An extension of a critical Hardy--Rellich inequality: explicit constants and the sharp weight range

Technical Summary: An Extension of a Critical Hardy–Rellich Inequality

Problem Statement
This paper addresses the critical Hardy–Rellich inequalities recently established by Castro [1], specifically focusing on the case where the exponent $p$ equals the dimension $N$ ($p=N$). While classical Hardy and Rellich inequalities fail at the endpoint $p=N$ without logarithmic corrections, Castro proved a critical inequality involving the quantity $\nabla(u/|x|)$ and the Hessian or Laplacian of $u$. However, two significant limitations remained in Castro's results:

1. The constants derived were not explicit.
2. The range of the weight exponent $a$ in the weighted inequalities was restricted to $a < 1$, which the authors identify as far from optimal.

The central open problem addressed is the determination of the optimal constants and the sharp range of the weight exponent $a$ for which these inequalities hold, particularly for the case $p=N$.

Methodology
The authors recast the problem using the Emden–Fowler transformation, introducing the variable $t = \log r$ (where $r = |x|$). This transformation allows the radial and angular components of the inequality to be analyzed through a single, classical one-dimensional weighted Hardy inequality.

Key methodological steps include:

• Decomposition: The gradient $\nabla(u/|x|)$ is decomposed into radial and tangential (spherical) components. The authors utilize pointwise bounds relating these components to the Hessian $|D^2u|$.
• One-Dimensional Reduction: By setting $f(t) = \phi(e^t)$, the problem reduces to applying the sharp weighted Hardy inequality in the form:
  $$\int_{\mathbb{R}} |f(t)|^N e^{\mu t} dt \leq \left( \frac{N}{|\mu|} \right)^N \int_{\mathbb{R}} |f'(t)|^N e^{\mu t} dt$$
  This replaces the auxiliary boundedness results for integral operators used in [1], which relied on Jensen's inequality.
• Convexity Splitting: For $N \geq 2$, the authors employ a convexity inequality $(A+B)^{N/2} \leq 2^{N/2-1}(A^{N/2} + B^{N/2})$ to decouple the radial and angular contributions, estimating them separately before combining the results.
• Counterexample Construction: To prove the sharpness of the range, the authors construct a sequence of radial functions in the Emden–Fowler variable that violates the inequality at the critical exponent $a=N$.

Key Contributions and Results

1. Extension of the Weight Range:
   The paper proves that the weighted inequality holds for all $a \neq N$, significantly extending the previous range of $a < 1$. Specifically, for $N \geq 2$ and $a \neq N$, the following inequality is established for $u \in C^\infty_c(\mathbb{R}^N \setminus \{0\})$:
   $$\int_{\mathbb{R}^N} \left| \nabla \left( \frac{u(x)}{|x|} \right) \right|^N |x|^a dx \leq C_{N,a} \int_{\mathbb{R}^N} |D^2 u|^N |x|^a dx$$
   The inequality fails if and only if $a = N$.

2. Explicit Constants:
   The authors derive an explicit constant $C_{N,a}$ for $N \geq 2$ and $a \notin \{0, N\}$:
   $$C_{N,a} = 2^{N/2-1} \left[ \left( \frac{N}{|a|} \right)^N + \left( \frac{N}{|N-a|} \right)^N \right]$$
   For the one-dimensional case ($N=1$), the sharp constant is $1/|1-a|$. The authors note that while this constant is explicit, it is not claimed to be the optimal (best) constant due to the convexity step used to separate radial and angular terms.

3. Sharpness of the Range:
   The paper establishes that $a=N$ is the unique critical value. At this value, the inequality fails even on the subspace of radial functions. This failure is attributed to the vanishing of the exponential weight in the equivalent one-dimensional problem, analogous to the failure of the classical Hardy inequality at $p=N$.

4. Extension to the Laplacian (Muckenhoupt Range):
   By combining the main result with the weighted Calderón–Zygmund inequality, the authors extend the result to the Laplacian formulation ($\Delta u$) for the Muckenhoupt range $-N < a < N(N-1)$ (with $a \neq N$). This provides a partial solution to Open Problem 1 in [1].

Significance and Scope
The paper claims to provide a partial solution to the open problems regarding the critical Hardy–Rellich inequalities posed in [1]. The primary significance lies in:

• Removing Artificial Restrictions: The use of the sharp weighted Hardy inequality, rather than Jensen-based estimates, removes the artificial constraint $a < 1$, revealing that the inequality holds for all $a > N$ as well.
• Explicitness: Providing explicit constants where previous results were non-constructive.
• Clarifying Criticality: Precisely identifying $a=N$ as the unique critical exponent where the inequality breaks down.

Limitations and Open Problems
The authors remain modest regarding the optimality of the derived constants, noting that the factor $2^{N/2-1}$ arises from a decoupling step that does not simultaneously extremize radial and angular contributions. Consequently, determining the best (optimal) constants for the general case remains an open problem (Open Problem 1 of [1]).

Furthermore, the paper does not resolve the fractional analogues (Open Problems 2 and 3 of [1]), noting that the current argument relies heavily on the structure of the second derivative $\partial_{rr}$, which lacks a direct nonlocal substitute for fractional operators. Finally, the validity of the Laplacian formulation (1.6) outside the Muckenhoupt range ($a \leq -N$ or $a \geq N(N-1)$) remains open.