Sharp forms and quantitative stability for general weighted discrete $p$-Hardy inequalities

Abstract: In this paper, we provide a sharp remainder term for the general weighted discrete $p$-Hardy inequality. By simply choosing weights and specifying $1<p<\infty$, we are able to recover the identity by Krej{č}i{ř}{\'ı}k-Štampach [KS22, Theorem 1], obtain the sharp form of the $p$-Hardy inequality by Fischer-Keller-Pogorzelski [FKP23, Theorem 1] and generalize the power weighted inequality by Gupta [Gup22, Theorem 2.1]{gupta2022discrete} with sharp remainder. In addition, we prove a quantitative stability result, thereby showing that any minimizing sequence of the discrete $p$-Hardy inequality must approach the family of non-trivial minimizers.

• The paper establishes a unified sharp identity for weighted discrete p-Hardy inequalities, incorporating explicit remainder terms for stability analysis.
• The paper introduces a flexible framework based on the C_p-functional that parametrizes weights and auxiliary functions to unify and extend known inequalities.
• The paper provides sharp quantitative stability estimates that control the distance to extremality with explicit convergence rates for minimizing sequences.

Sharp Forms and Quantitative Stability for General Weighted Discrete $p$-Hardy Inequalities

Introduction and Background

This paper develops sharp forms and quantitative stability results for general weighted discrete $p$-Hardy inequalities. The starting point is the classical discrete $p$-Hardy inequality, which for sequences $\{a_n\}_{n=1}^\infty \subset [0, \infty)$ and $1 < p < \infty$ asserts:

$$\sum_{n=1}^{\infty} a_n^p \geq \left(\frac{p-1}{p}\right)^p \sum_{n=1}^{\infty} \left(\frac{a_1 + a_2 + \ldots + a_n}{n}\right)^p$$

with sharp constant and equality for the trivial sequence. While the continuous Hardy inequality and its spectral consequences are extensively explored, their discrete analogs present unique features. For instance, the optimality of weights and remainder terms fundamentally diverges from the continuous situation, due to the absence of calculus-based tools and the nonexistence of analogs like polar coordinates.

Sophisticated advancements in the past decades (e.g., [krejcirik2022sharp], [fischer2023improved]) have delivered sharp constants and improved weights, showing that the optimal weight in the discrete $p$-Hardy setting strictly exceeds the naive analog of the continuous weight. Further, the algebraic structure in the discrete case often allows identities with sharp non-negative remainder terms, facilitating equality characterization and leading to refined stability analysis. These differences have led to novel concepts, such as the $C_p$-functional, for quantifying remainders in discrete inequalities.

Main Contributions

The authors present several key advances:

1. A unified sharp remainder formula for general weighted discrete $p$-Hardy inequalities: This result incorporates and strengthens recent sharp inequalities by Krejčiřík–Štampach [krejcirik2022sharp], Fischer–Keller–Pogorzelski [fischer2023improved], and Gupta [gupta2022discrete].
2. A flexible framework based on the $C_p$-functional: The identity is parameterized by weights $p$0, an auxiliary function $p$1, and a weight $p$2, subject to a structural difference inequality. This setup encompasses a broad family of discrete Hardy-type inequalities, recovers known sharp cases as specializations, and allows systematic derivation of new optimal inequalities for complex and real sequences.
3. Sharp quantitative stability estimates: The authors establish stability in the sense of quantitative remainder lower bounds, analogous to the celebrated Bianchi–Egnell results for Sobolev inequalities but in the discrete $p$3-Hardy context. In particular, any minimizing sequence for the $p$4-Hardy inequality (subject to compact support and boundary conditions) must converge, with explicit rate, toward the (formal) extremal profile---typically $p$5---modulo normalization.

The General Weighted Discrete $p$6-Hardy Identity

The core result is the following: Suppose $p$7 and $p$8 are non-negative functions on $p$9, and $p$0 is positive, non-decreasing, and satisfies $p$1. If

$p$2

then for any compactly supported $p$3 with $p$4, the identity holds:

$p$5

where $p$6 is a sharp, non-negative remainder term defined via the $p$7-functional:

$p$8

Moreover, if the structural condition on $p$9 is met with equality, then the Hardy identity holds with equality. Specializing to weights and $\{a_n\}_{n=1}^\infty \subset [0, \infty)$0 corresponding to known inequalities recovers and extends the results of previous works, including the sharp forms for weights of the type $\{a_n\}_{n=1}^\infty \subset [0, \infty)$1, weighted Copson inequalities, and others.

Explicit Stability and Remainder Analysis

Utilizing recent lower bounds on the $\{a_n\}_{n=1}^\infty \subset [0, \infty)$2-functional (e.g., [cazacu2024hardy]), the remainder can be quantitatively estimated to yield a lower bound involving the $\{a_n\}_{n=1}^\infty \subset [0, \infty)$3-distance of $\{a_n\}_{n=1}^\infty \subset [0, \infty)$4 to the extremal profile in a weighted norm:

$\{a_n\}_{n=1}^\infty \subset [0, \infty)$5

for an explicit constant $\{a_n\}_{n=1}^\infty \subset [0, \infty)$6. Thus, not only is the optimal constant for the discrete Hardy inequality strictly attained, but the "distance to extremality" can be controlled by the excess over the sharp constant. For $\{a_n\}_{n=1}^\infty \subset [0, \infty)$7, this quantifies stability in the classical discrete Hardy setting.

Special Cases and Recovery of Previous Results

The framework is sufficiently general to recover:

• The sharp discrete $\{a_n\}_{n=1}^\infty \subset [0, \infty)$8-Hardy identity by Fischer–Keller–Pogorzelski [fischer2023improved];
• The Krejčiřík–Štampach identity for $\{a_n\}_{n=1}^\infty \subset [0, \infty)$9 with improved weights and sharp remainder [krejcirik2022sharp];
• Power-type weighted inequalities, sharpening and generalizing the work of Gupta [gupta2022discrete];
• Discrete Copson inequalities with explicit and optimal weights;
• The identity of Huang–Ye for $1 < p < \infty$0 in the context of discrete Laplacians.

The $1 < p < \infty$1 framework is shown to be homogeneous of degree $1 < p < \infty$2 and possesses structural properties aligning with both complex and real sequence settings.

Implications and Prospects

Theoretical Significance

The main contribution is an algebraic, rather than variational, understanding of discrete $1 < p < \infty$3-Hardy inequalities: the $1 < p < \infty$4-functional provides a route to sharp identities with non-trivial and computable remainders. This advances the discrete analysis parallel to the best results in the continuous case, but avoids the limitations posed by the absence of calculus and lack of classical tools for discrete symmetries.

The abstract and parameterized identity encourages further generalization: higher-order discrete Hardy–Rellich–Birman inequalities, different boundary geometries, and connections to discrete (weighted) Sobolev spaces. The existence of an explicit, sharp remainder may prove crucial for spectral analysis, stability of nonlinear PDEs defined on graphs or discrete domains, and in developing discrete variational methods.

Practical and Mathematical Applications

• Spectral Theory of Discrete Operators: Optimal Hardy inequalities underpin lower bounds for spectra and sharp constants in uncertainty principles on graphs or lattices, with implications in mathematical physics.
• Analysis of Algorithms and Random Walks: The inequalities play a role in assessing boundary-adjacent influences and "escape" rates for random walk processes on graphs.
• Numerical Analysis: The sharpness and explicit control of remainder terms could refine error estimates for finite-difference schemes or discrete-time approximations.
• Discrete Optimization and Data Science: Weighted inequalities and the underlying extremal profiles (revealed by stability) can serve as benchmarks for regularization-penalty design and for understanding concentration phenomena in sequence spaces.

Future Research Directions

• Extension to General Discrete Structures: The abstract $1 < p < \infty$5-based approach should allow analogs of Hardy-type inequalities on more general graphs and hypergraphs, with different connectivity and weight structures.
• Sharpness and Nonlinear Equations: The explicit remainders may serve in establishing regularity, uniqueness, and blowup properties for nonlinear discrete equations connected to the $1 < p < \infty$6-Laplacian.
• Connections to Probabilistic Methods: Given the role of Hardy inequalities in continuous probability and martingale analysis, their discrete sharp forms may foster refinements in discrete probability, particularly for Markov chains and spatially inhomogeneous random walks.
• Bridging to Continuous Case via Scaling Limits: The precise structure of extremals and remainder terms raises the possibility of new scaling limit theorems relating discrete and continuous sharp inequalities.

Conclusion

The paper provides a general, sharp, and constructive framework for weighted discrete $1 < p < \infty$7-Hardy inequalities, including explicit remainder terms and quantitative stability. By establishing a flexible identity dependent on weight and auxiliary function choices, the authors encompass and extend the results of several significant works in the field, while delivering new explicit stability quantifications. The algebraic and functional-analytic perspectives introduced---notably the systematic use of the $1 < p < \infty$8-functional and remainder---set the stage for future advances in discrete analysis, spectral theory, and applications involving finite or countable sequence spaces.

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References:

• "Sharp forms and quantitative stability for general weighted discrete $1 < p < \infty$9-Hardy inequalities" (2604.02229)
• Additional references cited in the paper: [krejcirik2022sharp], [fischer2023improved], [gupta2022discrete], [cazacu2024hardy], [barki2024sharp], among others.