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Critical fractional Hardy inequalities

This paper provides a roadmap for studying critical fractional Hardy inequalities for Dirichlet and Sobolev–Bregman forms on the real line, specifically proving criticality results for the fractional Laplacian and the Servadei–Valdinoci form.

Original authors: Krzysztof Bogdan, Bartłomiej Dyda, Antoni Szczukiewicz

Published 2026-08-26

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CC BY 4.0

Original authors: Krzysztof Bogdan, Bartłomiej Dyda, Antoni Szczukiewicz

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

In the vast landscape of mathematics, there are rules that govern how things spread, balance, and settle. Imagine a system where a value at one point is not determined solely by its immediate neighbors, but is influenced by the entire landscape around it, reaching out across distances. This is the world of nonlocal interactions, a concept that appears in physics when modeling how particles jump or how heat diffuses in complex materials. Within this world, mathematicians study inequalities—fundamental rules that set limits on how much energy a system can hold or how sharply a value can change. One of the most famous of these rules is the Hardy inequality. It acts like a safety net, ensuring that a function does not become too large too quickly near a specific point, usually the center of the system. For decades, researchers have known the exact limits of this safety net for simple, local systems, but when the interactions become long-range and fractional, the rules become much harder to pin down.

The question that drives this new work is whether we can find the sharpest possible version of this safety net for these complex, long-range systems, and specifically, whether we can balance the rules on the left and right sides of the number line independently. In the standard version of this problem, the rules are often symmetric, treating the positive and negative sides of the number line exactly the same. However, real-world systems are rarely so perfectly balanced. The researchers wanted to know if they could adjust the rules to favor one side over the other, perhaps making the safety net tighter on the left while loosening it on the right, without breaking the fundamental laws of the system. They also wanted to understand what happens when the system is pushed to its absolute limit, a state where the safety net is so tight that it cannot be made any stronger without causing the system to collapse.

To answer these questions, the authors developed a new roadmap for analyzing these fractional systems. They focused on a specific type of mathematical form that measures the energy of a system based on the differences between values at different points. By introducing a clever mathematical tool that acts like a lens, they were able to rewrite the energy of the system in a way that separates the unavoidable minimum energy from the extra energy caused by the shape of the function. This separation allowed them to identify the exact weight, or the strength of the safety net, that makes the inequality as tight as possible. They found that for these systems, there is indeed a critical point where the weight cannot be increased any further. If you try to make the rule stricter, the inequality fails.

A major discovery in this work is that the balance between the two sides of the number line is not fixed. The researchers showed that by adjusting a specific parameter, they could shift the strength of the safety net from one side to the other. On one side, the rule could be made very strict, while on the other, it could be relaxed, or even become negative, meaning the system behaves in a counterintuitive way where the usual limits are inverted. This flexibility is crucial because it allows the mathematical model to fit a much wider variety of real-world scenarios where symmetry is broken. They proved that for a wide class of these systems, there is always a specific setting where the safety net is perfectly balanced and symmetric, but they also showed that this is just one of many possible configurations.

The paper also addresses a specific, well-known form of this problem that has been used in recent years to study partial differential equations on restricted domains. The authors confirmed that the limits previously proposed for this form were indeed the sharpest possible, but they went further by showing that the rules could be improved by adding an extra term that accounts for the asymmetry of the function being studied. This means that if a function is not perfectly balanced between the positive and negative sides, the system has even more room to hold energy than previously thought. The researchers demonstrated that this extra term is always non-negative, providing a more precise lower bound for the energy of the system.

Throughout their investigation, the team worked with functions that can be very rough or irregular, ensuring their results hold true even for the most chaotic inputs. They relied on rigorous proofs rather than simulations, establishing that their findings are absolute truths within the mathematical framework they defined. They also clarified the role of the reference measure, which is the standard against which the size of the functions is judged. They showed that using a specific weight that decays as you move away from the center is the most natural and effective way to define the size of these functions, allowing their results to apply to a broader range of parameters than previous studies.

In the end, this work provides a complete guide for understanding the limits of these fractional inequalities. It resolves long-standing questions about the one-dimensional case, proving that the critical weights are indeed the best possible and showing exactly how they can be tuned. The findings offer a new level of precision for mathematicians and physicists working with nonlocal operators, giving them a reliable toolkit to analyze systems where long-range interactions play a dominant role. By mapping out the exact boundaries of what is possible, the authors have turned a complex and abstract problem into a clear and navigable landscape, revealing the hidden flexibility within the rigid laws of mathematical inequality.

Technical Summary: Critical Fractional Hardy Inequalities

Problem Statement
The paper addresses the study of critical fractional Hardy inequalities, specifically focusing on the one-dimensional case for Dirichlet and Sobolev–Bregman forms. The central problem involves establishing ground state representations for quadratic and non-quadratic forms associated with symmetric kernels $k(x,y)$ that are homogeneous of order $-\alpha-1$ and bounded by the standard fractional kernel $\nu_\alpha(x,y) = |x-y|^{-\alpha-1}$ .

While sharp Hardy inequalities and ground state representations are known for the fractional Laplacian in $L^2$ (quadratic case) and for specific domains, this work seeks to:

Extend ground state representations to general symmetric kernels $k \in \mathcal{K}_\alpha$ and to $L^p$ settings ( $p \in (1, \infty)$ ) via Sobolev–Bregman forms.
Determine the criticality of the resulting Hardy weights, meaning proving that the weights cannot be increased on any set of positive Lebesgue measure.
Resolve specific cases of interest, including the Servadei–Valdinoci form (integration over $x>0$ or $y>0$ ) and the balancing of weights on the positive and negative half-lines.

Methodology
The authors employ a combination of algebraic identities, integral analysis, and functional analytic techniques:

Ground State Representation: The core methodology relies on a generalized "Doob conditioning" (or $h$ -transformation). For a function $u$ , the authors substitute $u = v \cdot h$ , where $h$ is a specific ground state candidate. They utilize an algebraic identity (Equation 32) for the Bregman divergence $F_p$ to decompose the Sobolev–Bregman form $E_{k,p}[u]$ into a weighted $L^p$ term and a non-negative remainder term.
Kernel Analysis: The kernels $k$ are assumed to be symmetric, homogeneous of degree $-\alpha-1$ , and bounded by $\nu_\alpha$ . The authors define specific constants ( $\gamma_\pm, C_\pm, D_\pm$ ) derived from integrals of the kernel against power functions to characterize the Hardy weights.
Criticality Proofs: To prove that a weight is critical, the authors construct a sequence of functions $v_n$ (approximating the constant function 1 on compact sets) and demonstrate that the ratio of the form to the weighted integral converges to zero. This involves establishing the boundedness of these sequences in a specific Hilbert space $H_{\alpha,p}$ and using weak convergence arguments (Banach–Alaoglu and Mazur's lemma) to show that any improvement to the weight would violate the inequality.
Balancing and Optimization: The paper introduces an asymmetry parameter $s$ and an exponent $\beta$ to balance the Hardy weights on the positive and negative half-lines. By optimizing over $s$ , the authors derive "improved" critical Hardy inequalities that include an additional non-negative term quantifying the asymmetry of the function $u$ .

Key Contributions and Results

General Ground State Representation (Theorem 3.1):
The paper establishes an exact identity for the Sobolev–Bregman form $E_{k,p}[u]$ for $p \in (1, \infty)$ and general kernels $k \in \mathcal{K}_\alpha$ . The representation expresses the form as the sum of a weighted $L^p$ norm and a non-negative remainder involving the ground state $h_{\beta,s}$ . This generalizes previous results known only for the fractional Laplacian or specific $p=2$ cases.
Criticality of Hardy Weights (Theorem 3.3):
The authors prove that the weights $W_{k,p,s}$ derived from the ground state representation are critical. Specifically, for any $s \in (0, \infty)$ , the weight cannot be increased on any set of positive measure without violating the inequality. This resolves the criticality question for a broad class of kernels in the $L^p$ setting.
Balancing and Asymmetry (Theorem 3.5 & Corollary 3.8):
The study reveals that for irreducible kernels, one can balance the Hardy weights on the positive ( $x>0$ ) and negative ( $x<0$ ) half-lines by adjusting the parameter $s$ .
- There exists a specific $s_0$ where the weights become symmetric ( $D_+ = D_-$ ), yielding a weight of the form $D|x|^{-\alpha}$ .
- The paper provides an "improved" critical Hardy inequality (Corollary 3.8) which adds a non-negative term $G_{k,p}(u)$ to the right-hand side. This term vanishes if $u$ is symmetric but is strictly positive if $u$ is asymmetric, offering a sharper bound even in the critical case.
Resolution of the Servadei–Valdinoci Form (Section 5.2):
The paper applies the general theory to the Servadei–Valdinoci form, which integrates over the domain where $x>0$ or $y>0$ . The authors verify the sharpness of previously conjectured constants and explicitly construct symmetric critical Hardy weights for this form, resolving questions regarding the optimality of constants found in prior literature (e.g., Bogdan et al. [8]).
Role of the Reference Measure (Section 5.3):
The authors clarify the necessity of using the weighted measure $|x|^{-\alpha}dx$ as the reference space. They demonstrate that while results can be transferred to the unweighted $L^2(dx)$ space, this imposes a restriction $\alpha < 1$ . The weighted space allows the results to hold for the full range $\alpha \in (0, 2)$ .

Significance and Claims
The paper positions itself as providing a "roadmap" for the study of critical fractional Hardy inequalities. Its primary significance lies in:

Generality: Extending ground state representations and criticality results from the specific case of the fractional Laplacian ( $p=2$ ) to general symmetric kernels and the full range of $p \in (1, \infty)$ (Sobolev–Bregman forms).
Exactness: Unlike some previous works where ground state representations for $p \neq 2$ were inequalities, this paper provides exact identities for all $p \in (1, \infty)$ .
Criticality: Providing rigorous proofs of criticality for these generalized forms, a topic noted as a matter of recent progress in the literature.
Unification: Unifying the treatment of the quadratic case ( $p=2$ ) and the general $p$ -case, showing that the latter is not merely a consequence of the former but requires distinct algebraic handling (via the Bregman divergence).

The authors explicitly state that while the results are definitive for the class of kernels considered, extensions to higher dimensions or other kernel classes remain open problems to be addressed in future work following the outlined roadmap.

Technical Summary: Critical Fractional Hardy Inequalities

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