authors address whether this deficit can bound the distance in the strong norms associated with the inequality's terms, rather than just in a weaker sense or for specific cases.

A critical preliminary finding of the work is that stability cannot be established for the individual norms $\|\cdot\|_{H^p_b}$ or $\|\cdot\|_{L^p_a}$ separately. The authors prove that for any non-negative constants, there exist functions where the deficit is small, yet the distance to the minimizer set in either individual norm remains large. Consequently, the study focuses on establishing "strong-form" stability involving the product or weighted sum of these norms.

Methodology
The authors employ a combination of variational analysis, spectral properties of minimizers, and sharp vector inequalities.

Variational Analysis and Orthogonality:
- The paper establishes the attainability of the infimum distance from a function $u$ to the set of minimizers $M_{p,a,b}$ under the condition that the deficit is sufficiently small (Proposition 2.1).
- For the case $p=2$ , the authors utilize the fact that the minimizer $v$ attaining the infimum distance satisfies specific orthogonality conditions with respect to $u$ in the $H^2_b$ , $L^2_a$ , and $L^2_c$ norms. These conditions allow for a direct derivation of the Pythagorean-like identity $\|u-v\|^2 = \|u\|^2 - \|v\|^2$ , which is crucial for the stability estimate.
- For $p > 2$ , the orthogonality conditions are more complex. The authors introduce a specific subset of minimizers $P_u$ (Definition 3.3) defined by a maximization problem involving the duality pairing. They prove that for functions with small deficits, a maximizer exists (Proposition 3.4) and satisfies necessary orthogonality conditions (Proposition 3.6).
Sharp Vector Inequalities:
- To handle the nonlinearity for $p > 2$ , the authors utilize sharp inequalities for vectors in $\mathbb{R}^n$ (Lemma 3.2 from [17]). These inequalities provide lower bounds for $|y+z|^p$ that include a term proportional to $|z|^p$ , allowing the authors to control the distance terms $\|u-v\|$ by the deficit.
Transformation and Reduction:
- The authors use scaling transformations to normalize the problem. For $p > 2$ , they employ a specific change of variables to reduce the general CKN parameters $(a, b)$ to a canonical case $(a, b) = (-\frac{1}{p-1}, 0)$ , simplifying the analysis while preserving the structural properties of the inequality.
Second-Order Inequalities:
- For second-order CKN inequalities, the authors leverage an identity derived by Chen and Tang [7] (Theorem 1.6). This identity expresses the deficit as a sum of non-negative terms involving the Laplacian and gradient. By restricting to radial functions, they utilize a weighted $L^p$ -Poincaré inequality (Lemma 4.1) to establish weak stability, which is then upgraded to strong stability using the methods developed for the first-order case.

Key Results

Impossibility of Separate Stability (Proposition 1.7): The authors rigorously prove that no universal constant $C$ exists such that the deficit controls the distance in the $H^p_b$ norm or the $L^p_a$ norm individually. This necessitates the use of a combined norm or product form for stability estimates.
Strong-Form Stability for $p=2$ (Theorem 1.8): For the case $p=2$ , under specific conditions on parameters $a$ and $b$ , the authors establish that:
$\frac{\inf_{v \in M_{2,a,b}} \|u - v\|_{H^2_b} \|u - v\|_{L^2_a}}{\|u\|_{H^2_b} \|u\|_{L^2_a}} \leq C \delta_{2,a,b}(u)$
The exponent $t=1$ is shown to be sharp.
Strong-Form Stability for $p>2$ (Theorem 1.9): For $p > 2$ , the authors prove a similar result but with a different exponent:
$\frac{\inf_{v \in M_{p,a,b}} \|u - v\|_{H^p_b} \|u - v\|_{L^p_a}^{p-1}}{\|u\|_{H^p_b} \|u\|_{L^p_a}^{p-1}} \leq C \delta_{p,a,b}(u)^{1/p}$
To achieve this, they introduce a stronger deficit $\tilde{\delta}_{p,a,b}(u)$ (Equation 1.11) and prove that it controls the distance in the weighted sum of norms (Theorem 1.10).
Second-Order CKN Stability (Theorems 1.11 & 1.12): The paper extends these results to second-order CKN inequalities involving the Laplacian. While the general case remains open, the authors establish both weak and strong-form stability results for radial functions. The strong-form result for radial functions mirrors the first-order case, with the deficit controlling the distance in terms of $\|\Delta(u-v)\|_{L^p_b}$ and $\|u-v\|_{H^p_a}$ .

Significance and Claims
The paper claims to provide the first strong-form stability results for a broad class of CKN inequalities where the interpolation term ( $\theta \neq 1$ ) is present and parameters are non-zero.

Resolution of the "Separate Norm" Issue: The work clarifies that stability must be formulated in terms of the product or weighted sum of the norms, correcting the assumption that individual norm stability might hold.
Sharpness: The authors demonstrate that the exponent $t=1$ for $p=2$ is sharp, meaning the linear dependence on the deficit cannot be improved.
Methodological Advancement: By constructing specific orthogonality conditions for $p>2$ via the set $P_u$ , the paper overcomes the difficulty of applying standard variational techniques to non-quadratic functionals.
Extension to Second-Order: The results for second-order inequalities, particularly for radial functions, represent a significant step forward in understanding the stability of higher-order weighted Sobolev-type inequalities, a topic previously less explored than the first-order case.

The paper concludes that while the general non-radial case for second-order inequalities remains challenging, the established methods provide a robust framework for analyzing stability in weighted interpolation inequalities.

The Main Discovery: A Stronger Grip on Stability

What the Paper Rules Out

How Sure Are They?

The Story in a Nutshell

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