Optimal Hardy Weight Analysis

• Optimal Hardy weight is a function or sequence that is maximal in a Hardy inequality, meaning any increase invalidates the inequality due to a loss of null-criticality.
• It is constructed through the ground-state (or supersolution) transform, using positive solutions of the underlying operator to ensure criticality and optimality at infinity.
• These weights play a crucial role in determining sharp spectral bounds and functional inequalities across Euclidean domains, manifolds, graphs, and fractional operator settings.

An optimal Hardy weight is a nonnegative function or sequence that appears in a Hardy-type inequality and is maximal in the sense that the corresponding inequality is valid, but fails if the weight is increased at any non-negligible subset (often formalized as ground-state criticality and null-criticality). Optimal Hardy weights are central in the analysis of Schrödinger operators, Dirichlet forms, and their discrete, continuum, and fractional analogues across Euclidean domains, manifolds, graphs, and metric measure spaces. The associated constants and weight functions dictate sharp spectral, analytic, and functional-inequality properties.

1. Definition and Criticality Characterization

Let $P$ denote an elliptic operator (classical case: $P=-\Delta$), $L$ a (possibly nonlocal) operator, or $h$ an energy functional, defined on a function space over a domain $\Omega\subset\mathbb{R}^n$, a manifold, or a discrete set such as $\mathbb{Z}^d$. A Hardy weight $W\geq0$ (or $w:\mathbb{Z}^d\to[0,\infty)$ in the discrete case) is optimal if:

• The Hardy inequality holds: e.g.,

$$\int_\Omega |\nabla \varphi|^2 \,dx \geq \int_\Omega W(x) |\varphi(x)|^2 dx, \quad \forall \varphi\in C_c^\infty(\Omega)$$

or its discrete/weighted variant.

• The shifted operator ($P-W$, $L-w$, or $h-w$) is critical, i.e., nonnegative but ceases to be so under any strictly larger addition to $W$ in a nontrivial set.
• Null-criticality: The ground state (unique positive solution to $(P-W)u=0$ or its discrete analogue) does not belong to $L^2(\Omega,Wdx)$; equivalently, the Rayleigh quotient cannot be minimized—no equality is attained by any nontrivial function/sequences.
• Optimality at infinity: In unbounded domains or graphs, any attempt to increase $W$ at infinity destroys the inequality.

In many frameworks, this definition coincides with that of null-criticality as in Agmon, Ancona, and Fraas-Pinchover's theories (Devyver et al., 2012, Keller et al., 2016, Fischer, 2022).

2. Construction Methods: Ground-State/Supersolution Transform

The universal construction strategy employs positive solutions ("ground states") of $P u=0$ (or $L u=0$) via the ground-state (supersolution) transform:

• Continuum (domains/manifolds): Given two linearly independent positive solutions $u_0,u_1$ of $P u=0$, define:

$$v(x) = \sqrt{u_0(x)u_1(x)}, \qquad W(x) = \frac{P v(x)}{v(x)} = \frac{1}{4}|\nabla_A \log(u_0/u_1)|^2$$

The maximal value is achieved for $a=1/2$ in the convex combination $v_a=u_0^{1-a}u_1^a$ (Devyver et al., 2012, Pinchover et al., 2019).

• Discretized settings (graphs/lattices): Use the minimal positive Green’s function $G(x)$ and/or auxiliary solution $u(x)$:

$$w(x) = \frac{L\psi(x)}{\psi(x)},\qquad \psi(x) = \sqrt{G(x)u(x)}$$

For operators with finite propagation (combinatorial Laplacians), the "difference form" analogue is used (Keller et al., 2021, Keller et al., 2016, Fischer, 2022).

• Nonlinear ($p$-Laplacian, quasilinear): For suitable domains and $1<p<\infty$,

$W(x) = (p-1)^p \frac{|\nabla G(x)|^p}{G(x)^p}$

where $G$ is a suitable fundamental solution (Versano, 2021, Devyver et al., 2013, Hou, 24 Dec 2025).

• Fractional Laplacian (continuous/discrete): Use (discrete or continuous) Riesz kernels as surrogates for ground-states and construct

$$w_{\sigma, \alpha}(x) := \frac{\kappa_{\sigma-\alpha}(x)}{\kappa_{-\alpha}(x)}$$

with $\kappa_\beta$ the (fractional) Riesz kernel; optimality corresponds to the precise threshold $\alpha_0$ of null-criticality (Keller et al., 2022, Hake et al., 31 Dec 2025, Das et al., 9 Jul 2025).

3. Canonical Formulas and Examples

Table: Representative Optimal Hardy Weights

Setting	Explicit Weight	Criticality Notes
$\mathbb{R}^n$, $-\Delta$, $n\ge3$	$W(x) = \frac{(n-2)^2}{4|x|^2}$	Classical, null-critical (Devyver et al., 2012)
Discrete (half-line)	$w_n = 2-\frac{\sqrt{n-1}+\sqrt{n+1}}{\sqrt{n}}$	$w_n\sim 1/(4n^2)+5/(64n^4)$; strictly stronger than naive $1/(4n^2)$ (Štampach et al., 2024)
$\mathbb{Z}^d$, $d\ge3$	$w(x)\sim \frac{(d-2)^2}{4|x|^2}$	Robust under general elliptic weights (Keller et al., 2021)
$p$-Laplacian ($1	$W(x)=\Big|\frac{n-p}{p}\Big|^p|x|^{-p}$	Critical, null-critical (Devyver et al., 2013)
Fractional Laplacian on $\mathbb{Z}$	$w_\sigma(n)=c_\sigma |n|^{-2}$, $$c_\sigma=4^\sigma\left[\Gamma\left(\frac{1+2\sigma}{4}\right)/\Gamma\left(\frac{1-2\sigma}{4}\right)\right]^2$$	Null-critical at boundary value (Keller et al., 2022, Hake et al., 31 Dec 2025)
Finsler $p$-Dirichlet	$$W(x) = \left(\frac{p-1}{p}\right)^p |\nabla \mathcal{G}/\mathcal{G}|_\mathcal{A}^p$$	Generalizes to variable norm (Hou, 24 Dec 2025)

Context: These weights control best-possible lower order terms in quadratic forms, admit no further improvement, and determine sharp operator domains and spectral gaps.

4. Boundary and Weighted Hardy Inequalities

Boundary Hardy inequalities are a central theme: e.g., let $$d_\Gamma(x) = \operatorname{dist}(x,\partial\Omega)$$.

• Weighted form:

$$\|d_\Gamma^{\delta/2-1}\varphi\|_2 \leq a_\delta\,\|d_\Gamma^{\delta/2}\nabla \varphi\|_2, \qquad \varphi \in C_c^\infty(I_r)$$

with $a_\delta = 2/|\delta-1|$ optimal in $C^{1,1}$ or convex $\Omega$, $\delta\ne1$ (Robinson, 2021). If $\Omega$ is a uniform domain with Ahlfors-regular boundary, $a_\delta \geq 2/|d-d_H+\delta-2|$.

• In the presence of corners, lower-dimensional faces, or acute dihedral angles, optimal constants may be strictly greater than the universal $2/|\delta-1|$ (Robinson, 2021).
• Analogous constructions carry over to cones and mixed geometry via separation of variables and reduction to spherical eigenvalue problems (Devyver et al., 2015).

5. Discrete, Weighted, and Fractional Variants

• Discrete lattices/graphs: Optimal Hardy weights have explicit expressions in terms of the Green function:

$$w_G(x) = \frac{1}{G(x)} \sum_{y\sim x} b(x,y) (G(x)-G(y))$$

and are optimal in the sense that any larger weight destroys the inequality or removes ground-state null-criticality (Keller et al., 2021, Keller et al., 2016).

• Fractional Laplacians: On both the integer lattice and the half-line, entire families of weights can be constructed, and the threshold for optimality is determined via asymptotic kernel expansions and null-criticality arguments (Keller et al., 2022, Hake et al., 31 Dec 2025, Das et al., 9 Jul 2025).
• Graphs with exponential growth (trees, spherically symmetric graphs): Optimal Hardy weights increase at infinity compared to classical ones, and attain their maxima via the ground-state transform on suitable superharmonic models (Berchio et al., 2020, Fischer et al., 30 Jan 2025).

6. Applications: Spectral Theory and Operator Self-Adjointness

• The explicit knowledge of the optimal Hardy weight provides sharp lower bounds for spectral forms, governing the spectral threshold and essential spectrum of Schrödinger and divergence-form operators. For example, in weighted boundary-degenerate elliptic operators, the knowledge of $a_\delta(\partial\Omega)$ yields optimal self-adjointness criteria (Robinson, 2021).
• For $-\operatorname{div}(C(x)\nabla)$ with $C(x)\simeq d_\Omega(x)^\delta$, $H$ is essentially self-adjoint iff $\delta > 3/2$ under additional regularity, and the critical constants are dictated by the boundary optimal Hardy constant.
• In rearrangement-invariant function spaces, the optimal domain and target for the classical (weighted) Hardy operator are explicitly characterized in terms of the weight that just allows boundedness, leading to deep interpolation and extrapolation consequences (Mihula, 2021).

7. Further Extensions, Open Problems, and Recent Developments

• The theory has been extended to operators with mixed boundary conditions (Pinchover et al., 2021), Finslerian $p$-Laplacians (Hou, 24 Dec 2025), and Hardy inequalities in cones for subcritical potentials (Devyver et al., 2015).
• In criticality theory, open problems remain concerning the optimality and precise randomness-induced fluctuations in i.i.d. random environments (see [(Keller et al., 2021), Remark 4.7]).
• Higher-order discrete Rellich-Birman analogues, corresponding to higher powers of discrete Laplacians, have led to optimal weights, improving on earlier conjectures and previous best-known constants (Štampach et al., 2024). Connections with Herglotz-Nevanlinna functions reveal deep links between extremal weights and integral transforms (Štampach et al., 25 Mar 2025).
• In $L^p$ and quasilinear settings, similar criteria via null-criticality and Picone-type identities yield optimal Hardy-weights (Devyver et al., 2013, Versano, 2021, Hou, 24 Dec 2025, Fischer, 2022). These weights guarantee the largest possible constant in nonlinear Hardy-type inequalities and extend to complex geometries and measure spaces.

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References:

• Devyver, Fraas, Pinchover, "Optimal Hardy Weight for Second-Order Elliptic Operator" (Devyver et al., 2012)
• Keller, Pinchover, Pogorzelski, "Optimal Hardy inequalities for Schrödinger operators on graphs" (Keller et al., 2016)
• Keller, Lemm, "Optimal Hardy weights on the Euclidean lattice" (Keller et al., 2021)
• Štampach, Wacławek, "Optimal discrete Hardy-Rellich-Birman inequalities" (Štampach et al., 2024)
• Fischer, "On the Optimality and Decay of $p$-Hardy Weights on Graphs" (Fischer, 2022)
• Pinchover, Versano, "On families of optimal Hardy-weights..." (Pinchover et al., 2019)
• Mihula, "Optimal behavior of weighted Hardy operators on rearrangement-invariant spaces" (Mihula, 2021)
• Hake, Keller, Pogorzelski, "Optimal Hardy Inequality for Fractional Laplacians on the Lattice" (Hake et al., 31 Dec 2025)
• Berchio, Santagati, Vallarino, "Poincaré and Hardy inequalities on homogeneous trees" (Berchio et al., 2020)
• Hou, "Optimal Hardy-weights for the Finsler $p$-Dirichlet integral..." (Hou, 24 Dec 2025)
• Gerhat, Krejčířik, Štampach, Huang-Ye, and others on higher order discrete Rellich-Birman problems (Štampach et al., 2024)

For in-depth proofs and further generalizations, consult the cited arXiv papers.