The weighted $L^2$-Caffarelli-Kohn-Nirenberg inequalities for the curl-free vector fields and second order derivatives: The sharp constants and stability estimates

Technical Summary: Weighted $L^2$-Caffarelli-Kohn-Nirenberg Inequalities for Curl-Free Vector Fields and Second Order Derivatives

Problem Statement
This paper investigates the weighted $L^2$-Caffarelli-Kohn-Nirenberg (CKN) inequalities, specifically focusing on two distinct but related contexts:

1. Curl-free vector fields: Establishing sharp constants and stability estimates for inequalities involving the gradient of a vector field $U$ (where $U = \nabla u$) and weighted norms.
2. Second-order derivatives: Proving sharp inequalities and stability estimates involving the Laplacian $\Delta u$ and the gradient $\nabla u$ with general weight parameters $a$ and $b$.

While sharp constants for first-order CKN inequalities and specific cases of second-order inequalities (e.g., the Heisenberg-Pauli-Weyl uncertainty principle) were previously established by authors such as Cazacu, Flynn, Lam, and others, this work addresses the gap in proving these inequalities for general parameters $a$ and $b$ in the second-order setting and provides the first stability estimates for the weighted CKN inequalities of curl-free vector fields.

Methodology
The authors employ a rigorous analytical approach centered on three main tools:

1. Spherical Harmonic Decomposition: The primary method involves decomposing functions $u$ and vector fields $U$ into radial and non-radial components using spherical harmonics. This reduces the multidimensional inequalities to a family of one-dimensional integral inequalities indexed by the degree $k$ of the spherical harmonics.
2. One-Dimensional Integral Inequalities: The core of the proof relies on establishing sharp weighted inequalities for functions of a single variable. The authors utilize a specific lemma (Lemma 2.2) that provides sharp constants for inequalities of the form:
   $$\left( \int_0^\infty (f')^2 r^{A-1} dr + C \int_0^\infty f^2 r^{A-3} dr \right) \int_0^\infty f^2 r^{B-1} dr \geq K \left( \int_0^\infty f^2 r^{\frac{A+B}{2}-2} dr \right)^2$$
   They determine the optimal constant $K$ and the extremal functions explicitly.
3. Stability Analysis via Orthogonal Decomposition: To prove stability (quantifying how close a function is to the set of extremals when the inequality is nearly saturated), the authors decompose the function into a radial part ($u_r$) and a part orthogonal to radial functions ($u_o$). They demonstrate that the sharp constant can be strictly improved for the non-radial part (Lemma 4.1 and Lemma 5.1). This "gap" allows them to bound the distance of the function from the set of extremals in terms of the deficit of the inequality.

Key Contributions and Results

1. Sharp Weighted Second-Order CKN Inequalities (Theorem 1.1)
The authors prove a new family of sharp weighted $L^2$-CKN inequalities for second-order derivatives. For $n \geq 2$ and parameters $a, b$ satisfying specific conditions (including $(n-2a-2)^2 \geq 8(1+a)(1+2a)$), they establish:
$$\int_{\mathbb{R}^n} \frac{|\Delta u|^2}{|x|^{2a}} dx \int_{\mathbb{R}^n} \frac{|\nabla u|^2}{|x|^{2b}} dx \geq C_2(n, a, b)^2 \left( \int_{\mathbb{R}^n} \frac{|\nabla u|^2}{|x|^{a+b+1}} dx \right)^2$$

• Sharp Constant: $C_2(n, a, b) = \frac{|a-b+1| + |n+2a|}{2}$.
• Extremal Functions: The equality is attained by specific radial functions defined via integrals involving exponential and power terms (Equation 1.11).
• Significance: This result complements previous work by Cazacu, Flynn, and Lam (2023) and Duong and Nguyen (2025) by covering a broader range of parameters and explicitly handling the second-order Laplacian term rather than just the gradient of the vector field.

2. Stability for Curl-Free Vector Fields (Theorem 1.3)
The paper establishes a stability version for the weighted CKN inequality for curl-free vector fields $U = \nabla u$. It proves that if the "deficit" (the difference between the left and right sides of the inequality) is small, the vector field $U$ is close to the set of extremal vector fields $M_{a,b}$ in a weighted $L^2$ norm.

• Result: $\delta(U) \geq C \inf_{V \in M_{a,b}} \frac{\|U-V\|_{L^2(|x|^{a+b+1})}^2}{\|U\|_{L^2(|x|^{a+b+1})}^2}$.
• Consequences: This yields stability versions for the weighted Heisenberg-Pauli-Weyl uncertainty principle and the weighted Hydrogen uncertainty principle for curl-free vector fields (Corollaries 1.4 and 1.5).

3. Stability for Second-Order Derivatives (Theorem 1.7)
Extending the stability analysis to the second-order inequality proved in Theorem 1.1, the authors show a similar stability estimate for functions $u \in Y_{a,b}(\mathbb{R}^n)$.

• Result: $\delta_{CKN}(u) \geq C \inf_{v \in N_{a,b}} \frac{\|\nabla u - \nabla v\|_{L^2(|x|^{a+b+1})}^2}{\|\nabla u\|_{L^2(|x|^{a+b+1})}^2}$.
• Consequences: This provides stability estimates for the weighted second-order Heisenberg-Pauli-Weyl and Hydrogen uncertainty principles (Corollaries 1.8 and 1.9).

Significance and Claims
The authors position this work as a significant advancement in the theory of functional inequalities:

• Generalization: Unlike previous results that were often restricted to specific parameter ranges (e.g., $b = -a-1$) or relied on methods like "expanding the square" which are difficult to generalize, this paper uses spherical harmonic decomposition to handle general parameters $a$ and $b$.
• Stability: The paper claims to provide the first stability estimates for the weighted CKN inequalities specifically for curl-free vector fields and for the second-order derivative case with general weights. Previous stability results were largely confined to the standard Sobolev inequality or specific uncertainty principles without weights or with restricted parameters.
• Methodological Rigor: The work demonstrates the power of combining spherical harmonic decomposition with one-dimensional variational problems to derive sharp constants and stability estimates, a technique the authors note is effective for studying sharp forms of functional inequalities.

The paper concludes by noting that while the constant $C$ in the stability estimates is explicitly computable, it is not necessarily optimal, leaving room for future refinement. The results recover known cases (such as the standard uncertainty principles when $a=0$) as special instances of the more general framework presented.