Advances in Hardy Improving Potentials

• Hardy improving potentials are critical enhancements that add positive remainder terms or spectral uplift to the classical Hardy inequality framework.
• They leverage techniques such as atomic decompositions, variational constraints, and Calderón–Zygmund methods to attain sharp coercivity and optimal constants.
• Applications span settings like the Heisenberg group sub-Laplacians, magnetic perturbations, boundary singularity refinements, and fractional dispersive regimes.

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: (Rocha, 22 May 2026, Cassano et al., 2021, Takahashi et al., 14 Jun 2026, Barbatis et al., 2017, Felli et al., 2022, Felli et al., 2012), and (Mizutani et al., 2020). 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 (Rocha, 22 May 2026, Cassano et al., 2021, Takahashi et al., 14 Jun 2026, Barbatis et al., 2017, Felli et al., 2022, Felli et al., 2012, Pietra et al., 2012, Mizutani et al., 2020).

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

$A_N=\left(\frac{N-p}{p}\right)^p,$

independent of the anisotropy $H$, with weight $(H^o(x))^{-p}$, and the best constant is not attained (Pietra et al., 2012). 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 (Barbatis et al., 2017). 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 (Felli et al., 2022). In semilinear elliptic problems with inverse-square singularity, the borderline coefficient is the sharp Hardy constant

$1

and precisely at this value the quadratic form ceases to be coercive in the standard $1Felli et al., 2012).

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 (Barbatis et al., 2017). It may denote a coercive effect of the Hardy term itself against lower-order gradient growth, quantified by explicit thresholds on the coupling parameter $1Pietra et al., 2012). It may denote a spectral or subcriticality effect produced by magnetic fields in the Heisenberg group (Cassano et al., 2021). It may also refer to a potential-theoretic mapping phenomenon: on $1Rocha, 22 May 2026). 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" (Rocha, 22 May 2026), the improvement mechanism is formulated through the Heisenberg sub-Laplacian

$1

on $1

$1

For bounded compactly supported $1$A_N=\left(\frac{N-p}{p}\right)^p,$0 satisfies $A_N=\left(\frac{N-p}{p}\right)^p,$1 in the distributional sense. The paper introduces the Orlicz-Hardy space $A_N=\left(\frac{N-p}{p}\right)^p,$2, defined by the grand maximal function $A_N=\left(\frac{N-p}{p}\right)^p,$3, and the Orlicz-Calderón–Hardy space $A_N=\left(\frac{N-p}{p}\right)^p,$4, defined through the quotient $A_N=\left(\frac{N-p}{p}\right)^p,$5 and the maximal quantity $A_N=\left(\frac{N-p}{p}\right)^p,$6. When $A_N=\left(\frac{N-p}{p}\right)^p,$7, the latter spaces coincide with the classical Calderón–Hardy spaces $A_N=\left(\frac{N-p}{p}\right)^p,$8.

The central theorem states that if $A_N=\left(\frac{N-p}{p}\right)^p,$9 and

$H$0

then

$H$1

is a bijection, and there exist constants $H$2 such that

$H$3

for all $H$4. Hence every $H$5 admits a unique $H$6 such that $H$7 in $H$8, with norm equivalence in both directions. In this setting, the Hardy-improving effect is the fact that $H$9, realized by convolution with $(H^o(x))^{-p}$0, maps $(H^o(x))^{-p}$1 into $(H^o(x))^{-p}$2. The target space records two derivatives through the exponent $(H^o(x))^{-p}$3, 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 $(H^o(x))^{-p}$4 is a $(H^o(x))^{-p}$5-atom supported in a ball $(H^o(x))^{-p}$6 and $(H^o(x))^{-p}$7, then the class $(H^o(x))^{-p}$8 satisfies

$(H^o(x))^{-p}$9

which is the key estimate used to pass from atomic data to potential control. The same paper establishes a sharp obstruction: if $\Omega\subset\mathbb R^n$0 and $\Omega\subset\mathbb R^n$1, then $\Omega\subset\mathbb R^n$2. 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" (Cassano et al., 2021). There the setting is $\Omega\subset\mathbb R^n$3, with horizontal distribution $\Omega\subset\mathbb R^n$4, contact form

$\Omega\subset\mathbb R^n$5

and Folland–Stein sub-Laplacian

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

Magnetic vector potentials are horizontal $\Omega\subset\mathbb R^n$7-forms modulo $\Omega\subset\mathbb R^n$8, $\Omega\subset\mathbb R^n$9, and the horizontal magnetic field is the Rumin differential $0\in\partial\Omega$0, a horizontal closed $0\in\partial\Omega$1-form. The magnetic sub-Laplacian is

$0\in\partial\Omega$2

with quadratic form

$0\in\partial\Omega$3

The baseline nonmagnetic Hardy–Garofalo–Lanconelli inequality is

$0\in\partial\Omega$4

equivalently $0\in\partial\Omega$5 in $0\in\partial\Omega$6, 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 $0\in\partial\Omega$7, the bottom of the spectrum is uplifted: $0\in\partial\Omega$8 for a universal constant $0\in\partial\Omega$9, so $n^2/4$0 in quadratic-form sense. For Aharonov–Bohm potentials $n^2/4$1 on $n^2/4$2, where $n^2/4$3, one has the sharp Hardy inequality

$n^2/4$4

and, under natural symmetry assumptions, the stronger remainder estimate

$n^2/4$5

For smooth horizontal fields nonzero on a bounded Lipschitz domain $n^2/4$6, or for AB fields with $n^2/4$7, the shifted critical operator becomes locally subcritical: $n^2/4$8

The paper explicitly characterizes a potential as Hardy-improving when one of three mechanisms occurs: the associated horizontal magnetic field $n^2/4$9 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 $$\Lambda_{N,s}=2^{2s}\frac{\Gamma^2\!\big(\frac{N+2s}{4}\big)}{\Gamma^2\!\big(\frac{N-2s}{4}\big)},$$0 is optimal; the coefficient $$\Lambda_{N,s}=2^{2s}\frac{\Gamma^2\!\big(\frac{N+2s}{4}\big)}{\Gamma^2\!\big(\frac{N-2s}{4}\big)},$$1 in the vertical-drift inequality $$\Lambda_{N,s}=2^{2s}\frac{\Gamma^2\!\big(\frac{N+2s}{4}\big)}{\Gamma^2\!\big(\frac{N-2s}{4}\big)},$$2 is optimal; and the spectral uplift $$\Lambda_{N,s}=2^{2s}\frac{\Gamma^2\!\big(\frac{N+2s}{4}\big)}{\Gamma^2\!\big(\frac{N-2s}{4}\big)},$$3 is optimal with respect to scaling.

4. Constrained, quasilinear, and anisotropic formulations

In "Some quasilinear minimization problems involving Hardy potentials" (Takahashi et al., 14 Jun 2026), the improvement is variational and constrained rather than operator-theoretic. The setting is $$\Lambda_{N,s}=2^{2s}\frac{\Gamma^2\!\big(\frac{N+2s}{4}\big)}{\Gamma^2\!\big(\frac{N-2s}{4}\big)},$$4 on a smooth bounded domain $$\Lambda_{N,s}=2^{2s}\frac{\Gamma^2\!\big(\frac{N+2s}{4}\big)}{\Gamma^2\!\big(\frac{N-2s}{4}\big)},$$5 with $$\Lambda_{N,s}=2^{2s}\frac{\Gamma^2\!\big(\frac{N+2s}{4}\big)}{\Gamma^2\!\big(\frac{N-2s}{4}\big)},$$6, under weighted mean-zero constraints. For the origin singularity,

$$\Lambda_{N,s}=2^{2s}\frac{\Gamma^2\!\big(\frac{N+2s}{4}\big)}{\Gamma^2\!\big(\frac{N-2s}{4}\big)},$$7

and similarly one defines $$\Lambda_{N,s}=2^{2s}\frac{\Gamma^2\!\big(\frac{N+2s}{4}\big)}{\Gamma^2\!\big(\frac{N-2s}{4}\big)},$$8 for $$\Lambda_{N,s}=2^{2s}\frac{\Gamma^2\!\big(\frac{N+2s}{4}\big)}{\Gamma^2\!\big(\frac{N-2s}{4}\big)},$$9 and $\mu<K_s\Lambda_{N,s}$0 for the critical logarithmic weight $\mu<K_s\Lambda_{N,s}$1. The associated constrained Hardy quotients are $\mu<K_s\Lambda_{N,s}$2, $\mu<K_s\Lambda_{N,s}$3, and $\mu<K_s\Lambda_{N,s}$4. The main result is that strict inequality below the classical Hardy threshold yields attainment: if $\mu<K_s\Lambda_{N,s}$5, then the infimum is attained in $\mu<K_s\Lambda_{N,s}$6; if $\mu<K_s\Lambda_{N,s}$7, then the infimum is attained in $\mu<K_s\Lambda_{N,s}$8; if $\mu<K_s\Lambda_{N,s}$9, then the infimum is attained in $1

$1

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" (Pietra et al., 2012). There the prototype operator is

$1

with Hardy-type zero-order term $1

$1

and the quantity

$1

determines the effective threshold. If $1

$1

then a weak solution $1

$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" (Barbatis et al., 2017) gives a systematic theory of improving potentials at a boundary singularity. Let $1

$1

implies the sharp boundary Hardy inequality

$1

If $1

The improvement theory proceeds in two directions. First, one obtains Hardy–Sobolev improvements and successive logarithmic corrections. For $1

$1

and the exponent $1

$1

with every coefficient $1

Second, the paper defines admissible and maximal improving potentials. A nonnegative $1

$1

Its concentration level near the singularity is

$1

The potential is subcritical if $1

$1

If $1

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

In "On fractional parabolic equations with Hardy-type potentials" (Felli et al., 2022), the model equation is

$1

on $1

$1

with finite limit as $1

$1

The same framework yields strong unique continuation: if the solution vanishes at $1

The borderline elliptic counterpart appears in "On semilinear elliptic equations with borderline Hardy potentials" (Felli et al., 2012). For

$1

the standard $1

$1

and uses the Emden–Fowler transform

$1

to convert the problem to the cylinder $1

$1

with $1

A dispersive analogue is given in "Kato smoothing, Strichartz and uniform Sobolev estimates for fractional operators with sharp Hardy potentials" (Mizutani et al., 2020). There the operator is

$1

with model potential $1Strichartz estimates for $1

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 (Rocha, 22 May 2026, Cassano et al., 2021, Barbatis et al., 2017, Felli et al., 2022).