Weighted Hardy Inequality

• Weighted Hardy inequalities are coercive estimates where singular or degenerate weights modify energy or potential terms and define optimal constants.
• They generalize classical Hardy inequalities to settings including boundary-distance, discrete, nonlocal, and variable exponent frameworks.
• Applications span singular Schrödinger operators, degenerate elliptic equations, and variational problems, where constants determine spectral thresholds and existence of extremals.

A weighted Hardy inequality is a coercive estimate in which a singular or degenerate weight modifies either the energy term, the potential term, or both. In its most familiar forms, it compares a weighted gradient norm, a weighted difference quotient, or a weighted averaging operator against a weighted $L^p$-quantity involving distance to a point, the boundary, or a lower-dimensional singular set. The subject includes local and nonlocal inequalities, continuous and discrete operators, and variational problems in which the optimal Hardy constant governs coercivity, spectral thresholds, and existence of extremals (Goel et al., 2020, Fall et al., 2012, Dyda et al., 2022).

1. Classical prototypes and general weighted formulations

The classical Euclidean Hardy inequality in $\mathbb{R}^N$ has the form

$$\int_{\mathbb{R}^N} |\nabla u|^2\,dx \ge \left(\frac{N-2}{2}\right)^2 \int_{\mathbb{R}^N} \frac{|u|^2}{|x|^2}\,dx, \qquad u\in C_c^\infty(\mathbb{R}^N),$$

with best constant $\bigl(\frac{N-2}{2}\bigr)^2$ and non-attainment in $H^1$ (Fall et al., 2012). A boundary-distance analogue replaces $|x|$ by $d_\Omega(x)=\operatorname{dist}(x,\partial\Omega)$. For arbitrary domains $\Omega\subsetneqq\mathbb{R}^n$, $1<p<\infty$, and $\alpha+p>n$, one has

$\mathbb{R}^N$0

and the constant $\mathbb{R}^N$1 is sharp (Goel et al., 2020).

In the $\mathbb{R}^N$2 boundary-distance setting, the distance function again supplies the weight, but the structure is first-order rather than quadratic. For open $\mathbb{R}^N$3, $\mathbb{R}^N$4, and $\mathbb{R}^N$5,

$\mathbb{R}^N$6

for all $\mathbb{R}^N$7; under $\mathbb{R}^N$8, this simplifies to

$\mathbb{R}^N$9

with sharp constant $$\int_{\mathbb{R}^N} |\nabla u|^2\,dx \ge \left(\frac{N-2}{2}\right)^2 \int_{\mathbb{R}^N} \frac{|u|^2}{|x|^2}\,dx, \qquad u\in C_c^\infty(\mathbb{R}^N),$$0 (Psaradakis, 2013).

A parallel operator-theoretic formulation is built from Hardy averaging operators. On $$\int_{\mathbb{R}^N} |\nabla u|^2\,dx \ge \left(\frac{N-2}{2}\right)^2 \int_{\mathbb{R}^N} \frac{|u|^2}{|x|^2}\,dx, \qquad u\in C_c^\infty(\mathbb{R}^N),$$1,

$$\int_{\mathbb{R}^N} |\nabla u|^2\,dx \ge \left(\frac{N-2}{2}\right)^2 \int_{\mathbb{R}^N} \frac{|u|^2}{|x|^2}\,dx, \qquad u\in C_c^\infty(\mathbb{R}^N),$$2

Weighted Hardy inequalities then appear as boundedness statements such as $$\int_{\mathbb{R}^N} |\nabla u|^2\,dx \ge \left(\frac{N-2}{2}\right)^2 \int_{\mathbb{R}^N} \frac{|u|^2}{|x|^2}\,dx, \qquad u\in C_c^\infty(\mathbb{R}^N),$$3 or $$\int_{\mathbb{R}^N} |\nabla u|^2\,dx \ge \left(\frac{N-2}{2}\right)^2 \int_{\mathbb{R}^N} \frac{|u|^2}{|x|^2}\,dx, \qquad u\in C_c^\infty(\mathbb{R}^N),$$4, characterized by Muckenhoupt-type conditions $$\int_{\mathbb{R}^N} |\nabla u|^2\,dx \ge \left(\frac{N-2}{2}\right)^2 \int_{\mathbb{R}^N} \frac{|u|^2}{|x|^2}\,dx, \qquad u\in C_c^\infty(\mathbb{R}^N),$$5 and $$\int_{\mathbb{R}^N} |\nabla u|^2\,dx \ge \left(\frac{N-2}{2}\right)^2 \int_{\mathbb{R}^N} \frac{|u|^2}{|x|^2}\,dx, \qquad u\in C_c^\infty(\mathbb{R}^N),$$6 on the weight $$\int_{\mathbb{R}^N} |\nabla u|^2\,dx \ge \left(\frac{N-2}{2}\right)^2 \int_{\mathbb{R}^N} \frac{|u|^2}{|x|^2}\,dx, \qquad u\in C_c^\infty(\mathbb{R}^N),$$7 (Barza et al., 2017). In this operator language, weighted Cesàro and Copson spaces encode Hardy inequalities as norm equivalences rather than only one-sided estimates.

2. Domain geometry and boundary-distance inequalities

A central theme is that boundary regularity can often be weakened to quantitative thickness. For $$\int_{\mathbb{R}^N} |\nabla u|^2\,dx \ge \left(\frac{N-2}{2}\right)^2 \int_{\mathbb{R}^N} \frac{|u|^2}{|x|^2}\,dx, \qquad u\in C_c^\infty(\mathbb{R}^N),$$8, the weighted $$\int_{\mathbb{R}^N} |\nabla u|^2\,dx \ge \left(\frac{N-2}{2}\right)^2 \int_{\mathbb{R}^N} \frac{|u|^2}{|x|^2}\,dx, \qquad u\in C_c^\infty(\mathbb{R}^N),$$9-Hardy inequality

$\bigl(\frac{N-2}{2}\bigr)^2$0

holds on bounded Lipschitz domains for all $\bigl(\frac{N-2}{2}\bigr)^2$1. More generally, if the boundary satisfies the uniform density condition

$\bigl(\frac{N-2}{2}\bigr)^2$2

for some $\bigl(\frac{N-2}{2}\bigr)^2$3 and all $\bigl(\frac{N-2}{2}\bigr)^2$4, then $\bigl(\frac{N-2}{2}\bigr)^2$5 admits the $\bigl(\frac{N-2}{2}\bigr)^2$6-Hardy inequality for all

$\bigl(\frac{N-2}{2}\bigr)^2$7

This range is stated to be optimal in the sense described in the source, and the theory extends to metric spaces with doubling measure and a Poincaré inequality (Lehrbäck, 2012).

For weighted $\bigl(\frac{N-2}{2}\bigr)^2$8 boundary inequalities, the paper on the weighted Hardy constant studies

$\bigl(\frac{N-2}{2}\bigr)^2$9

where $H^1$0, $H^1$1, and $H^1$2 (Robinson, 2021). For $H^1$3, this inequality is equivalent to a weighted version of Davies’ weak Hardy inequality, with equality of optimal constants: $H^1$4 If $H^1$5 is a uniform domain with Ahlfors regular boundary of Hausdorff dimension $H^1$6, then the inequality holds for all $H^1$7, except possibly at

$H^1$8

and the optimal constant satisfies

$H^1$9

If $|x|$0 is $|x|$1 or convex, then

$|x|$2

while for complements of convex domains the same conclusion holds for $|x|$3, but for $|x|$4 the constant can be strictly larger than $|x|$5 (Robinson, 2021).

The $|x|$6 theory shows that mean curvature and reach enter through $|x|$7. For $|x|$8 domains with uniform interior sphere condition, $|x|$9 is a signed Radon measure whose absolutely continuous part is bounded below by $d_\Omega(x)=\operatorname{dist}(x,\partial\Omega)$0, where $d_\Omega(x)=\operatorname{dist}(x,\partial\Omega)$1 is the infimum of the mean curvature on $d_\Omega(x)=\operatorname{dist}(x,\partial\Omega)$2. Mean convexity is equivalent to $d_\Omega(x)=\operatorname{dist}(x,\partial\Omega)$3 in distributions, and for strictly mean convex bounded domains one obtains a sharp homogeneous remainder term

$d_\Omega(x)=\operatorname{dist}(x,\partial\Omega)$4

with

$d_\Omega(x)=\operatorname{dist}(x,\partial\Omega)$5

(Psaradakis, 2013).

3. Higher-dimensional singularities and variational Hardy constants

Weighted Hardy inequalities also arise when the singular set is a submanifold rather than a point. In a smooth bounded domain $d_\Omega(x)=\operatorname{dist}(x,\partial\Omega)$6, $d_\Omega(x)=\operatorname{dist}(x,\partial\Omega)$7, with a closed smooth submanifold $d_\Omega(x)=\operatorname{dist}(x,\partial\Omega)$8 of dimension $d_\Omega(x)=\operatorname{dist}(x,\partial\Omega)$9, the principal weight is

$\Omega\subsetneqq\mathbb{R}^n$0

For

$\Omega\subsetneqq\mathbb{R}^n$1

with $\Omega\subsetneqq\mathbb{R}^n$2, $\Omega\subsetneqq\mathbb{R}^n$3, $\Omega\subsetneqq\mathbb{R}^n$4, $\Omega\subsetneqq\mathbb{R}^n$5, $\Omega\subsetneqq\mathbb{R}^n$6 on $\Omega\subsetneqq\mathbb{R}^n$7, and $\Omega\subsetneqq\mathbb{R}^n$8, the critical Hardy constant is

$\Omega\subsetneqq\mathbb{R}^n$9

There exists $1<p<\infty$0 such that

$1<p<\infty$1

Moreover, the infimum is attained for $1<p<\infty$2, not attained for $1<p<\infty$3, and at $1<p<\infty$4 it is attained if and only if

$1<p<\infty$5

This gives a full existence/nonexistence criterion for minimizers in terms of a boundary integral over the singular manifold (Fall et al., 2012).

The same variational pattern appears on compact Riemannian manifolds. If $1<p<\infty$6 is a smooth compact manifold of dimension $1<p<\infty$7 and $1<p<\infty$8 is a closed submanifold of dimension $1<p<\infty$9, with $\alpha+p>n$0, the weighted Hardy quotient

$\alpha+p>n$1

has critical value

$\alpha+p>n$2

There exists $\alpha+p>n$3 such that $\alpha+p>n$4 equals this critical value for $\alpha+p>n$5, drops below it for $\alpha+p>n$6, is attained for $\alpha+p>n$7, and is not attained for $\alpha+p>n$8. At $\alpha+p>n$9, attainment holds if and only if

$\mathbb{R}^N$00

The analysis uses Fermi coordinates, logarithmically corrected virtual ground states, and local Hardy inequalities with remainder terms of the form $\mathbb{R}^N$01 (Thiam, 2015).

These variational problems show that weighted Hardy inequalities are not only norm inequalities. They also define sharp thresholds for singular Schrödinger-type operators, with minimizers appearing or disappearing according to geometric codimension, lower-order perturbations, and the behavior of the coefficient ratio $\mathbb{R}^N$02 or $\mathbb{R}^N$03 on the singular set (Fall et al., 2012, Thiam, 2015).

4. Nonlocal, nonlinear, and variable-exponent extensions

In the fractional setting, the gradient energy is replaced by a weighted Gagliardo seminorm. For $\mathbb{R}^N$04, $\mathbb{R}^N$05, and suitable $\mathbb{R}^N$06, the weighted fractional Hardy inequality takes the form

$\mathbb{R}^N$07

Sharp constants are identified for the half-space $\mathbb{R}^N$08, convex domains, and $\mathbb{R}^N$09 (Dyda et al., 2022). For the half-space, the sharp constant is $\mathbb{R}^N$10; for the punctured space, it is $\mathbb{R}^N$11. In convex domains, the same half-space constant remains sharp. The paper also derives weighted fractional Hardy–Sobolev–Maz’ya inequalities and remainder estimates for $\mathbb{R}^N$12 based on the non-linear ground state representation of Frank and Seiringer (Dyda et al., 2022).

For the weighted $\mathbb{R}^N$13-Laplacian with Robin boundary conditions, a general abstract inequality is obtained from a positive supersolution $\mathbb{R}^N$14 satisfying

$\mathbb{R}^N$15

Then, for $\mathbb{R}^N$16,

$\mathbb{R}^N$17

plus a nonnegative remainder term involving $\mathbb{R}^N$18, with different forms for $\mathbb{R}^N$19 and $\mathbb{R}^N$20 (Kombe et al., 2021). This framework produces power, logarithmic, exponential, non-radial, Maz’ya-type, and Heisenberg–Pauli–Weyl-type inequalities with explicit interior and boundary weights.

A different extension concerns variable exponent spaces $\mathbb{R}^N$21 with $\mathbb{R}^N$22. These spaces are quasi-Banach and not locally convex, yet two-weight Hardy-type inequalities can still be proved for monotone functions. With

$\mathbb{R}^N$23

and exponents $\mathbb{R}^N$24, the paper proves estimates of the form

$\mathbb{R}^N$25

for nonnegative decreasing or increasing $\mathbb{R}^N$26, under the embedding condition

$\mathbb{R}^N$27

An analogous estimate holds for $\mathbb{R}^N$28 (Bandaliev, 2012). This places Hardy inequalities inside a non-locally convex variable-exponent regime where duality-based methods are unavailable.

5. Discrete, mean, and factorization formulations

The discrete weighted Hardy inequality is classical in the form

$\mathbb{R}^N$29

A weighted discrete version,

$\mathbb{R}^N$30

is characterized by the Andersen–Heinig condition

$\mathbb{R}^N$31

with $\mathbb{R}^N$32 (Bui et al., 2020). In the cited application, this criterion is used to solve $\mathbb{R}^N$33 in weighted Sobolev spaces on a planar cusp domain, yielding estimates relevant for Stokes and Korn theory (Bui et al., 2020).

A recent abstraction replaces prefix sums by nested averages over measurable partitions of a measure space. For

$\mathbb{R}^N$34

the inequality

$\mathbb{R}^N$35

holds if and only if the testing quantity

$\mathbb{R}^N$36

is finite, and then

$\mathbb{R}^N$37

This recovers the classical discrete Hardy inequality when $\mathbb{R}^N$38, $\mathbb{R}^N$39, $\mathbb{R}^N$40, $\mathbb{R}^N$41, and $\mathbb{R}^N$42 (Bouthat et al., 23 Jun 2026).

In the continuous one-weight setting, factorization theory refines Hardy inequalities by decomposing function spaces. If $\mathbb{R}^N$43 and $\mathbb{R}^N$44 are the Hardy and dual Hardy operators, then weighted Cesàro and Copson norms are

$\mathbb{R}^N$45

Under $\mathbb{R}^N$46,

$\mathbb{R}^N$47

where $\mathbb{R}^N$48 is the infimum of $\mathbb{R}^N$49 over factorizations $\mathbb{R}^N$50. An analogous statement holds for $\mathbb{R}^N$51 with $\mathbb{R}^N$52 and $\mathbb{R}^N$53 (Barza et al., 2017). These factorizations recover the best forms of the weighted Hardy inequalities for $\mathbb{R}^N$54 and $\mathbb{R}^N$55.

A distinct mean-theoretic generalization replaces arithmetic means by an abstract weighted mean $\mathbb{R}^N$56. Given a weight sequence $\mathbb{R}^N$57 and $\mathbb{R}^N$58, the $\mathbb{R}^N$59-weighted Hardy constant $\mathbb{R}^N$60 is the smallest $\mathbb{R}^N$61 such that

$\mathbb{R}^N$62

For symmetric, monotone, Jensen-concave weighted means satisfying the weighted Kedlaya inequality and $\mathbb{R}^N$63, this constant is

$\mathbb{R}^N$64

provided $\mathbb{R}^N$65. Moreover, for symmetric monotone means, the largest possible weighted Hardy constant over all admissible weight sequences is achieved for the constant sequence $\mathbb{R}^N$66 (Páles et al., 2017).

For non-increasing sequences, still another discrete theory studies

$\mathbb{R}^N$67

with $\mathbb{R}^N$68 non-increasing. This holds if and only if

$\mathbb{R}^N$69

and the best constants satisfy

$\mathbb{R}^N$70

improving an earlier bound in that range (Gao, 2014).

6. Techniques, sharp constants, and applications

Several proof architectures recur across the literature. One is the ground-state or supersolution method: identify a positive comparison function $\mathbb{R}^N$71, compute the action of the relevant operator on $\mathbb{R}^N$72, and derive the Hardy inequality from positivity. This method underlies the weighted $\mathbb{R}^N$73-Laplacian with Robin boundary conditions (Kombe et al., 2021), the criticality-theoretic proof of the boundary-distance inequality with sharp constant $\mathbb{R}^N$74 (Goel et al., 2020), and the fractional weighted inequalities via the non-linear ground state representation (Dyda et al., 2022).

A second framework is localization near the singular geometry. Fermi coordinates and expansions of the metric or Laplacian are used near boundary submanifolds and manifold singular sets, leading to model solutions such as

$\mathbb{R}^N$75

local improved Hardy inequalities with logarithmic remainders, and barrier constructions that distinguish existence from concentration at the singular set (Fall et al., 2012, Thiam, 2015).

A third route is direct one-dimensional integration by parts. In the elementary approach to weighted Hardy-type inequalities, one differentiates a composite quantity involving the weight and the Hardy transform, then applies Hölder’s inequality to get explicit constants. This yields the optimal constant

$\mathbb{R}^N$76

in the power-weight case and, by iteration, an infinite sequence of Birman–Hardy–Rellich-type inequalities; operator-valued versions follow by trace or positive-operator arguments (Chuah et al., 2019).

The applications are correspondingly broad. Weighted Hardy inequalities give lower bounds for singular Schrödinger and $\mathbb{R}^N$77-Laplacian forms, determine spectral thresholds and critical couplings, and govern the existence of minimizers or ground states (Fall et al., 2012, Kombe et al., 2021). In the Kolmogorov setting with invariant measure, the critical weighted Hardy inequality yields the existence/nonexistence threshold for positive exponentially bounded solutions of a parabolic problem with potential

$\mathbb{R}^N$78

in dimension two, with critical coefficient $\mathbb{R}^N$79 (Sano et al., 2018). For degenerate elliptic diffusion operators with coefficients $\mathbb{R}^N$80, the boundary Hardy constant controls essential self-adjointness, with threshold $\mathbb{R}^N$81 in the $\mathbb{R}^N$82, convex, and exterior-convex geometries covered by the theory (Robinson, 2021). In weighted Sobolev analysis on cuspidal domains, discrete Hardy inequalities provide the gluing mechanism in local-to-global constructions for $\mathbb{R}^N$83, and the same machinery is linked in the cited work to Korn inequalities and the Stokes equations (Bui et al., 2020).

Weighted Hardy inequalities therefore form a unified family of sharp coercive estimates whose geometry is encoded by distance functions, codimension, curvature, Hausdorff dimension, or nested averaging structure. Their constants are often optimal, their failure is frequently tied to critical thresholds, and their modern formulations connect PDE, spectral theory, metric geometry, and discrete analysis in a common variational language.