---
title: Sharp Fractional Hardy Inequality
url: https://www.emergentmind.com/topics/sharp-fractional-hardy-inequality
type: topic
---

# Sharp Fractional Hardy Inequality

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, \(1<p<\infty\), \(0<s<1\), and \(sp>1\), 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 [1109.6570].

## 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
\[
[u]_{W^{s,p}(\Omega)}^p
=
\iint_{\Omega\times\Omega}
\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}
\,dx\,dy,
\]
and the full norm is
\[
\|u\|_{W^{s,p}(\Omega)}^p
=
\|u\|_{L^p(\Omega)}^p+[u]_{W^{s,p}(\Omega)}^p.
\]
The space \(W^{s,p}_0(\Omega)\) is the closure of \(C_c^\infty(\Omega)\) with respect to the Gagliardo seminorm [1109.6570].

Under the assumptions \(1<p<\infty\), \(0<s<1\), and \(sp>1\), the sharp fractional Hardy inequality reads
\[
\int_{\Omega} |u(x)|^p\,\operatorname{dist}(x,\partial\Omega)^{-sp}\,dx
\le
D_{n,p,s}
\iint_{\Omega\times\Omega}
\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}
\,dx\,dy,
\qquad u\in W^{s,p}_0(\Omega).
\]
Here
\[
\operatorname{dist}(x,\partial\Omega)=\inf\{|x-z|:z\notin\Omega\}.
\]
The terminology “sharp” means that the coefficient \(D_{n,p,s}\) is the best, i.e. largest, constant for which the inequality holds [1109.6570].

A closely related global Sobolev–Slobodeckiĭ formulation replaces the domain-restricted seminorm by
\[
[u]_{W^{s,p}(\mathbb R^n)}^p
=
\iint_{\mathbb R^n\times\mathbb R^n}
\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}
\,dx\,dy
\]
and defines the sharp Hardy constant variationally by
\[
\mathfrak h_{s,p}(\Omega)
:=
\inf\Bigl\{
[u]_{W^{s,p}(\mathbb R^n)}^p:
\int_\Omega |u|^p/d_\Omega^{sp}=1,\ u\in C_0^\infty(\Omega)
\Bigr\},
\]
where \(d_\Omega(x)=\operatorname{dist}(x,\partial\Omega)\) [2209.03012].

## 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 \(D_{n,p,s}\) is independent of \(\Omega\) and agrees with the half-space value. An explicit representation is
\[
D_{n,p,s}
=
2\int_{0}^{1}
r^{\,n-1}\bigl(1-r^{(ps-1)/p}\bigr)^p(1-r)^{-1-ps}\,dr
\]
[1109.6570].

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
\[
\mathfrak h_{s,p}(\Omega)
=
\mathfrak h_{s,p}(\mathbb H^n_+)
=:
C_{n,sp}\,\Lambda_{s,p},
\]
with
\[
\Lambda_{s,p}
=
2\int_0^1
\frac{|1-t^{(sp-1)/p}|^p}{(1-t)^{1+sp}}\,dt
+
\frac{2}{sp},
\]
and, for \(n\ge2\),
\[
C_{n,sp}
=
(n-1)\,\omega_{n-1}\,\mathcal I(n-2;sp),
\qquad
\mathcal I(k;\alpha)
=
\int_0^\infty t^k(1+t^2)^{-(k+2+\alpha)/2}\,dt,
\]
while \(C_{1,sp}=1\) [2209.03012].

The same work computes \(\mathfrak h_{s,p}(\Omega)\) exactly in three regimes: the half-space \(\Omega=\mathbb H^n_+\) for all \(1<p<\infty\), \(0<s<1\); any convex \(\Omega\) whenever \(sp\ge1\); and, in the Hilbertian case \(p=2\), every convex \(\Omega\) for the whole range \(0<s<1\) [2209.03012]. 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 \(\mathbb R^n_+\). For \(u\) supported in \(\{x_n>0\}\), one introduces
\[
v(x)=x_n^{-(ps-1)/p}u(x),
\]
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 \(D_{n,p,s}\), which is why the argument delivers the sharp constant rather than a lower bound with loss [1109.6570].

The passage from the half-space to balls uses an analogous “ground-state” weight
\[
w(x)=(1-|x|^2)^{(ps-1)/p}.
\]
The extension from model domains to arbitrary \(\Omega\) 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 [1109.6570].

A different but related mechanism appears in the convex-domain theory based on positive local weak supersolutions of the nonlocal \(p\)-Laplace equation. There one tests powers of the distance function \(d_\Omega^\beta\), and the optimal exponent is
\[
\beta=\frac{sp-1}{p}.
\]
This gives a direct route to the sharp constant in the regimes where the supersolution method is valid [2209.03012].

## 4. Extremals, non-attainment, and asymptotic optimizers

In bounded domains, equality is not attained by any nontrivial \(u\in W^{s,p}_0(\Omega)\). 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 [1109.6570].

What does exist is a family of extremal sequences. These sequences “blow up” near \(\partial\Omega\) and asymptotically realize the half-space profile
\[
u(x)=x_n^{(ps-1)/p}\phi(x).
\]
On the half-space itself, equality is likewise not attained in the energy space, but approximate extremals can be described explicitly in the form
\[
u_\varepsilon(x)
=
x_n^{(ps-1)/p}\,\chi_{x_n>\varepsilon}\,\psi(x').
\]
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 [1109.6570].

In weighted nonlocal problems the same pattern persists. For singularities supported on a flat submanifold \(K=\{x_k=0\}\subset\mathbb R^d\), minimizing sequences either drift toward \(K\) 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 [2503.19057].

## 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
\[
\iint_{\Omega\times\Omega}
\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}
\,dx\,dy
-
D_{n,p,s}
\int_{\Omega}
\frac{|u(x)|^p}{\operatorname{dist}(x,\partial\Omega)^{ps}}
\,dx
\ge
C_{n,p,s}
\Bigl(\int_\Omega |u|^q\Bigr)^{p/q},
\]
with
\[
q=\frac{np}{n-ps},
\]
and \(C_{n,p,s}>0\) independent of \(\Omega\). The important point is that subtracting the sharp Hardy term still leaves an energy that controls the critical \(L^q\)-norm, while the Hardy coefficient itself remains the exact sharp one [1109.6570].

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
\[
a,B,a+B\in(-1,sp),
\qquad
1+a+B+sp>0,
\]
then the sharp constant is
\[
D(d,s,p,a,B)
=
\frac{2\,\pi^{(d-1)/2}}{p}\,
\frac{\Gamma\!\bigl(\tfrac{1+sp}{2}\bigr)}{\Gamma\!\bigl(\tfrac{d+sp}{2}\bigr)}
\int_{0}^{1}
(1-t)^{sp-1}\,t^{a+B}\,
\bigl|1-t^{1+a+B-sp}\bigr|^{p}\,dt,
\]
and for \(p\ge2\), \(sp<d\), a weighted fractional Hardy–Sobolev–Maz’ya inequality follows with this same sharp Hardy constant on the left-hand side [2210.06760].

For \(1<p<2\), 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
\[
C_p
=
\min_{x>0}
\frac{(p-1)x^p+1}{(x^{p/2}+1)^2}
>0,
\]
and when \(u\ge0\) this can be improved to \(p-1\). As \(p\to2\), the remainder formula becomes an identity; as \(p\to1\), the remainder vanishes [2301.11263].

## 6. Convex geometry, \(p=1\), and open-set lower bounds

The convex-domain theory reveals that sharpness is sensitive to the interaction between geometry and the parameter \(sp\). The supersolution method based on \(d_\Omega^\beta\) computes the exact constant for any convex \(\Omega\) when \(sp\ge1\), and for all \(0<s<1\) in the case \(p=2\). The same source exhibits a simple one-dimensional counterexample suggesting that this method cannot work for \(sp<1\) and \(\Omega\) different from a half-space. For \(1<p\neq2\) and \(0<s<1/p\), whether every convex \(\Omega\) still satisfies
\[
\mathfrak h_{s,p}(\Omega)
=
\mathfrak h_{s,p}(\mathbb H^n_+)
=
C_{n,sp}\Lambda_{s,p}
\]
is left as an open problem [2209.03012].

In the limit case \(p=1\), the sharp constant acquires a geometric interpretation. For convex non-empty \(\Omega\),
\[
h_{s,1}(\Omega)
=
\inf_{E\subset\Omega,\ |E|>0}
\frac{P_s(E)}{V_{s,\Omega}(E)}
=:g_s(\Omega),
\]
where
\[
P_s(E)=[1_E]_{W^{s,1}(\mathbb R^n)},
\qquad
V_{s,\Omega}(E)=\int_E d(x)^{-s}\,dx.
\]
Thus \(h_{s,1}(\Omega)\) is the Cheeger constant for the fractional perimeter and the weighted volume \(d(x)^{-s}\,dx\). In dimension one, if \(I\) is an open interval of length \(\ell\), then
\[
h_{s,1}(I)=2\,\ell^{-s},
\]
and for the unit interval,
\[
h_{s,1}((0,1))=2^{2-s}.
\]
The same geometric approach also gives new one-dimensional lower bounds for non-convex sets, some of them optimal when \(p=1\) [2407.08373].

For general open sets with \(sp>N\), the sharp Hardy constant of the punctured space \(\mathbb R^N\setminus\{0\}\) provides an optimal lower bound for the Hardy constant \(h_{s,p}(\Omega)\). 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 \(s\nearrow1\) and as \(p\nearrow\infty\), and yields a lower bound for the nonlocal eigenvalue \(\lambda_{s,p}(\Omega)\) in terms of \(h_{s,p}\) [2407.06568].

## 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
\[
\mathbb H^n_+=\{\xi=(x_1,x',y,t)\in\mathbb H^n:x_1>0\},
\]
Roy established
\[
C\int_{\mathbb H^n_+}\frac{|f(\xi)|^p}{x_1^{sp+\alpha}}\,d\xi
\le
\iint_{\mathbb H^n\times\mathbb H^n}
\frac{|f(\xi)-f(\xi')|^p}
{d(\xi^{-1}\!\circ\xi')^{Q+sp}|z-z'|^\alpha}
\,d\xi'\,d\xi,
\]
for all \(f\in C_c^\infty(\mathbb H^n_+)\), and computed the sharp constant in closed form when \(sp+\alpha>1\). The proof introduces a weighted nonlocal \(p\)-Laplacian on \(\mathbb H^n\), uses the weak harmonicity of \(x_1^s\) in the unweighted case, reduces the variational quotient to the \(x_1\)-variable, and then applies one-dimensional sharp fractional Hardy results [2504.05949].

For weighted singularities on a flat submanifold \(K=\{x_k=0\}\subset\mathbb R^d\), the sharp constant \(C(d,s,p,k,a,B)\) is given explicitly by an integral involving
\[
P_{m,s,p}(r)
=
\int_{S^{m-1}}
\frac{d\omega}{|e_m-r\omega|^{m+sp}},
\qquad m=d-k,
\]
and in the point-singularity case \(k=d\), \(p=2\), \(a=B=0\), one recovers the classical constant
\[
C(d,s)=2^{2s}\,\frac{\Gamma\!\bigl(\tfrac{d+2s}{2}\bigr)}{\Gamma\!\bigl(\tfrac{d-2s}{2}\bigr)}
\]
for
\[
\int_{\mathbb R^d}\!\int_{\mathbb R^d}
\frac{|u(x)-u(y)|^2}{|x-y|^{d+2s}}\,dx\,dy
\ge
C(d,s)\int_{\mathbb R^d}\frac{|u(x)|^2}{|x|^{2s}}\,dx.
\]
The proof again relies on a Frank–Seiringer type ground-state representation and concentrating test functions [2503.19057].

Discrete analogues exhibit the same sharpness structure in a different form. On \(\mathbb Z\), for \(0<\sigma<\tfrac12\), Keller and Nietschmann constructed an explicit weight \(w_\sigma\) such that
\[
\sum_{n\in\mathbb Z} w_\sigma(n)|u(n)|^2
\le
\langle u,(-\Delta)^\sigma u\rangle,
\qquad u\in C_c(\mathbb Z),
\]
and proved that \(w_\sigma\) is critical and null-critical, so the extremal constant is \(1\) in the normalization above [2207.12097]. On \(\mathbb Z^d\), Hake, Keller, and Pogorzelski identified a one-parameter family of Hardy weights \(w_{\sigma,d;\alpha}\), showed that the threshold \(\alpha_0=(d/2+\sigma)/2\) yields the optimal weight, and obtained the sharp constant
\[
C_{\sigma,d}
=
4^\sigma
\frac{\Gamma\!\bigl(\tfrac d4+\tfrac\sigma2\bigr)^2}
{\Gamma\!\bigl(\tfrac d4-\tfrac\sigma2\bigr)^2},
\]
with null-criticality at \(\alpha_0\) [2601.00902].

A further extension replaces one-body or boundary singularities by genuine interaction potentials. For \(s\in(0,1)\) and \(d\ge4-2s\), a sharp three-particle inequality holds on the collision-free configuration space \(\Omega_3\subset\mathbb R^{3d}\):
\[
\int_{\mathbb R^{3d}}
u(x)\Bigl[( -\Delta_{x_1})^s+( -\Delta_{x_2})^s+( -\Delta_{x_3})^s\Bigr]u(x)\,dx
\ge
C_{fH}(d,s)
\int_{\mathbb R^{3d}}V_{s,3}(x_1,x_2,x_3)|u(x)|^2\,dx,
\]
where
\[
C_{fH}(d,s)
=
2^{2s}\,
\frac{\Gamma^2((d+2s)/4)}{\Gamma^2((d-2s)/4)}.
\]
Here the sharp two-particle fractional Hardy constant survives unchanged, while the potential \(V_{s,3}\) captures genuine three-body effects [2605.30586].

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.

Source: https://www.emergentmind.com/topics/sharp-fractional-hardy-inequality