Improved quantitative stability for the critical Hardy inequality

Abstract: We establish a quantitative stability estimate for the critical Hardy inequality on bounded domains containing the origin. Our result improves the existing quantitative stability estimate by reducing the exponent in the distance function from $N<sup>{2}$ to $N$, replacing the Lorentz-Zygmund framework with the Luxemburg norm of the critical exponential Orlicz space $\operatorname{Exp}L<sup>{\frac{N}{N-1}}(Ω)$, and avoiding any cut-off modification of the virtual extremizers. As a consequence, we also obtain an improved quantitative stability estimate for the critical Hardy inequality with the logarithmic weight considered by Cianchi and Ferone, where the exponent is likewise reduced from $N<sup>{2}$ to $N$ and the distance is measured directly from the virtual extremizer without truncation. The proof is completely rearrangement-free and relies on a critical Hardy inequality with a remainder term, scale-invariant Sobolev inequalities, and a refined dyadic summation argument. These ingredients yield stronger quantitative stability estimates for both forms of the critical Hardy inequality.

• The paper proves that the critical Hardy deficit controls the exponential-Orlicz distance to the virtual extremizer family with distance exponent N, improving the previous exponent N².
• It combines a Hardy inequality with remainder, scale-invariant Sobolev estimates on annuli, and dyadic summation to establish stability without rearrangement arguments.
• The results apply to both a more singular logarithmic weight and the Cianchi–Ferone weight, use untruncated extremizers, and leave the optimality of exponent N open.

Background and context

The sharp Hardy inequality on $\mathbb{R}^N$, valid for $1$((N-p)/p)^p$ that is not attained in $\mathcal{D}^{1,p}(\mathbb{R}^N)$. Its formal extremals, the virtual extremizers $v_a(x)=a|x|^{-(N-p)/p}$, lie outside the natural energy space but belong to the Marcinkiewicz space $L^{p^*,\infty}$. Cianchi and Ferone exploited this to prove a first quantitative stability result: the Hardy deficit controls the distance from the virtual extremizer family with exponent $2p^*$, later improved by Banerjee, Ganguly and Sahu to $\max\{4,2p\}$ using rearrangement-free methods (2608.19732).

In the critical case $p=N$ the weight $|x|^{-N}$ is neither locally integrable at the origin nor at infinity, so the inequality must be reformulated on bounded domains with a logarithmic correction. Cianchi and Ferone treated the weight $1

The paper under review establishes a substantially sharper stability estimate for the critical Hardy inequality of Ioku and Ishiwata, whose logarithmic weight $1

Main results

The central object is the virtual extremizer

$1

which satisfies the Euler–Lagrange equation for the critical Hardy inequality with constant $1$((N-p)/p)^p$0, the Orlicz space generated by $((N-p)/p)^p$1. The paper proves two theorems.

Theorem 1. For any bounded open $((N-p)/p)^p$2 containing the origin, $((N-p)/p)^p$3, there exists $((N-p)/p)^p$4 such that for all $((N-p)/p)^p$5,

$((N-p)/p)^p$6

where $((N-p)/p)^p$7 is the Hardy deficit and $((N-p)/p)^p$8 is the Luxemburg-norm-based Orlicz distance from the family $((N-p)/p)^p$9.

Theorem 2. The same estimate holds for the Cianchi–Ferone weight $\mathcal{D}^{1,p}(\mathbb{R}^N)$0, with exponent $\mathcal{D}^{1,p}(\mathbb{R}^N)$1 instead of $\mathcal{D}^{1,p}(\mathbb{R}^N)$2, distance measured directly from the untruncated extremizer $\mathcal{D}^{1,p}(\mathbb{R}^N)$3, and validity extended to all $\mathcal{D}^{1,p}(\mathbb{R}^N)$4 rather than only $\mathcal{D}^{1,p}(\mathbb{R}^N)$5.

Relative to Cianchi and Ferone's estimate, these results improve three aspects simultaneously: the exponent drops from $\mathcal{D}^{1,p}(\mathbb{R}^N)$6 to $\mathcal{D}^{1,p}(\mathbb{R}^N)$7; the normalization uses the Luxemburg norm of $\mathcal{D}^{1,p}(\mathbb{R}^N)$8 instead of the Lorentz–Zygmund norm — a genuine strengthening since $\mathcal{D}^{1,p}(\mathbb{R}^N)$9, so the denominator of the distance functional is smaller and the right-hand side correspondingly larger; and no cut-off modification of the extremizers is required. A summary comparison:

Aspect	Cianchi–Ferone	This paper
Exponent on distance	$v_a(x)=a|x|^{-(N-p)/p}$0	$v_a(x)=a|x|^{-(N-p)/p}$1
Normalizing space	$v_a(x)=a|x|^{-(N-p)/p}$2	$v_a(x)=a|x|^{-(N-p)/p}$3
Extremizers used	Truncated ($v_a(x)=a|x|^{-(N-p)/p}$4)	Untruncated $v_a(x)=a|x|^{-(N-p)/p}$5, $v_a(x)=a|x|^{-(N-p)/p}$6
Method	Rearrangement-based	Rearrangement-free

Key ingredients of the proof

Three components combine into a fully local argument. First, a critical Hardy inequality with remainder: writing $v_a(x)=a|x|^{-(N-p)/p}$7 and applying the convexity estimate $v_a(x)=a|x|^{-(N-p)/p}$8 together with the Euler–Lagrange equation satisfied by $v_a(x)=a|x|^{-(N-p)/p}$9, the cross terms cancel exactly and one obtains

$L^{p^*,\infty}$0

with $L^{p^*,\infty}$1.

Second, a scale-invariant Sobolev inequality on annuli: for bounded $L^{p^*,\infty}$2 domains and every $L^{p^*,\infty}$3,

$L^{p^*,\infty}$4

with constant independent of the scaling parameter $L^{p^*,\infty}$5 and explicit $L^{p^*,\infty}$6-dependence. This is derived from the global Sobolev embedding plus Poincaré, applied to rescaled functions.

Third, a dyadic summation argument over annuli $L^{p^*,\infty}$7: the deviation of $L^{p^*,\infty}$8 from its threshold level on the "bad" set $L^{p^*,\infty}$9 is controlled annulus-by-annulus, using an elementary lemma bounding differences of averages over disjoint sets, then summed against the weights $2p^*$0 via the bound $2p^*$1. Summing the resulting moment estimates $2p^*$2 as a series yields, by Stirling's approximation, exponential integrability with respect to $2p^*$3:

$2p^*$4

and a complementary bound on $2p^*$5 follows directly since there $2p^*$6. The case split $2p^*$7 versus $2p^*$8, extension to sign-changing functions via $2p^*$9 and convexity of $\max\{4,2p\}$0, removal of the normalization by scaling, and a density/Fatou argument extend the result from nonnegative compactly supported functions to all of $\max\{4,2p\}$1.

The proof of Theorem 2 reduces to Theorem 1 via the identity $\max\{4,2p\}$2 for $\max\{4,2p\}$3 in the sub-unitary-deficit case, and via the continuous embedding $\max\{4,2p\}$4 (a consequence of the strict inclusion $\max\{4,2p\}$5) in the super-unitary case, where the normalization is now taken in the Lorentz–Zygmund scale.

It is worth noting that the exponent $\max\{4,2p\}$6 arises structurally from the dyadic summation; it is not tuned to be sharp. The paper explicitly leaves open whether the exponent can be further reduced below $\max\{4,2p\}$7 — indeed whether $\max\{4,2p\}$8 is optimal remains unresolved.

Limitations and open questions

Two limitations are inherent in the framework. The main theorem requires the strict condition $\max\{4,2p\}$9: the weight $p=N$0 must remain integrable up to the boundary of $p=N$1, and the case $p=N$2 is excluded for Theorem 1 (though covered for the milder Cianchi–Ferone weight in Theorem 2). Second, the stability constants depend on $p=N$3, $p=N$4, and $p=N$5 through quantities such as $p=N$6, which deteriorate as $p=N$7; no uniform-in-$p=N$8 statement is claimed. The optimality of the exponent $p=N$9 — both for the distance power and for the choice of the $|x|^{-N}$0 normalization — is left as an open problem, as is any analogue for the fractional or higher-order critical Hardy inequalities.

Conclusion

This work improves the known quantitative stability theory for the critical Hardy inequality on three independent axes: the distance exponent is lowered from $|x|^{-N}$1 to $|x|^{-N}$2, the normalization is strengthened from the Lorentz–Zygmund norm to the Luxemburg norm of the exponential Orlicz space containing the true virtual extremizers, and truncation of the extremizers is eliminated. Achieving this while working with a strictly more singular Hardy weight than the classical Cianchi–Ferone setting, and without any recourse to symmetrization, demonstrates that rearrangement-free techniques based on scale-invariant Sobolev estimates, remainder-form Hardy inequalities, and dyadic summation are capable of producing sharper stability estimates than rearrangement-based ones. The principal question the paper does not settle is whether the exponent $|x|^{-N}$3 is optimal.