Fractional Hardy Inequalities

Updated 28 December 2025

Fractional Hardy inequalities are mathematical statements that link Hardy-type potentials with fractional Sobolev seminorms, capturing singular behavior and regularity.
They play a crucial role in nonlocal PDE analysis, spectral theory, and variational problems by extending classical Hardy inequalities to include weighted and submanifold cases.
This topic covers sharp constants, logarithmic corrections in critical cases, and geometric conditions that ensure optimality and applicability in complex analytical settings.

A fractional Hardy inequality is an inequality connecting a Hardy-type potential—typically singular near the boundary or a submanifold—against a nonlocal (fractional) Sobolev seminorm, quantifying the relationship between singular behavior and regularity of functions, often with sharp or optimal constants. These inequalities are fundamental in nonlinear analysis, the theory of nonlocal PDEs, and harmonic analysis, providing insight into function spaces, spectral theory, geometry, and variational problems involving nonlocal operators.

1. Core Definitions and Model Inequalities

Let $s\in(0,1)$ , $p>1$ , and $u$ a function (with appropriate vanishing or compact support conditions) in a domain $\Omega\subset\mathbb{R}^d$ or a more general measure-metric space. The prototypical (unweighted) fractional Hardy inequality in $\Omega$ states that

$\int_\Omega \frac{|u(x)|^p}{\delta(x)^{sp}} \, dx \leq C \iint_{\Omega\times\Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{d+sp}} dxdy,$

where $\delta(x)$ is the distance to the boundary $\partial\Omega$ (or, more generally, to a submanifold or singular set). The double integral is the Gagliardo seminorm defining the fractional Sobolev space $W^{s,p}(\Omega)$ . The best constant $C$ depends on $(d,s,p)$ and the geometry of $\Omega$ .

Weighted and submanifold variants appear frequently:

For a flat submanifold $K$ of codimension $k$ , $\mathrm{dist}(x,K) = |x_k|$ , the inequality becomes

$\int_{\mathbb{R}^d} \frac{|u(x)|^p}{|x_k|^{sp}} dx \leq C \iint_{\mathbb{R}^d\times\mathbb{R}^d} \frac{|u(x)-u(y)|^p}{|x-y|^{d+sp}} dxdy,$

and the critical regime $sp=k$ necessitates a logarithmic correction.

Fractional Hardy inequalities generalize local (first-order) Hardy inequalities and are intimately connected to potential theory for nonlocal operators, spectral theory, and embedding results for fractional Sobolev spaces.

2. Geometric and Analytical Conditions

The validity and sharpness of fractional Hardy inequalities depend on both analytical exponents and the geometry of the domain or singular set:

Fatness and Ahlfors/Assouad dimension conditions: Uniform $(s,p)$ -fatness (quantified using capacities or Hausdorff content) is frequently necessary and sufficient for Hardy inequalities in irregular or fractal domains (Ihnatsyeva et al., 2013, Sk, 2022). Fatness alone, however, can be insufficient without so-called “visibility” conditions or content regularity (Ihnatsyeva et al., 2013).
Visibility and John Domains: Visibility of the boundary (conditioned via curves or their accessible points) ensures the local Hardy inequality holds, not just the global variant (Ihnatsyeva et al., 2013, Dyda et al., 2017).
Capacity-density and open-endedness: A fractional capacity-density condition (in terms of sets attaining fractional capacities uniformly at small scales) characterizes the validity of pointwise and integral Hardy inequalities, and possesses a self-improvement property: if it holds for $(s,p,q)$ , it holds for values in a neighborhood (Ihnatsyeva et al., 8 Apr 2024).

On general metric spaces with doubling and reverse-doubling measure, combining boundary capacitary Poincaré inequalities with Mazʹya-type arguments yields both pointwise and localized Hardy inequalities, showing sharp dependence on boundary codimension and scaling (Dyda et al., 2021).

3. Extensions: Weighted, Submanifold, and Critical Inequalities

Fractional Hardy inequalities have been systematically extended in several directions:

Weighted Inequalities and Singular Weights: By introducing weights of the form $d(x)^{-\alpha}$ or $|x_k|^{-a}$ in both the seminorm and potential term, one obtains sharp weighted inequalities with singularities not only on the boundary but on arbitrary flat submanifolds. The optimal parameter regime and exact constants were analyzed in detail in (Dyda et al., 2022, Sahu, 19 Sep 2024, Kijaczko et al., 24 Mar 2025).
Critical Cases and Logarithmic Corrections: When the homogeneity exponent $sp$ matches the codimension $k$ (i.e., $sp=k$ ), the classical power-law potential must be replaced by a potential with an optimal logarithmic correction. Such corrections are shown to be sharp via explicit construction of near-extremal functions (Adimurthi et al., 15 Jul 2024, Sahu, 19 Sep 2024).
Supercritical Regimes ( $sp>n$ ): In the supercritical regime, fractional Hardy inequalities hold without further geometric conditions: every proper open subset of $\mathbb{R}^n$ admits the inequality (Sk, 2022). In this case, the optimal constant depends only on exponents and dimension.

4. Best Constants and Sharpness

Sharp constants in fractional Hardy inequalities are of fundamental significance for applications to spectral theory and extremal problems:

Spectral Methods and Ground-State Representations: The extremal value for the inequality is often computed using ground-state transform, semigroup representation, or explicit harmonic analysis, e.g., as in the works of Frank–Lieb–Seiringer (Ciaurri et al., 2016, Dyda et al., 2022, Aldovardi et al., 2022), and for Dunkl and Hermite operators via $h$ -harmonic expansion.
Remainder Terms: For $p\geq 2$ , exact ground-state splitting yields not only the optimal constant but also an explicit remainder (deficit) term, yielding fractional Hardy–Sobolev–Mazʹya inequalities (Kijaczko et al., 24 Mar 2025, Dyda et al., 2022). The strict positivity (for nontrivial $u$ ) of the remainder quantifies the gap to attainability.
Limit Cases: Optimal constants recover the sharpest known local (first-order) Hardy constants as $s\uparrow 1$ , and degenerate to explicit multiplicative constants as $s\downarrow 0$ (Brasco et al., 2018, Dyda et al., 2022).

5. Broader Settings: Metric Measure Spaces and Non-Euclidean Structures

Fractional Hardy inequalities extend to a wide class of analytic and geometric settings:

Metric Measure Spaces with Doubling (Reverse-Doubling) Measures: The inequalities are formulated via metric balls, capacities, and Poincaré inequalities, reflecting the interplay between measure growth, local geometry, and function space properties (Dyda et al., 2013, Dyda et al., 2021, Kassymov et al., 21 Jul 2024, Ihnatsyeva et al., 8 Apr 2024).
Homogeneous Groups and Non-Euclidean Structures: Homogeneous Lie groups and the Heisenberg group admit such inequalities via polar decomposition, with explicit constants dependent on the homogeneous dimension and quasi-norm (Kassymov et al., 10 Oct 2024, Kassymov et al., 21 Jul 2024).
Fractional Hardy–Rellich Inequalities: Higher-order variants (fractional Hardy–Rellich) are obtained for powers of the Laplacian, often using nonlocal Pohozaev or integration-by-parts identities. These yield explicit integral representations and relate to spectral properties of fractional Laplacians (Nitti et al., 2023).

Fractional Hardy inequalities are central to several related themes:

Fractional Hardy–Sobolev–Mazʹya Inequalities: The Hardy deficit controls an additional Sobolev hot term, interpolating between Hardy and Sobolev embeddings. Such inequalities are sharp on half-spaces, convex domains, and John domains (possibly weighted), with sharp constants and explicit range of exponents (Filippas et al., 2011, Dyda et al., 2017, Dyda et al., 2022, Kijaczko et al., 24 Mar 2025).
Trace Inequalities: Trace Hardy inequalities describe boundary behavior of nonlocal Dirichlet or Neumann problems, with explicit constants in terms of the extension method (Caffarelli–Silvestre) and geometric (convexity/mean-convexity) conditions (Filippas et al., 2011).
Spectral Theory, Uncertainty Principles: Fractional Hardy inequalities underpin uncertainty principles for fractional Laplacians and influence spectral gaps, heat kernel estimates, and stability in nonlocal evolution equations (Kassymov et al., 10 Oct 2024, Keller et al., 2022).
Extension Operators and Removability: The existence and boundedness of extension operators for fractional Sobolev spaces hinges on the validity of sharp Hardy inequalities, as does the removability theory for singular sets (Ihnatsyeva et al., 2013).

7. Open Problems and Research Directions

Key unresolved questions and recent advances include:

Optimal Constants in General Settings: For general Orlicz and variable exponent spaces, the precise sharp constant remains unknown except in special cases (Salort, 2020). Higher-dimensional, anisotropic, or double-phase variants are largely open.
Necessity and Sufficiency of Geometric Conditions: Full characterization of the minimal geometric (fatness, visibility, accessibility) requirements for Hardy inequalities on irregular or fractal domains is an active area (Ihnatsyeva et al., 2013, Ihnatsyeva et al., 8 Apr 2024).
Extensions to Nonlocal Operators: Ongoing work addresses fractional Hardy inequalities for Dunkl-Laplacians, harmonic oscillators, graphs and quantum graphs, as well as for nonlocal Bregman–type forms (Ciaurri et al., 2016, Kijaczko et al., 2021, Keller et al., 2022).
Boundary Trace and Nonlocal Boundary Regularity: The interaction of fractional Hardy potentials with nonlocal boundary data, trace spaces, and boundary regularity conditions is central for nonlocal PDE and free boundary problems (Filippas et al., 2011, Brasco et al., 2018).
Logarithmic Improvements and Criticality: The necessity and sharpness of logarithmic corrections in the critical case ( $sp=k$ for codimension $k$ singularities) are established, but the full spectrum of endpoint behaviors in various geometries is incompletely understood (Adimurthi et al., 15 Jul 2024, Kijaczko et al., 24 Mar 2025, Sahu, 19 Sep 2024).