---
title: Advances in Hardy Improving Potentials
url: https://www.emergentmind.com/topics/hardy-improving-potentials
type: topic
---

# Advances in Hardy Improving Potentials

Searching arXiv for the cited papers and closely related work on Hardy improving potentials.
I’m checking arXiv metadata for the core papers on Hardy improving potentials and related Hardy-type improvements.
Using arXiv search to confirm the primary sources: 2605.24194, 2110.13775, 2606.15849, 1701.06336, 2212.03744, 1209.4852, and 2002.02163.
Hardy improving potentials are potentials, magnetic fields, or potential operators that strengthen a baseline Hardy framework by producing a positive remainder term, a spectral uplift, a coercive gap below a sharp Hardy threshold, or an improvement of the target function space for a potential map. In current analysis this theme appears in several distinct but related settings: sub-Laplacian potentials on the Heisenberg group, magnetic perturbations of the Folland–Stein operator, constrained quasilinear minimization with Hardy weights, boundary-singularity Hardy–Sobolev refinements, and fractional or borderline inverse-square regimes in elliptic and parabolic problems [2605.24194] [2110.13775] [2606.15849] [1701.06336] [2212.03744] [1209.4852] [1209.0928] [2002.02163].

## 1. Critical Hardy structures and the meaning of “improvement”

A recurrent feature of the subject is that the unperturbed Hardy inequality is critical. In the anisotropic \(W^{1,p}_0(\Omega)\) setting with \(1<p<N\), Van Schaftingen’s anisotropic Hardy inequality has best constant
\[
A_N=\left(\frac{N-p}{p}\right)^p,
\]
independent of the anisotropy \(H\), with weight \((H^o(x))^{-p}\), and the best constant is not attained [1209.0928]. For boundary singularities in a bounded domain \(\Omega\subset\mathbb R^n\) with \(0\in\partial\Omega\), the sharp boundary Hardy constant is \(n^2/4\) under a quantitative exterior ball condition, and this constant is sharp if an interior ball condition also holds [1701.06336]. In the fractional parabolic setting, the critical inverse-square coefficient is controlled by the optimal Hardy constant
\[
\Lambda_{N,s}=2^{2s}\frac{\Gamma^2\!\big(\frac{N+2s}{4}\big)}{\Gamma^2\!\big(\frac{N-2s}{4}\big)},
\]
with the subcritical regime \(\mu<K_s\Lambda_{N,s}\) singled out as the coercive range [2212.03744]. In semilinear elliptic problems with inverse-square singularity, the borderline coefficient is the sharp Hardy constant
\[
\mu^*=\frac{(N-2)^2}{4},
\]
and precisely at this value the quadratic form ceases to be coercive in the standard \(H^1\) framework [1209.4852].

Within this critical landscape, “improvement” has several technically different meanings. It may denote a sharper inequality obtained by adding a nonnegative remainder, as in boundary Hardy–Sobolev and logarithmic refinements [1701.06336]. It may denote a coercive effect of the Hardy term itself against lower-order gradient growth, quantified by explicit thresholds on the coupling parameter \(\lambda\) [1209.0928]. It may denote a spectral or subcriticality effect produced by magnetic fields in the Heisenberg group [2110.13775]. It may also refer to a potential-theoretic mapping phenomenon: on \(\mathbb H^n\), the inverse sub-Laplacian improves data from an Orlicz-Hardy space to an Orlicz-Calderón–Hardy class [2605.24194]. This suggests that the expression “Hardy improving potentials” is not confined to a single inequality, but indexes a family of mechanisms through which critical Hardy structures acquire additional regularity, coercivity, or spectral positivity.

## 2. Potential-theoretic Hardy improvement on the Heisenberg group

In "On potentials of distributions in Orlicz-Hardy type spaces on the Heisenberg group" [2605.24194], the improvement mechanism is formulated through the Heisenberg sub-Laplacian
\[
\mathcal L=-\sum_{i=1}^{2n}X_i^2
\]
on \(\mathbb H^n\), where the homogeneous dimension is \(Q=2n+2\), the Korányi gauge is \(\rho(z)=(|x|^4+t^2)^{1/4}\), and Folland’s fundamental solution is
\[
\Gamma(z)=c_n\rho(z)^{-2n}.
\]
For bounded compactly supported \(a\), the potential \(b=a*\Gamma\) satisfies \(\mathcal L b=a\) in the distributional sense. The paper introduces the Orlicz-Hardy space \(H^\Phi(\mathbb H^n)\), defined by the grand maximal function \(\mathcal M_N f\), and the Orlicz-Calderón–Hardy space \(\mathcal H^\Phi_{q,\gamma}(\mathbb H^n)\), defined through the quotient \(E_k^q=L^q_{\mathrm{loc}}(\mathbb H^n)/\mathbb P_k\) and the maximal quantity \(N_{q,\gamma}(G;\cdot)\). When \(\Phi(t)=t^p\), the latter spaces coincide with the classical Calderón–Hardy spaces \(\mathcal H^p_{q,\gamma}\).

The central theorem states that if \(1<q<(n+1)/n\) and
\[
Q(2+Q/q)^{-1}<i(\Phi)\le I(\Phi)<\infty,
\]
then
\[
\mathcal L:\mathcal H^\Phi_{q,2}(\mathbb H^n)\to H^\Phi(\mathbb H^n)
\]
is a bijection, and there exist constants \(c_1,c_2>0\) such that
\[
c_1\|G\|_{\mathcal H^\Phi_{q,2}}\le \|\mathcal LG\|_{H^\Phi}\le c_2\|G\|_{\mathcal H^\Phi_{q,2}}
\]
for all \(G\in\mathcal H^\Phi_{q,2}(\mathbb H^n)\). Hence every \(f\in H^\Phi(\mathbb H^n)\) admits a unique \(G\in\mathcal H^\Phi_{q,2}(\mathbb H^n)\) such that \(\mathcal LG=f\) in \(\mathcal S'(\mathbb H^n)\), with norm equivalence in both directions. In this setting, the Hardy-improving effect is the fact that \(\mathcal L^{-1}\), realized by convolution with \(\Gamma=c_n\rho^{-2n}\), maps \(H^\Phi(\mathbb H^n)\) into \(\mathcal H^\Phi_{q,2}(\mathbb H^n)\). The target space records two derivatives through the exponent \(\gamma=2\), so the improvement is encoded in the maximal control of the potential class rather than in a remainder inequality.

The proof is driven by atomic decomposition, Calderón–Zygmund/Whitney machinery, and a pointwise estimate for potentials of atoms. If \(a\) is a \((\Phi,\infty,m)\)-atom supported in a ball \(B\) and \(b=a*(c_n\rho^{-2n})\), then the class \(\widetilde b\in E_1^q\) satisfies
\[
N_{q,2}(\widetilde b; z) \lesssim \|\chi_B\|_{\Phi}^{-1} [M\chi_B(z)]^{(2+Q/q)/Q} + \chi_{4\beta^2B}(z) M a(z) + \chi_{4\beta^2B}(z) \sum_{d(I)=2} T_I^* a(z),
\]
which is the key estimate used to pass from atomic data to potential control. The same paper establishes a sharp obstruction: if \(1<q<(n+1)/n\) and \(0<I(\Phi)<Q(2+Q/q)^{-1}\), then \(\mathcal H^\Phi_{q,2}(\mathbb H^n)=\{0\}\). The lower-type threshold therefore quantifies exactly when the potential improvement is nontrivial.

## 3. Magnetic Hardy improvements on the Heisenberg group

A different use of the term appears in "Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group" [2110.13775]. There the setting is \(H^1\), with horizontal distribution \(\mathcal E=\mathrm{span}\{X,Y\}\), contact form
\[
\omega=dz-\tfrac12(x\,dy-y\,dx),
\]
and Folland–Stein sub-Laplacian
\[
L_0=-X^2-Y^2.
\]
Magnetic vector potentials are horizontal \(1\)-forms modulo \(\omega\), \(A\in\Omega^1(H^1)/\mathrm{span}\{\omega\}\), and the horizontal magnetic field is the Rumin differential \(B=DA\), a horizontal closed \(2\)-form. The magnetic sub-Laplacian is
\[
L_A=(-i\nabla_H+A)^2=-(X+iA_x)^2-(Y+iA_y)^2,
\]
with quadratic form
\[
Q_A(u)=\int_{H^1}|(\nabla_H+iA)u|^2\,dq.
\]

The baseline nonmagnetic Hardy–Garofalo–Lanconelli inequality is
\[
\int_{H^1}|\nabla_Hu|^2\,dq\ge \int_{H^1}\frac{r^2}{\rho^4}|u|^2\,dq,\qquad u\in C_c^\infty(H^1\setminus\{0\}),
\]
equivalently \(-\Delta\ge r^2/\rho^4\) in \(L^2(H^1)\), and it is optimal in the critical sense that no strictly positive term can be added on the right-hand side. Magnetic perturbations improve this picture in several distinct ways. For a uniform horizontal magnetic field \(B=b_1\,dx\wedge\omega+b_2\,dy\wedge\omega\), the bottom of the spectrum is uplifted:
\[
\inf \sigma(-\Delta_A)=c|B|^{2/3}
\]
for a universal constant \(c>0\), so \(-\Delta_A\ge c|B|^{2/3}\) in quadratic-form sense. For Aharonov–Bohm potentials \(A_\alpha=\alpha\,d\phi \mod \omega\) on \(H^1\setminus\mathcal Z\), where \(\mathcal Z=\{(0,0,z):z\in\mathbb R\}\), one has the sharp Hardy inequality
\[
-\Delta_{A_\alpha}\ge \frac{d(\alpha,\mathbb Z)^2}{r^2}
\quad\text{in }L^2(H^1\setminus\mathcal Z),
\]
and, under natural symmetry assumptions, the stronger remainder estimate
\[
-\Delta_{A_\alpha}-\frac{r^2}{\rho^4}\ge d(\alpha,\mathbb Z)^2\,\frac{1-|\nabla\rho|^4}{r^2}.
\]
For smooth horizontal fields nonzero on a bounded Lipschitz domain \(\Omega\), or for AB fields with \(\alpha\notin\mathbb Z\), the shifted critical operator becomes locally subcritical:
\[
-\Delta_A-\frac{r^2}{\rho^4}\ge c(A,\Omega)\chi_\Omega.
\]

The paper explicitly characterizes a potential as Hardy-improving when one of three mechanisms occurs: the associated horizontal magnetic field \(B\) is not identically zero on some region, the potential is AB-type with non-integer flux, or the field is uniform. This notion is sharp in several senses. The base Hardy inequality is critical; the AB constant \(d(\alpha,\mathbb Z)^2\) is optimal; the coefficient \(1-\alpha^2\) in the vertical-drift inequality \(L_\alpha\ge (1-\alpha^2)r^2/\rho^4\) is optimal; and the spectral uplift \(c|B|^{2/3}\) is optimal with respect to scaling.

## 4. Constrained, quasilinear, and anisotropic formulations

In "Some quasilinear minimization problems involving Hardy potentials" [2606.15849], the improvement is variational and constrained rather than operator-theoretic. The setting is \(W^{1,p}(\Omega)\) on a smooth bounded domain \(\Omega\subset\mathbb R^N\) with \(0\in\Omega\), under weighted mean-zero constraints. For the origin singularity,
\[
X_p=\left\{u\in W^{1,p}(\Omega):\int_\Omega u(x)|x|^{-p}\,dx=0\right\},
\]
and similarly one defines \(Y_p\) for \(d(x,K)^{-p}\) and \(X_N\) for the critical logarithmic weight \(|x|^{-N}(\log(1/|x|))^{-N}\). The associated constrained Hardy quotients are \(\bar H_p(\Omega)\), \(\bar J_{K,p}(\Omega)\), and \(\bar C_N(\Omega)\). The main result is that strict inequality below the classical Hardy threshold yields attainment: if \(\bar H_p(\Omega)<H_p=((N-p)/p)^p\), then the infimum is attained in \(X_p\); if \(\bar J_{K,p}(\Omega)<J_{K,p}=((k-p)/p)^p\), then the infimum is attained in \(Y_p\); if \(\bar C_N(\Omega)<C_N=((N-1)/N)^N\), then the infimum is attained in \(X_N\). The key analytic input is a Cherrier-type inequality such as
\[
(J_{K,p}-\varepsilon)\int_\Omega |u|^p d(x,K)^{-p}\,dx
\le
\int_\Omega |\nabla u|^p\,dx + C(\varepsilon)\int_\Omega |u|^p\,dx,
\]
together with Ekeland’s principle and the Valeriola–Willem criterion. In this framework, Hardy potentials are improving because the weighted mean-zero constraint removes the saturating direction, lowers the constrained variational level below the Hardy constant, and restores compactness.

The anisotropic counterpart is developed in "Anisotropic elliptic equations with general growth in the gradient and Hardy-type potentials" [1209.0928]. There the prototype operator is
\[
Q u := -\operatorname{div}\big(H(Du)^{p-1}D H(Du)\big), \qquad 1<p<N,
\]
with Hardy-type zero-order term \(\lambda (H^o(x))^{-p}|u|^{p-2}u\) and lower-order gradient growth \(B(u)H(Du)^q\), \(p-1<q\le p\). The sharp anisotropic Hardy constant is again
\[
A_N=\left(\frac{N-p}{p}\right)^p,
\]
and the quantity
\[
B_\infty:=\int_0^{+\infty}\big(B(s)\big)^{\frac{1}{p-q}}\,s^{-\frac{p-1}{p-q}}\,ds
\]
determines the effective threshold. If \(f\in L((p^*)',p')\) and
\[
0\le \lambda < A_N e^{-B_\infty},
\]
then a weak solution \(u\in W^{1,p}_0(\Omega)\) exists. For \(f\in L(m,\theta)\) above or below the energy level, one has the finer threshold
\[
\lambda(m)=e^{-B_\infty}F(a_m),\qquad F(a)=-(p-1)a^p+(N-p)a^{p-1},\qquad a_m=\frac{N-mp}{m(p-1)},
\]
which is optimal for the Lorentz estimates obtained. The paper stresses that it does not produce a remainder-type improved Hardy inequality; rather, the Hardy weight itself is the improving mechanism that compensates the loss of coercivity caused by gradient-dependent lower-order terms and yields a priori estimates and compactness.

## 5. Boundary singularities, maximal improving potentials, and logarithmic refinements

"Sharp Hardy and Hardy--Sobolev inequalities with point singularities on the boundary" [1701.06336] gives a systematic theory of improving potentials at a boundary singularity. Let \(\Omega\subset\mathbb R^n\) be bounded, \(n\ge2\), with \(0\in\partial\Omega\). If \(\Omega\) satisfies an exterior ball condition at \(0\) with radius \(p\), and \(D=\sup_{x\in\Omega}|x|\), then there exists \(\tau_n>0\) such that
\[
p>\frac{D}{\tau_n}
\]
implies the sharp boundary Hardy inequality
\[
\int_\Omega |\nabla u|^2\,dx \ge \frac{n^2}{4}\int_\Omega \frac{u^2}{|x|^2}\,dx,\qquad u\in C_c^\infty(\Omega).
\]
If \(\Omega\) also satisfies an interior ball condition at \(0\), the constant \(n^2/4\) is sharp. The paper also shows that the large exterior ball condition is genuinely necessary: for annuli \(\mathcal A(e_n;p,p(1+\tau))\), the best constant \(\Lambda_\tau\) equals \(n^2/4\) for \(0<\tau<\tau_n\), but drops below \(n^2/4\) for \(\tau>\tau_n\), with \(\Lambda_\tau\downarrow (n-2)^2/4\) as \(\tau\to+\infty\).

The improvement theory proceeds in two directions. First, one obtains Hardy–Sobolev improvements and successive logarithmic corrections. For \(n\ge3\), under a sufficiently large exterior ball condition,
\[
\int_{\Omega} |\nabla u|^2\,dx \ge \frac{n^2}{4} \int_{\Omega} \frac{u^2}{|x|^2}\,dx + C_n \left(\int_{\Omega} X_1^{\frac{2n-2}{n-2}}\!\left(\frac{|x|}{3D}\right)|u|^{\frac{2n}{n-2}}\,dx\right)^{\frac{n-2}{n}},
\]
and the exponent \((2n-2)/(n-2)\) is sharp. More strongly,
\[
\int_{\Omega} |\nabla u|^2\,dx \ge \frac{n^2}{4} \int_{\Omega} \frac{u^2}{|x|^2}\,dx + \frac{1}{4}\sum_{i=1}^{\infty}\int_{\Omega}\frac{u^2}{|x|^2}\,X_1\cdots X_i\,dx,
\]
with every coefficient \(1/4\) sharp under the interior ball condition. Analogous finite- and infinite-logarithmic improvements are obtained in cones, where the Hardy constant becomes \(((n-2)^2/4)+\mu_1(\omega)\), and the improved Sobolev constant depends on the cone opening \(\omega\).

Second, the paper defines admissible and maximal improving potentials. A nonnegative \(V\in L^{n/2}(\Omega\setminus\{0\})\) is admissible if there exist \(\lambda\ge0\) and \(C>0\) such that
\[
\lambda \int_{\Omega} \frac{u^2}{|x|^2}\,dx + \int_{\Omega} |\nabla u|^2\,dx
\ge
\frac{n^2}{4}\int_{\Omega}\frac{u^2}{|x|^2}\,dx + C\int_{\Omega}V(x)u^2\,dx.
\]
Its concentration level near the singularity is
\[
C_r(V)=\inf_{u\in C_c^\infty(\Omega\cap B_r)}
\frac{\int_{\Omega\cap B_r} |\nabla u|^2\,dx - \frac{n^2}{4}\int_{\Omega\cap B_r}\frac{u^2}{|x|^2}\,dx}
{\int_{\Omega\cap B_r}V(x)u^2\,dx},
\qquad
C_0(V)=\lim_{r\to0^+}C_r(V).
\]
The potential is subcritical if \(C_0(V)=+\infty\), and a sufficient condition is
\[
\int_\Omega V^{n/2}(x)\,X_1^{\,1-n}\!\left(\frac{|x|}{D}\right)\,dx<+\infty.
\]
If \(b(\lambda)<C_0(V)\), then the best constant is attained; if \(V\) is subcritical, one obtains a maximal potential in the sense that no further nonnegative potential can be added to improve the inequality. The paper extends the same structure to \(m\)-admissible and \(m\)-maximal potentials after \(m\) logarithmic corrections.

## 6. Fractional, dispersive, and borderline inverse-square regimes

In "On fractional parabolic equations with Hardy-type potentials" [2212.03744], the model equation is
\[
H^s(w)=\mu |x|^{-2s}w + g\,w
\]
on \(\mathbb R^N\times (t_0-T,t_0)\), with \(s\in(0,1)\), \(N>2s\), and \(\mu<K_s\Lambda_{N,s}\). Through a Caffarelli–Silvestre-type extension, the problem is lifted to a degenerate parabolic equation in \(\mathbb R^{N+1}_+\) with weight \(y^{1-2s}\). The Hardy potential enters the boundary form and remains coercive in the subcritical regime. This permits an Almgren–Poon frequency function
\[
\mathcal N(t)=(t_0-t)\frac{D(t)}{H(t)}
\]
with finite limit as \(t\to t_0^-\), a blow-up analysis in Gaussian spaces, and classification of local asymptotic profiles by the Hardy-modified Ornstein–Uhlenbeck spectrum
\[
\gamma_{m,k}=m-\alpha_k.
\]
The same framework yields strong unique continuation: if the solution vanishes at \((0,t_0)\) of infinite space-like order, then it is identically zero. Here the inverse-square Hardy term is “improving” in the sense that subcritical strength supports coercivity, monotonicity, and profile classification rather than obstructing them.

The borderline elliptic counterpart appears in "On semilinear elliptic equations with borderline Hardy potentials" [1209.4852]. For
\[
-\Delta u - \mu^* |x|^{-2}u = h(x)u + f(x,u),\qquad \mu^*=\frac{(N-2)^2}{4},
\]
the standard \(H^1\) theory breaks down exactly at the critical Hardy coefficient. The paper therefore introduces the Hilbert space \(H(\omega)\), defined by the scalar product
\[
(u,v)_{H(\omega)}
=
\int_{\omega}\nabla u\cdot \nabla v\,dx
-
\mu^*\int_{\omega}|x|^{-2}uv\,dx
+
\frac{N-2}{2}\int_{\partial\omega}|x|^{-2}uv(x\cdot \nu)\,dS,
\]
and uses the Emden–Fowler transform
\[
Tu(t,\theta)=e^{(N-2)t/2}u(e^{-t}\theta)
\]
to convert the problem to the cylinder \(\mathbb R\times S^{N-1}\). An Almgren-type frequency function then yields the exact asymptotic behavior
\[
u(r\theta)\approx r^{-(N-2)/2+\sqrt{\lambda_{\ell_0}}}\,\psi(\theta),
\]
with \(\psi\) in the \(\lambda_{\ell_0}\)-eigenspace of \(-\Delta_{S^{N-1}}\), and strong unique continuation follows. The assumptions \(|h(x)|\le C_h|x|^{-2+\varepsilon}\) and subcritical growth for \(f\) imply that these perturbations contribute only lower-order terms in the frequency analysis; in the paper’s terminology, they are compatible with the borderline Hardy structure and do not alter the leading homogeneous asymptotics.

A dispersive analogue is given in "Kato smoothing, Strichartz and uniform Sobolev estimates for fractional operators with sharp Hardy potentials" [2002.02163]. There the operator is
\[
H=(-\Delta)^\sigma + V(x),
\]
with model potential \(V(x)=a|x|^{-2\sigma}\), \(0<\sigma<n/2\), and subcritical coupling \(a>-C_{\sigma,n}\). Under Assumption A, the paper proves uniform Kato–Yajima resolvent bounds,
global Kato smoothing, Strichartz estimates for \(\sigma>1/2\), and uniform Sobolev estimates for \(\sigma\ge n/(n+1)\). In the authors’ formulation, the scale-invariant Hardy potential yields enhanced uniform resolvent control with the natural weight \(|x|^{-\sigma}\), improved Strichartz estimates with gain of regularity for \(1<\sigma<n/2\), and persistence of the same estimates at critical coupling on the nonradial subspace \(\Prad^\perp\). The improvement is thus dispersive and spectral rather than variational.

Taken together, these developments show that Hardy improving potentials do not form a single class tied to one operator or one inequality. They are a family of critical or subcritical perturbations that, depending on the setting, sharpen Hardy inequalities, create spectral gaps, recover coercivity at or below sharp thresholds, or transport data into better-controlled potential spaces. The common structural feature is that the Hardy scale remains decisive: every genuine improvement is quantified relative to a sharp constant, a critical homogeneity, or an exact threshold, and collapse or loss of coercivity occurs once that threshold is crossed [2605.24194] [2110.13775] [1701.06336] [2212.03744].

Source: https://www.emergentmind.com/topics/hardy-improving-potentials