Sharp Fractional Hardy Inequality

• Sharp fractional Hardy inequality is a nonlocal estimate that bounds singular weighted L^p norms by fractional Dirichlet energy using the best constant obtainable in the half-space model.
• It employs ground-state representations and variational methods to derive the optimal constant, which remains independent of the specific domain.
• The theory extends to weighted, non-Euclidean, and discrete contexts, influencing Hardy–Sobolev–Maz’ya inequalities and many-body interaction studies.

The sharp fractional Hardy inequality is a nonlocal estimate that bounds a singular weighted $L^p$ term by a fractional Dirichlet-type energy with the largest admissible constant. In the Euclidean domain setting, if $\Omega\subset\mathbb R^n$ is open, $1<p<\infty$, $0$sp>1$, then for every $u\in W^{s,p}_0(\Omega)$ one has a boundary-singular inequality with weight $\operatorname{dist}(x,\partial\Omega)^{-sp}$, and the optimal constant is independent of $\Omega$ and coincides with the half-space value. In the formulation of D’Yda and Frank, this sharp Hardy term can be subtracted from the fractional energy while still leaving coercivity strong enough to control the Sobolev-critical norm, yielding a fractional Hardy–Sobolev–Maz’ya inequality with the same Hardy constant (Dyda et al., 2011).

1. Precise formulation in fractional Sobolev spaces

Let $\Omega\subset\mathbb R^n$ be open. For $u:\Omega\to\mathbb R$, the Gagliardo seminorm is

$\Omega\subset\mathbb R^n$0

and the full norm is

$\Omega\subset\mathbb R^n$1

The space $\Omega\subset\mathbb R^n$2 is the closure of $\Omega\subset\mathbb R^n$3 with respect to the Gagliardo seminorm (Dyda et al., 2011).

Under the assumptions $\Omega\subset\mathbb R^n$4, $\Omega\subset\mathbb R^n$5, and $\Omega\subset\mathbb R^n$6, the sharp fractional Hardy inequality reads

$\Omega\subset\mathbb R^n$7

Here

$\Omega\subset\mathbb R^n$8

The terminology “sharp” means that the coefficient $\Omega\subset\mathbb R^n$9 is the best, i.e. largest, constant for which the inequality holds (Dyda et al., 2011).

A closely related global Sobolev–Slobodeckiĭ formulation replaces the domain-restricted seminorm by

$1<p<\infty$0

and defines the sharp Hardy constant variationally by

$1<p<\infty$1

where $1<p<\infty$2 (Bianchi et al., 2022).

2. Exact constant and universality of the half-space value

A central feature of the Euclidean theory is that the best constant does not depend on the particular open set. By Loss–Sloane, the best constant $1<p<\infty$3 is independent of $1<p<\infty$4 and agrees with the half-space value. An explicit representation is

$1<p<\infty$5

(Dyda et al., 2011).

In the Sobolev–Slobodeckiĭ framework, Bianchi, Brasco, and Zagati identify the sharp constant on the half-space and, in the regimes where exact computation is available, on convex sets as

$1<p<\infty$6

with

$1<p<\infty$7

and, for $1<p<\infty$8,

$1<p<\infty$9

while $0Bianchi et al., 2022).

The same work computes $0Bianchi et al., 2022). This universality of the half-space constant is one of the defining structural properties of the sharp theory.

3. Ground-state representation and sharpness mechanism

The proof strategy in the domain case is built around an exact ground-state representation on the half-space $sp>1$0. For $sp>1$1 supported in $sp>1$2, one introduces

$sp>1$3

and obtains a representation in which the fractional energy minus the Hardy term is expressed through a nonnegative remainder. In this identity, the coefficient of the Hardy term is precisely $sp>1$4, which is why the argument delivers the sharp constant rather than a lower bound with loss (Dyda et al., 2011).

The passage from the half-space to balls uses an analogous “ground-state” weight

$sp>1$5

The extension from model domains to arbitrary $sp>1$6 combines this with two additional ingredients: averaging over directions through the Loss–Sloane formula, and a one-dimensional reduction based on the Garsia–Rodemich–Rumsey inequality. The key lemmas are ground-state-representation formulas on half-spaces and balls, together with a one-dimensional Hardy–Sobolev inequality with remainder. Tracking constants through all steps shows that no loss occurs in the Hardy term, hence the half-space constant remains sharp on arbitrary domains (Dyda et al., 2011).

A different but related mechanism appears in the convex-domain theory based on positive local weak supersolutions of the nonlocal $sp>1$7-Laplace equation. There one tests powers of the distance function $sp>1$8, and the optimal exponent is

$sp>1$9

This gives a direct route to the sharp constant in the regimes where the supersolution method is valid (Bianchi et al., 2022).

4. Extremals, non-attainment, and asymptotic optimizers

In bounded domains, equality is not attained by any nontrivial $u\in W^{s,p}_0(\Omega)$0. Any extremal would have to concentrate at the boundary and escape the function space. Thus the sharp constant exists, but there is no genuine maximizer in the natural energy class (Dyda et al., 2011).

What does exist is a family of extremal sequences. These sequences “blow up” near $u\in W^{s,p}_0(\Omega)$1 and asymptotically realize the half-space profile

$u\in W^{s,p}_0(\Omega)$2

On the half-space itself, equality is likewise not attained in the energy space, but approximate extremals can be described explicitly in the form

$u\in W^{s,p}_0(\Omega)$3

This non-attainment phenomenon is characteristic of sharp Hardy inequalities: the best constant is encoded by a singular boundary profile rather than by a finite-energy extremizer (Dyda et al., 2011).

In weighted nonlocal problems the same pattern persists. For singularities supported on a flat submanifold $u\in W^{s,p}_0(\Omega)$4, minimizing sequences either drift toward $u\in W^{s,p}_0(\Omega)$5 or escape to infinity in the transverse directions, and the infimum is not attained in the weighted space, even though the constant is largest possible (Kijaczko et al., 24 Mar 2025).

5. Hardy–Sobolev–Maz’ya synthesis and remainder terms

The main result of D’Yda and Frank is not only the Hardy inequality itself, but the combined fractional Hardy–Sobolev–Maz’ya estimate

$u\in W^{s,p}_0(\Omega)$6

with

$u\in W^{s,p}_0(\Omega)$7

and $u\in W^{s,p}_0(\Omega)$8 independent of $u\in W^{s,p}_0(\Omega)$9. The important point is that subtracting the sharp Hardy term still leaves an energy that controls the critical $\operatorname{dist}(x,\partial\Omega)^{-sp}$0-norm, while the Hardy coefficient itself remains the exact sharp one (Dyda et al., 2011).

Weighted versions of this principle were later established for half-spaces, convex domains, and punctured space. In the weighted half-space and convex-domain setting, if

$\operatorname{dist}(x,\partial\Omega)^{-sp}$1

then the sharp constant is

$\operatorname{dist}(x,\partial\Omega)^{-sp}$2

and for $\operatorname{dist}(x,\partial\Omega)^{-sp}$3, $\operatorname{dist}(x,\partial\Omega)^{-sp}$4, a weighted fractional Hardy–Sobolev–Maz’ya inequality follows with this same sharp Hardy constant on the left-hand side (Dyda et al., 2022).

For $\operatorname{dist}(x,\partial\Omega)^{-sp}$5, Dyda and Kijaczko proved sharp weighted fractional Hardy inequalities with remainder and corresponding Hardy–Sobolev–Maz’ya inequalities. In their ground-state decomposition, the remainder constant is

$\operatorname{dist}(x,\partial\Omega)^{-sp}$6

and when $\operatorname{dist}(x,\partial\Omega)^{-sp}$7 this can be improved to $\operatorname{dist}(x,\partial\Omega)^{-sp}$8. As $\operatorname{dist}(x,\partial\Omega)^{-sp}$9, the remainder formula becomes an identity; as $\Omega$0, the remainder vanishes (Dyda et al., 2023).

6. Convex geometry, $\Omega$1, and open-set lower bounds

The convex-domain theory reveals that sharpness is sensitive to the interaction between geometry and the parameter $\Omega$2. The supersolution method based on $\Omega$3 computes the exact constant for any convex $\Omega$4 when $\Omega$5, and for all $\Omega$6 in the case $\Omega$7. The same source exhibits a simple one-dimensional counterexample suggesting that this method cannot work for $\Omega$8 and $\Omega$9 different from a half-space. For $\Omega\subset\mathbb R^n$0 and $\Omega\subset\mathbb R^n$1, whether every convex $\Omega\subset\mathbb R^n$2 still satisfies

$\Omega\subset\mathbb R^n$3

is left as an open problem (Bianchi et al., 2022).

In the limit case $\Omega\subset\mathbb R^n$4, the sharp constant acquires a geometric interpretation. For convex non-empty $\Omega\subset\mathbb R^n$5,

$\Omega\subset\mathbb R^n$6

where

$\Omega\subset\mathbb R^n$7

Thus $\Omega\subset\mathbb R^n$8 is the Cheeger constant for the fractional perimeter and the weighted volume $\Omega\subset\mathbb R^n$9. In dimension one, if $u:\Omega\to\mathbb R$0 is an open interval of length $u:\Omega\to\mathbb R$1, then

$u:\Omega\to\mathbb R$2

and for the unit interval,

$u:\Omega\to\mathbb R$3

The same geometric approach also gives new one-dimensional lower bounds for non-convex sets, some of them optimal when $u:\Omega\to\mathbb R$4 (Bianchi et al., 2024).

For general open sets with $u:\Omega\to\mathbb R$5, the sharp Hardy constant of the punctured space $u:\Omega\to\mathbb R$6 provides an optimal lower bound for the Hardy constant $u:\Omega\to\mathbb R$7. In that regime, the proof uses positive local weak supersolutions built from powers of the distance function. The same analysis computes the limit of the punctured-space constant as $u:\Omega\to\mathbb R$8 and as $u:\Omega\to\mathbb R$9, and yields a lower bound for the nonlocal eigenvalue $\Omega\subset\mathbb R^n$00 in terms of $\Omega\subset\mathbb R^n$01 (Cinti et al., 2024).

7. Weighted, non-Euclidean, discrete, and many-body extensions

The sharp fractional Hardy inequality has developed into a broad family of exact nonlocal estimates beyond the Euclidean boundary-distance model. In the Heisenberg-group half-space

$\Omega\subset\mathbb R^n$02

Roy established

$\Omega\subset\mathbb R^n$03

for all $\Omega\subset\mathbb R^n$04, and computed the sharp constant in closed form when $\Omega\subset\mathbb R^n$05. The proof introduces a weighted nonlocal $\Omega\subset\mathbb R^n$06-Laplacian on $\Omega\subset\mathbb R^n$07, uses the weak harmonicity of $\Omega\subset\mathbb R^n$08 in the unweighted case, reduces the variational quotient to the $\Omega\subset\mathbb R^n$09-variable, and then applies one-dimensional sharp fractional Hardy results (Roy, 8 Apr 2025).

For weighted singularities on a flat submanifold $\Omega\subset\mathbb R^n$10, the sharp constant $\Omega\subset\mathbb R^n$11 is given explicitly by an integral involving

$\Omega\subset\mathbb R^n$12

and in the point-singularity case $\Omega\subset\mathbb R^n$13, $\Omega\subset\mathbb R^n$14, $\Omega\subset\mathbb R^n$15, one recovers the classical constant

$\Omega\subset\mathbb R^n$16

for

$\Omega\subset\mathbb R^n$17

The proof again relies on a Frank–Seiringer type ground-state representation and concentrating test functions (Kijaczko et al., 24 Mar 2025).

Discrete analogues exhibit the same sharpness structure in a different form. On $\Omega\subset\mathbb R^n$18, for $\Omega\subset\mathbb R^n$19, Keller and Nietschmann constructed an explicit weight $\Omega\subset\mathbb R^n$20 such that

$\Omega\subset\mathbb R^n$21

and proved that $\Omega\subset\mathbb R^n$22 is critical and null-critical, so the extremal constant is $\Omega\subset\mathbb R^n$23 in the normalization above (Keller et al., 2022). On $\Omega\subset\mathbb R^n$24, Hake, Keller, and Pogorzelski identified a one-parameter family of Hardy weights $\Omega\subset\mathbb R^n$25, showed that the threshold $\Omega\subset\mathbb R^n$26 yields the optimal weight, and obtained the sharp constant

$\Omega\subset\mathbb R^n$27

with null-criticality at $\Omega\subset\mathbb R^n$28 (Hake et al., 31 Dec 2025).

A further extension replaces one-body or boundary singularities by genuine interaction potentials. For $\Omega\subset\mathbb R^n$29 and $\Omega\subset\mathbb R^n$30, a sharp three-particle inequality holds on the collision-free configuration space $\Omega\subset\mathbb R^n$31: $\Omega\subset\mathbb R^n$32 where

$\Omega\subset\mathbb R^n$33

Here the sharp two-particle fractional Hardy constant survives unchanged, while the potential $\Omega\subset\mathbb R^n$34 captures genuine three-body effects (Mahadevan et al., 28 May 2026).

These developments show that the sharp fractional Hardy inequality is not a single isolated estimate but a stable analytic template: exact constants emerge from ground-state structure, half-space or model-geometry reductions, and variational non-attainment, and the same pattern persists across weighted settings, convex domains, noncommutative groups, discrete lattices, and interacting many-particle systems.