Fractional Hardy Inequality Overview

• Fractional Hardy inequality is a fundamental analytic estimate that relates weighted Lp-norms to the energy of nonlocal differential operators.
• It establishes sharp optimal constants and critical weights by considering the influence of ambient geometry, irregular domains, and discrete settings.
• Methodologies based on ground state representations, capacity criteria, and spectral decompositions provide robust insights into nonlocal and fractal boundary behaviors.

The fractional Hardy inequality is a fundamental analytic estimate relating weighted $L^p$-norms of a function by the singularity of its support to the energy associated with a nonlocal, typically fractional, differential (or difference) operator. The precise structure and validity of such inequalities depend crucially on the ambient geometric setting, the regularity and geometry of the domain or underlying space, and, on discrete structures, on the specific features of the operator. Recent advances encompass sharp constants, optimality, characterizations, and generalizations to irregular domains, non-Euclidean contexts, and discrete settings. Below the principal variants, sharp forms, and the role of geometric and analytic features are summarized with rigorous detail and reference to major works.

1. Fundamental Forms and General Framework

The archetypal Euclidean fractional Hardy inequality asserts, for $0 < s < 1$, $1 < p < \infty$, and $sp < d$, that

$$\int_{\mathbb{R}^d} \frac{|u(x)|^p}{|x|^{sp}} \, dx \leq C \iint_{\mathbb{R}^d \times \mathbb{R}^d} \frac{|u(x)-u(y)|^p}{|x-y|^{d+sp}} \, dx\,dy \quad \forall u \in C_c^\infty(\mathbb{R}^d).$$

Here, the right side is the Gagliardo seminorm for the homogeneous fractional Sobolev space $W^{s,p}(\mathbb{R}^d)$, corresponding to the quadratic form of the fractional Laplacian $(-\Delta)^{s/2}$ (Aldovardi et al., 2022).

A unifying abstract framework treats metric measure spaces $(X, \rho, \mu)$ with Dirichlet forms of the type

$$\mathcal{E}[u] = \iint_{D \times D} |u(x) - u(y)|^p K(\rho(x,y)) \, \mu(dx) \, \mu(dy),$$

where $K$ is a suitable symmetric kernel and $D$ is an open subset of $X$ (Dyda et al., 2013). The general fractional Hardy inequality then reads

$$\int_{D} |u(x)|^p \phi(\delta(x)) \, dx \leq C \iint_{D \times D} |u(x) - u(y)|^p \phi(\rho(x, y)) \frac{dx\,dy}{\rho(x, y)^d},$$

with $\delta(x) = \dist(x, X \setminus D)$ and $\phi$ a regularly varying function (possibly a power), under thin/fat boundary and scaling conditions. This framework subsumes a wide class of nonlocal and weighted Hardy inequalities on irregular domains.

2. Optimal Constants and Critical Weights

The classical question of optimality seeks the largest admissible constant and, where relevant, the most singular (least decaying) weight such that the inequality holds. For the standard fractional Laplacian $(-\Delta)^{s/2}$ on $\mathbb{R}^d$, the best constant and the weight $|x|^{-sp}$ are optimal in the sense that any increase in the weight or constant causes the inequality to fail at infinity or near the origin (Kassymov et al., 2024, Aldovardi et al., 2022).

On the discrete half-line $\mathbb{N}$, Das & de la Fuente-Fernández established the optimal Hardy weight for the fractional Laplacian $(-\Delta_{\mathbb{N}})^\sigma$ ($\sigma \in (0,1]$), given by

$$W^{op}_\sigma(n) = 4^\sigma \frac{\Gamma\bigl(\frac{3+2\sigma}{4}\bigr)^2}{\Gamma\bigl(\frac{3-2\sigma}{4}\bigr)^2} \cdot \frac{\Gamma(n - \frac{1+2\sigma}{4}) \Gamma(n + \frac{5-2\sigma}{4})} {\Gamma(n + \frac{-1+2\sigma}{4}) \Gamma(n + \frac{5+2\sigma}{4})},$$

such that for all $f : \mathbb N \to \mathbb R$ with finite support and $f(0) = 0$,

$$\langle (-\Delta_{\mathbb{N}})^\sigma f, f\rangle_{\ell^2} \geq \sum_{n = 1}^\infty W^{op}_\sigma(n)\, |f(n)|^2.$$

This weight is critical (unique minimal ground state), null-critical (ground state not in the weighted $\ell^2$), and optimal in the sense that any strict increase invalidates the inequality even at large $n$ (Das et al., 9 Jul 2025). Analogous results for the lattice $\mathbb{Z}^n$ are formulated via discrete Riesz kernels, yielding a family of Hardy weights $w_{\sigma, \alpha}(x)$ parameterized by $\alpha$, with optimal threshold at $\alpha_0 = (n/2 + \sigma)/2$ (Hake et al., 31 Dec 2025).

3. Dependence on Geometry: Boundary Thickness, Fatness, and Visibility

The validity of the fractional Hardy inequality on a domain $G \subset \mathbb{R}^n$ is governed by fine boundary geometry. For open sets, fatness (in terms of Riesz capacity or Hausdorff content) is necessary but not sufficient (Ihnatsyeva et al., 2013). The essential sufficient condition is a conjunction of $(s,p)$-uniform fatness and a visibility property: for each $x \in G$, the "visual boundary" of $G$ from $x$ (measured by Hausdorff content in a John domain setting) must be sufficiently large relative to $\delta(x)$. This leads to a sharp characterization: $G$ admits the inequality

$$\int_G \frac{|u(x)|^p}{\delta(x)^{sp}}\, dx \leq C \iint_{G \times G} \frac{|u(x) - u(y)|^p}{|x-y|^{n+sp}}\, dx\,dy,$$

if and only if the boundary visible from $x$ satisfies a uniform lower bound in Hausdorff content of dimension $\lambda > n - sp$ (Ihnatsyeva et al., 2013). Mere fatness is shown, by explicit counterexamples, not to suffice in the fractional case.

4. Characterizations, Capacitary Criteria, and Self-Improvement

An influential result is the Maz'ya-type capacity characterization: a bounded domain $G$ supports the fractional Hardy inequality if and only if the fractional $(s,p)$-capacity is quasi-additive with respect to Whitney cubes and the zero extension operator from $C_c(G)$ to $W^{s,p}(\mathbb{R}^n)$ is bounded (Dyda et al., 2013). This capacity-theoretic approach emphasizes the role of local geometric and analytic structure and leads to robustness under null-set modifications and removability results.

Recent progress (Ihnatsyeva et al., 2024) establishes the equivalence of the pointwise fractional Hardy inequality (generalized Hardy-Poincaré in terms of metric balls and maximal functions), the integral (Gagliardo) Hardy inequality, and a fractional capacity density condition. Critical is the self-improvement property: if the capacity density holds for some $(s,p,q)$, it automatically holds in a full neighborhood of parameters, including $q$, reflecting deep stability.

5. Fractional Hardy on Convex, Uniform, and Fractal Domains

On open convex sets $K \subset \mathbb{R}^N$, the fractional Hardy inequality holds with controlled constant,

$$\mathcal{C}\, s(1-s) \int_K \frac{|u(x)|^p}{d_K(x)^{sp}} dx \leq [u]_{W^{s,p}(\mathbb R^N)}^p,$$

where $d_K(x)$ is the distance to the boundary and $\mathcal{C}$ is dimension- and exponent-dependent but stable as $s \nearrow 1$ (Brasco et al., 2018). The key analytic fact is that $d_K^s$ is nonlocally superharmonic.

In domains possessing only partial regularity or exhibiting fractal boundaries, the general framework of (Dyda et al., 2013) applies: the validity of the fractional Hardy inequality is regulated by the (upper or lower) Assouad dimension of the boundary in comparison to the scaling exponent of the weight.

For boundaries of codimension $k \geq 2$, the precise threshold is $sp = k$: in the critical case, a sharp logarithmic correction is needed; for $sp < k$ (subcritical), standard power weights suffice; and for $sp > k$ (supercritical), the weight can be further optimized via dimension shift phenomena (Adimurthi et al., 2024).

6. Extensions to Non-Euclidean and Discrete Settings

Fractional Hardy inequalities are established in wide generality on homogeneous Lie groups $G$ with homogeneous quasi-norm $|\cdot|$ and dimension $Q$. In the complementary ("supercritical") range $sp > Q$, for $u \in C_c(G \setminus \{e\})$,

$$\int_G \frac{|u(x)|^p}{|x|^{sp}}\, dx \leq C_H \iint_{G \times G} \frac{|u(x)-u(y)|^p}{|y^{-1}x|^{Q+sp}}\, dx\,dy,$$

with explicit constants depending on the group structure (Kassymov et al., 2024, Kassymov et al., 2024). This unifies known results on Euclidean spaces, Carnot groups, and the Heisenberg group, and relies on abstract polar decomposition and fractional integral inequalities.

On discrete structures, especially lattices $\mathbb{Z}^d$ and half-lines, both the form of the operator (as a graph Laplacian or difference operator) and the geometry (boundary or bulk, lattice dimension) impact the decay rate and optimal constants of the Hardy weight. Elementary dyadic shell arguments, discrete spectral calculus, and construction of ground state sequences underlie these results (Dyda, 9 Jun 2025, Keller et al., 2022, Hake et al., 31 Dec 2025, Das et al., 9 Jul 2025).

7. Methodological Principles and Proof Techniques

Sharpness and optimality proofs rest on ground state representations, spectral decomposition, and the construction of positive super-solutions. The Frank-Lieb-Seiringer abstract ground state method has been implemented across various settings (Euclidean, operator-theoretic, discrete, Lie group) and underpins the ability to track or fully compute best constants (Ciaurri et al., 2016, Dyda et al., 2022).

On domains, decompositions via Whitney cubes, layer-cake methods, interpolation, and capacity estimates yield control over the interplay between boundary geometry and functional inequalities (Dyda et al., 2013, Ihnatsyeva et al., 2013). Weighted and singular variants—including those with weights singular along submanifolds or at intersections of boundaries—are accessible along these lines (Adimurthi et al., 2024).

A final class of methods exploits rearrangement and Sobolev embeddings (Savin-Valdinoci, Michael–Simon–Sobolev framework) to interpolate between Hardy and Sobolev inequalities directly, and to derive optimal or universal constants that are robust with respect to convexity or perturbation of the domain (Csató et al., 2024).

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In summary, the theory of fractional Hardy inequalities articulates how singular potential-type terms are controlled by nonlocal energy forms, quantifies the influence of geometric and analytic features (including optimal constants), and extends with full technical rigor to discrete, non-Euclidean, and low-regularity settings. This landscape continues to expand with new criticality phenomena, capacity-theoretic advances, and specialized inequalities adapted to singular substructures and operator theory.