Hardy's Inequality with Power Weights

• Hardy's inequality with power weights is a family of sharp estimates comparing weighted norms of functions and their derivatives through power laws and logarithmic refinements.
• It encompasses both continuous and discrete frameworks, using optimal constants derived from scaling properties and precise admissible parameter ranges.
• Recent research extends the theory to two-weight criteria, boundary-distance and fractional variants, providing quantitative stability and explicit operator formulations.

Searching arXiv for recent and foundational papers on Hardy inequalities with power weights, including discrete, continuous, weighted, and logarithmic refinement variants. Hardy’s inequality with power weights is a family of estimates that compare a weighted norm of a function, an averaged primitive, or a difference quotient with a weighted norm of a derivative, gradient, Laplacian, or fractional increment. In the one-dimensional differential form, a representative sharp statement is

$$\int_0^b x^{\alpha}|f'(x)|^p\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^p \int_0^b x^{\alpha-p}|f(x)|^p\,dx,$$

valid for $0<b\le \infty$, $p\in[1,\infty)$, and $\alpha\in\mathbb{R}$, with best constant and equality only for the trivial function (Chuah et al., 2019). In current research, this theme appears in continuous, discrete, fractional, two-weight, boundary-distance, and geometric forms, with sharp constants, critical borderline replacements by logarithmic weights, and, in some settings, quantitative stability estimates (Barki, 2024).

1. One-dimensional weighted Hardy inequalities and operator forms

A basic power-weighted formulation on an interval is the differential inequality

$$\int_0^b x^{\alpha}|f'(x)|^p\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^p \int_0^b x^{\alpha-p}|f(x)|^p\,dx,$$

with sharp constant $(|\,\alpha-p+1\,|/p)^p$ (Chuah et al., 2019). The same work also gives the integral forms

$$\int_0^{b} x^{\alpha}\,\|F(x)\|_{B}^{p}\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^{p}\int_{0}^{b} x^{\alpha-p}\Big\| \int_{0}^{x} F(x')\,dx' \Big\|_{B}^{p}\,dx, \quad \alpha<p-1,$$

and

$$\int_0^{b} x^{\alpha}\,\|F(x)\|_{B}^{p}\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^{p}\int_{0}^{b} x^{\alpha-p}\Big\| \int_{x}^{b} F(x')\,dx' \Big\|_{B}^{p}\,dx, \quad \alpha>p-1,$$

which isolate the two one-sided Hardy operators (Chuah et al., 2019).

Iteration produces the one-dimensional Birman–Hardy–Rellich sequence

$$\int_0^b x^{\alpha}|f^{(n)}(x)|^p\,dx \ge \Bigg(\prod_{j=1}^{k}\Big(\frac{|\,\alpha-jp+1\,|}{p}\Big)^p\Bigg) \int_0^b x^{\alpha-kp}|f^{(n-k)}(x)|^p\,dx,$$

for $1\le k\le n$, again with optimal product constant (Chuah et al., 2019). In the quadratic case, a parallel $0<b\le \infty$0 theory with logarithmic refinements on finite interior and exterior intervals was developed for all integer orders $0<b\le \infty$1, with

$0<b\le \infty$2

as the principal power-weight constant and strict logarithmic improvements on the right-hand side (Gesztesy et al., 2020).

For the classical Hardy operator

$0<b\le \infty$3

power weights lead to sharp bounds of the form

$0<b\le \infty$4

and the operator $0<b\le \infty$5 admits a separate sharp weighted theory in $0<b\le \infty$6, with distinct optimal constants for general functions, positive functions, and positive decreasing functions (Strzelecki, 2019). That setting also yields sharp comparisons between $0<b\le \infty$7 and the dual Hardy operator $0<b\le \infty$8 under the same power weights (Strzelecki, 2019).

2. Two-weight criteria and power-weighted Hardy operators

A general two-weight Hardy framework replaces a single power by a pair of measures. On metric measure spaces with polar decomposition, the ball-integral operator

$0<b\le \infty$9

satisfies

$p\in[1,\infty)$0

if and only if the corresponding Muckenhoupt–Sawyer-type testing quantity

$p\in[1,\infty)$1

is finite, where $p\in[1,\infty)$2 and $p\in[1,\infty)$3 are the tail and ball integrals of $p\in[1,\infty)$4 and $p\in[1,\infty)$5, respectively (Ruzhansky et al., 2018). The same paper gives equivalent formulations $p\in[1,\infty)$6–$p\in[1,\infty)$7 and two-sided estimates relating them to the optimal constant (Ruzhansky et al., 2018).

For homogeneous groups with homogeneous dimension $p\in[1,\infty)$8, choosing $p\in[1,\infty)$9 and $\alpha\in\mathbb{R}$0 yields a precise admissibility condition: $\alpha\in\mathbb{R}$1 together with explicit bounds for the best constant (Ruzhansky et al., 2018). The Euclidean case is recovered by taking $\alpha\in\mathbb{R}$2 (Ruzhansky et al., 2018).

A mixed-norm two-weight formulation also appears in the measure-theoretic Hardy inequality

$\alpha\in\mathbb{R}$3

with

$\alpha\in\mathbb{R}$4

and, for power weights $\alpha\in\mathbb{R}$5, $\alpha\in\mathbb{R}$6, finiteness of $\alpha\in\mathbb{R}$7 is equivalent to

$\alpha\in\mathbb{R}$8

(Klaassen et al., 2020).

Within the theory of weighted means, another Hardy-type perspective fixes a weight sequence $\alpha\in\mathbb{R}$9 and studies

$$\int_0^b x^{\alpha}|f'(x)|^p\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^p \int_0^b x^{\alpha-p}|f(x)|^p\,dx,$$0

For symmetric, monotone weighted means, the largest weighted Hardy constant is achieved by the constant weight vector (Páles et al., 2017). For power weights $$\int_0^b x^{\alpha}|f'(x)|^p\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^p \int_0^b x^{\alpha-p}|f(x)|^p\,dx,$$1, the arithmetic mean has finite sharp constant precisely when $$\int_0^b x^{\alpha}|f'(x)|^p\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^p \int_0^b x^{\alpha-p}|f(x)|^p\,dx,$$2, in which case the constant is $$\int_0^b x^{\alpha}|f'(x)|^p\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^p \int_0^b x^{\alpha-p}|f(x)|^p\,dx,$$3; for $$\int_0^b x^{\alpha}|f'(x)|^p\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^p \int_0^b x^{\alpha-p}|f(x)|^p\,dx,$$4 the constant is infinite (Páles et al., 2017). By contrast, for power means $$\int_0^b x^{\alpha}|f'(x)|^p\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^p \int_0^b x^{\alpha-p}|f(x)|^p\,dx,$$5 with $$\int_0^b x^{\alpha}|f'(x)|^p\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^p \int_0^b x^{\alpha-p}|f(x)|^p\,dx,$$6, the sharp constant is independent of $$\int_0^b x^{\alpha}|f'(x)|^p\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^p \int_0^b x^{\alpha-p}|f(x)|^p\,dx,$$7 when $$\int_0^b x^{\alpha}|f'(x)|^p\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^p \int_0^b x^{\alpha-p}|f(x)|^p\,dx,$$8: $$\int_0^b x^{\alpha}|f'(x)|^p\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^p \int_0^b x^{\alpha-p}|f(x)|^p\,dx,$$9 for the geometric mean, $(|\,\alpha-p+1\,|/p)^p$0 for the harmonic mean, and $(|\,\alpha-p+1\,|/p)^p$1 for $(|\,\alpha-p+1\,|/p)^p$2 (Páles et al., 2017).

3. Discrete power weights on the half-line and on lattices

On the discrete half-line, the natural gradient Hardy form is

$(|\,\alpha-p+1\,|/p)^p$3

For general positive weights $(|\,\alpha-p+1\,|/p)^p$4 on $(|\,\alpha-p+1\,|/p)^p$5, a sharp discrete Muckenhoupt/Sawyer characterization is given by

$(|\,\alpha-p+1\,|/p)^p$6

with

$(|\,\alpha-p+1\,|/p)^p$7

and an analogous $(|\,\alpha-p+1\,|/p)^p$8 when the inequality is restricted to zero-sum sequences (Barki, 2024).

Specializing to power weights $(|\,\alpha-p+1\,|/p)^p$9 and $$\int_0^{b} x^{\alpha}\,\|F(x)\|_{B}^{p}\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^{p}\int_{0}^{b} x^{\alpha-p}\Big\| \int_{0}^{x} F(x')\,dx' \Big\|_{B}^{p}\,dx, \quad \alpha<p-1,$$0, one obtains the main sharp discrete $$\int_0^{b} x^{\alpha}\,\|F(x)\|_{B}^{p}\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^{p}\int_{0}^{b} x^{\alpha-p}\Big\| \int_{0}^{x} F(x')\,dx' \Big\|_{B}^{p}\,dx, \quad \alpha<p-1,$$1-Hardy inequality

$$\int_0^{b} x^{\alpha}\,\|F(x)\|_{B}^{p}\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^{p}\int_{0}^{b} x^{\alpha-p}\Big\| \int_{0}^{x} F(x')\,dx' \Big\|_{B}^{p}\,dx, \quad \alpha<p-1,$$2

with

$$\int_0^{b} x^{\alpha}\,\|F(x)\|_{B}^{p}\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^{p}\int_{0}^{b} x^{\alpha-p}\Big\| \int_{0}^{x} F(x')\,dx' \Big\|_{B}^{p}\,dx, \quad \alpha<p-1,$$3

and the constant is sharp in $$\int_0^{b} x^{\alpha}\,\|F(x)\|_{B}^{p}\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^{p}\int_{0}^{b} x^{\alpha-p}\Big\| \int_{0}^{x} F(x')\,dx' \Big\|_{B}^{p}\,dx, \quad \alpha<p-1,$$4 (Barki, 2024). The corresponding averaged Hardy operator inequality is

$$\int_0^{b} x^{\alpha}\,\|F(x)\|_{B}^{p}\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^{p}\int_{0}^{b} x^{\alpha-p}\Big\| \int_{0}^{x} F(x')\,dx' \Big\|_{B}^{p}\,dx, \quad \alpha<p-1,$$5

again sharp for $$\int_0^{b} x^{\alpha}\,\|F(x)\|_{B}^{p}\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^{p}\int_{0}^{b} x^{\alpha-p}\Big\| \int_{0}^{x} F(x')\,dx' \Big\|_{B}^{p}\,dx, \quad \alpha<p-1,$$6, $$\int_0^{b} x^{\alpha}\,\|F(x)\|_{B}^{p}\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^{p}\int_{0}^{b} x^{\alpha-p}\Big\| \int_{0}^{x} F(x')\,dx' \Big\|_{B}^{p}\,dx, \quad \alpha<p-1,$$7 (Barki, 2024).

At the critical value $$\int_0^{b} x^{\alpha}\,\|F(x)\|_{B}^{p}\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^{p}\int_{0}^{b} x^{\alpha-p}\Big\| \int_{0}^{x} F(x')\,dx' \Big\|_{B}^{p}\,dx, \quad \alpha<p-1,$$8, the correct discrete replacement is a Leray-type inequality: $$\int_0^{b} x^{\alpha}\,\|F(x)\|_{B}^{p}\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^{p}\int_{0}^{b} x^{\alpha-p}\Big\| \int_{0}^{x} F(x')\,dx' \Big\|_{B}^{p}\,dx, \quad \alpha<p-1,$$9 with the same sharp constant as in the continuum (Barki, 2024). For $$\int_0^{b} x^{\alpha}\,\|F(x)\|_{B}^{p}\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^{p}\int_{0}^{b} x^{\alpha-p}\Big\| \int_{x}^{b} F(x')\,dx' \Big\|_{B}^{p}\,dx, \quad \alpha>p-1,$$0, the discrete and continuous constants satisfy

$$\int_0^{b} x^{\alpha}\,\|F(x)\|_{B}^{p}\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^{p}\int_{0}^{b} x^{\alpha-p}\Big\| \int_{x}^{b} F(x')\,dx' \Big\|_{B}^{p}\,dx, \quad \alpha>p-1,$$1

and equality with the continuous constant can fail (Barki, 2024).

The quadratic case $$\int_0^{b} x^{\alpha}\,\|F(x)\|_{B}^{p}\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^{p}\int_{0}^{b} x^{\alpha-p}\Big\| \int_{x}^{b} F(x')\,dx' \Big\|_{B}^{p}\,dx, \quad \alpha>p-1,$$2 had been treated earlier on the half-line: for $$\int_0^{b} x^{\alpha}\,\|F(x)\|_{B}^{p}\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^{p}\int_{0}^{b} x^{\alpha-p}\Big\| \int_{x}^{b} F(x')\,dx' \Big\|_{B}^{p}\,dx, \quad \alpha>p-1,$$3,

$$\int_0^{b} x^{\alpha}\,\|F(x)\|_{B}^{p}\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^{p}\int_{0}^{b} x^{\alpha-p}\Big\| \int_{x}^{b} F(x')\,dx' \Big\|_{B}^{p}\,dx, \quad \alpha>p-1,$$4

with sharp constant, and for $$\int_0^{b} x^{\alpha}\,\|F(x)\|_{B}^{p}\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^{p}\int_{0}^{b} x^{\alpha-p}\Big\| \int_{x}^{b} F(x')\,dx' \Big\|_{B}^{p}\,dx, \quad \alpha>p-1,$$5 an improved expansion by positive remainder terms

$$\int_0^{b} x^{\alpha}\,\|F(x)\|_{B}^{p}\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^{p}\int_{0}^{b} x^{\alpha-p}\Big\| \int_{x}^{b} F(x')\,dx' \Big\|_{B}^{p}\,dx, \quad \alpha>p-1,$$6

was obtained (Gupta, 2021). The later $$\int_0^{b} x^{\alpha}\,\|F(x)\|_{B}^{p}\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^{p}\int_{0}^{b} x^{\alpha-p}\Big\| \int_{x}^{b} F(x')\,dx' \Big\|_{B}^{p}\,dx, \quad \alpha>p-1,$$7 theory extends the underlying two-weight criterion to all exponents and recovers sharp equality with the continuous constant for $$\int_0^{b} x^{\alpha}\,\|F(x)\|_{B}^{p}\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^{p}\int_{0}^{b} x^{\alpha-p}\Big\| \int_{x}^{b} F(x')\,dx' \Big\|_{B}^{p}\,dx, \quad \alpha>p-1,$$8, $$\int_0^{b} x^{\alpha}\,\|F(x)\|_{B}^{p}\,dx \ge \Big(\frac{|\,\alpha-p+1\,|}{p}\Big)^{p}\int_{0}^{b} x^{\alpha-p}\Big\| \int_{x}^{b} F(x')\,dx' \Big\|_{B}^{p}\,dx, \quad \alpha>p-1,$$9 (Barki, 2024).

On the full lattice $$\int_0^b x^{\alpha}|f^{(n)}(x)|^p\,dx \ge \Bigg(\prod_{j=1}^{k}\Big(\frac{|\,\alpha-jp+1\,|}{p}\Big)^p\Bigg) \int_0^b x^{\alpha-kp}|f^{(n-k)}(x)|^p\,dx,$$0, a separate non-fractional theory yields a complete characterization in the class of power weights $$\int_0^b x^{\alpha}|f^{(n)}(x)|^p\,dx \ge \Bigg(\prod_{j=1}^{k}\Big(\frac{|\,\alpha-jp+1\,|}{p}\Big)^p\Bigg) \int_0^b x^{\alpha-kp}|f^{(n-k)}(x)|^p\,dx,$$1. For $$\int_0^b x^{\alpha}|f^{(n)}(x)|^p\,dx \ge \Bigg(\prod_{j=1}^{k}\Big(\frac{|\,\alpha-jp+1\,|}{p}\Big)^p\Bigg) \int_0^b x^{\alpha-kp}|f^{(n-k)}(x)|^p\,dx,$$2, the critical exponent is $$\int_0^b x^{\alpha}|f^{(n)}(x)|^p\,dx \ge \Bigg(\prod_{j=1}^{k}\Big(\frac{|\,\alpha-jp+1\,|}{p}\Big)^p\Bigg) \int_0^b x^{\alpha-kp}|f^{(n-k)}(x)|^p\,dx,$$3; for $$\int_0^b x^{\alpha}|f^{(n)}(x)|^p\,dx \ge \Bigg(\prod_{j=1}^{k}\Big(\frac{|\,\alpha-jp+1\,|}{p}\Big)^p\Bigg) \int_0^b x^{\alpha-kp}|f^{(n-k)}(x)|^p\,dx,$$4, one needs $$\int_0^b x^{\alpha}|f^{(n)}(x)|^p\,dx \ge \Bigg(\prod_{j=1}^{k}\Big(\frac{|\,\alpha-jp+1\,|}{p}\Big)^p\Bigg) \int_0^b x^{\alpha-kp}|f^{(n-k)}(x)|^p\,dx,$$5; for $$\int_0^b x^{\alpha}|f^{(n)}(x)|^p\,dx \ge \Bigg(\prod_{j=1}^{k}\Big(\frac{|\,\alpha-jp+1\,|}{p}\Big)^p\Bigg) \int_0^b x^{\alpha-kp}|f^{(n-k)}(x)|^p\,dx,$$6, the inequality with $$\int_0^b x^{\alpha}|f^{(n)}(x)|^p\,dx \ge \Bigg(\prod_{j=1}^{k}\Big(\frac{|\,\alpha-jp+1\,|}{p}\Big)^p\Bigg) \int_0^b x^{\alpha-kp}|f^{(n-k)}(x)|^p\,dx,$$7 holds for functions vanishing at the origin; and for $$\int_0^b x^{\alpha}|f^{(n)}(x)|^p\,dx \ge \Bigg(\prod_{j=1}^{k}\Big(\frac{|\,\alpha-jp+1\,|}{p}\Big)^p\Bigg) \int_0^b x^{\alpha-kp}|f^{(n-k)}(x)|^p\,dx,$$8, the critical exponent is $$\int_0^b x^{\alpha}|f^{(n)}(x)|^p\,dx \ge \Bigg(\prod_{j=1}^{k}\Big(\frac{|\,\alpha-jp+1\,|}{p}\Big)^p\Bigg) \int_0^b x^{\alpha-kp}|f^{(n-k)}(x)|^p\,dx,$$9 (Dyda, 9 Jun 2025). The same paper establishes a parallel fractional theory with the critical threshold $1\le k\le n$0 and the expected $1\le k\le n$1-loss in the critical case (Dyda, 9 Jun 2025).

4. Euclidean, boundary-distance, and homogeneous higher-dimensional forms

In $1\le k\le n$2, the standard power-weighted Hardy inequality takes the form

$1\le k\le n$3

for $1\le k\le n$4, $1\le k\le n$5, and $1\le k\le n$6, with optimal constant and non-attainment for nonzero $1\le k\le n$7 (Gesztesy et al., 2024). A refined radial-derivative version replaces $1\le k\le n$8 by $1\le k\le n$9 and has the same optimal constant (Gesztesy et al., 2024).

A broad homogeneous-weight extension considers angular factors $0<b\le \infty$00 on the sphere and proves

$0<b\le \infty$01

If $0<b\le \infty$02 and $0<b\le \infty$03, then

$0<b\le \infty$04

and for $0<b\le \infty$05 this reduces to the sharp Caffarelli–Kohn–Nirenberg Hardy constant (Roy, 6 Sep 2025). The same work also gives a sharp fractional counterpart with $0<b\le \infty$06 and the sharp Frank–Seiringer constant $0<b\le \infty$07 (Roy, 6 Sep 2025).

Boundary-distance formulations replace $0<b\le \infty$08 by $0<b\le \infty$09. If an open set $0<b\le \infty$10 satisfies the inner boundary density condition

$0<b\le \infty$11

then $0<b\le \infty$12 admits the weighted Hardy inequality

$0<b\le \infty$13

for all $0<b\le \infty$14, and the endpoint is sharp in general (Lehrbäck, 2012). In particular, the codimension-one case $0<b\le \infty$15 yields the familiar range $0<b\le \infty$16 beyond Lipschitz domains (Lehrbäck, 2012).

A distinct anisotropic higher-dimensional model appears on the first orthant $0<b\le \infty$17, where

$0<b\le \infty$18

for $0<b\le \infty$19 and $0<b\le \infty$20 (Kömbe et al., 2021). This produces orthant Hardy–Sobolev and Hardy–Sobolev–Maz’ya inequalities with explicit remainder terms (Kömbe et al., 2021).

5. Critical exponents and logarithmic refinements

At critical exponents, the power-weighted Hardy constant vanishes or ceases to be coercive, and logarithmic terms replace the missing power gain. On the discrete half-line, the critical value $0<b\le \infty$21 gives the discrete Leray inequality

$0<b\le \infty$22

which is sharp and matches the continuum constant (Barki, 2024).

For one-sided one-dimensional boundary problems on $0<b\le \infty$23, the critical range is organized by the threshold $0<b\le \infty$24. In the noncritical case $0<b\le \infty$25,

$0<b\le \infty$26

with sharp endpoint coefficient (Liu et al., 2019). At the critical value $0<b\le \infty$27, the pure power term is replaced by a logarithmic weight: $0<b\le \infty$28 where $0<b\le \infty$29 (Liu et al., 2019).

For second-order inequalities on balls $0<b\le \infty$30, a logarithmic refinement of the power-weighted Hardy–Rellich inequality reads

$0<b\le \infty$31

valid for all $0<b\le \infty$32, all $0<b\le \infty$33, and $0<b\le \infty$34 (Gesztesy et al., 2024). This remains meaningful even when the principal spectral constant $0<b\le \infty$35 vanishes (Gesztesy et al., 2024).

The one-dimensional weighted Birman–Hardy–Rellich sequence also admits logarithmic improvements with iterated logarithms $0<b\le \infty$36 and normalized logarithms $0<b\le \infty$37, adding a cascade of lower-order positive terms while preserving the sharp power-weight constant $0<b\le \infty$38 (Gesztesy et al., 2020).

6. Sharpness, stability, and proof mechanisms

Sharpness is typically established by near-extremal power profiles. In the continuous one-dimensional theory, profiles of the form $0<b\le \infty$39 with $0<b\le \infty$40 compactly supported near the singular point approximate the sharp constant $0<b\le \infty$41 (Chuah et al., 2019). In the discrete half-line theory, sampling smooth compactly supported functions by $0<b\le \infty$42 and passing to $0<b\le \infty$43 shows that, for $0<b\le \infty$44, $0<b\le \infty$45, the discrete sharp constant matches the continuous one (Barki, 2024). In the $0<b\le \infty$46, $0<b\le \infty$47 discrete setting, asymptotics of the optimal discrete Hardy weight show that $0<b\le \infty$48 for every negative integer $0<b\le \infty$49 (Barki, 2024).

A quantitative stability theory is available in the discrete $0<b\le \infty$50-Hardy setting. For $0<b\le \infty$51, $0<b\le \infty$52, there exists a positive remainder weight $0<b\le \infty$53 such that

$0<b\le \infty$54

so the Hardy deficit controls a weighted $0<b\le \infty$55-distance to the unique optimizer $0<b\le \infty$56 and yields $0<b\le \infty$57-stability with $0<b\le \infty$58 (Barki, 2024). For $0<b\le \infty$59, the remainder is explicit and recovers the positive remainder found by Frank–Kovarik–Pinchover (Barki, 2024).

The proof technology varies by setting but is structurally consistent. Two-weight operator inequalities rely on Muckenhoupt–Sawyer testing quantities and quantitative upper-lower bounds for the optimal constant (Barki, 2024). Sharp power-weight constants on the discrete half-line use refined estimates for sums $0<b\le \infty$60, together with Jensen and Hermite–Hadamard inequalities (Barki, 2024). Higher-dimensional Euclidean Hardy and Hardy–Rellich inequalities are derived by factorization of weighted differential operators and, in the sharp spectral theory, by spherical harmonic decomposition (Gesztesy et al., 2024). Boundary-distance inequalities depend on Hausdorff-content thickness, Poincaré inequalities, Whitney-chain arguments, and maximal-function control (Lehrbäck, 2012). Fractional and Heisenberg-group variants employ ground-state representations and explicit integral kernels (Roncal et al., 2015).

Taken together, these results show that Hardy’s inequality with power weights is not a single estimate but a hierarchy of sharp coercive principles. The leading constant is often dictated by scaling, but the decisive analytic content lies in the admissible parameter range, the treatment of critical exponents, and the transition from pure power laws to logarithmic corrections, angular factors, anisotropic weights, or discrete graph and lattice geometries (Barki, 2024).