Sharp remainder formulae for general weighted Hardy and Rellich type inequalities for $1<p<\infty$

This paper extends the weighted $L^p$-Hardy inequalities and identities to the full range $1<p<\infty$, while also establishing a new sharp remainder formula for general weighted$L^p$-Rellich inequalities involving quasilinear second-order degenerate elliptic operators.

The Gist

Imagine you are trying to measure the "energy" or "effort" required to move a fluid through a complex pipe system. In mathematics, this is often done using equations called inequalities. These inequalities act like safety nets or speed limits; they tell us the minimum amount of energy needed to do a job, ensuring we don't underestimate the difficulty.

This paper is about sharpening those safety nets. The authors, Yerkin Shaimerdenov, Nurgissa Yessirkegenov, and Amir Zhangirbayev, have taken some very old, very famous rules (called Hardy and Rellich inequalities) and made them work perfectly for a much wider range of situations than before.

Here is a breakdown of what they did, using simple analogies:

1. The Old Rules vs. The New Rules

For decades, mathematicians had a set of rules to calculate the minimum energy needed to move things around in space.

• The Problem: These rules worked great if the "fluid" was thick and heavy (mathematically, when a number $p$ was 2 or larger). But if the fluid was thinner or behaved differently (when $1 < p < 2$), the old rules were either broken or gave a loose, inaccurate estimate.
• The Analogy: Imagine you have a ruler that works perfectly for measuring heavy bricks, but if you try to measure a feather with it, the ruler bends and gives you the wrong number.
• The Breakthrough: The authors found a new, flexible "ruler" (a specific algebraic identity) that works perfectly for both heavy bricks and light feathers. They extended the rules to work for all types of fluids between 1 and infinity, not just the heavy ones.

2. The "Perfect Fit" (Sharp Remainders)

In the past, when mathematicians used these inequalities, they often had to say, "The energy is at least this much." It was a rough estimate. Sometimes, the actual energy was much higher, but the rule didn't tell you how much higher.

• The Analogy: Imagine you are packing a suitcase. The old rule said, "You need at least 10 liters of space." But if you actually need 15 liters, the rule doesn't tell you that you are 5 liters short. You might think you're fine, but your suitcase won't close.
• The Innovation: This paper provides a "Sharp Remainder." This is like adding a precise label to the suitcase: "You need 10 liters, plus an extra 5 liters because of the shape of your clothes."
• Why it matters: They didn't just say "it's at least X." They gave an exact formula for the difference between the minimum estimate and the actual reality. This difference is called the "remainder." If you know the remainder, you know exactly how much "extra" energy is needed.

3. The "Ghost" Extremizers

One of the coolest parts of their discovery is the concept of "virtual extremizers."

• The Analogy: Imagine you are trying to find the perfect balance point on a seesaw. The math says the perfect balance happens if you sit exactly at a specific spot. However, that spot is actually a hole in the ground (a singularity) where you can't physically sit.
• The Insight: The authors found that while you can't physically sit in that perfect spot (because the math blows up there), the formula still works if you pretend you are sitting there. They call these "virtual" solutions. They are like ghosts that prove the rule is perfect, even though you can't touch them. This helps mathematicians understand the absolute limit of the system.

4. Where Does This Apply?

The authors didn't just fix the rules for flat, empty space (like a standard room). They fixed them for:

• Twisted Spaces: Imagine space where some directions are harder to move through than others (like walking through water vs. walking on land).
• Complex Operators: They applied this to "degenerate elliptic operators," which are fancy math names for machines that handle weird, distorted geometries (like the Baouendi-Grushin or Heisenberg operators).
• Real-World Impact: These geometries show up in quantum physics, image processing, and even in how heat spreads through materials that aren't uniform.

Summary

Think of this paper as upgrading the GPS for mathematicians.

• Before: The GPS worked well for highways (standard cases) but got lost on dirt roads (complex cases) and only gave you a rough estimate of the distance.
• Now: The new GPS works on every road type, from smooth highways to rocky trails. It doesn't just give you a distance; it tells you the exact extra fuel you'll need for the bumps in the road, ensuring you never run out of gas.

By proving these identities hold for all numbers between 1 and infinity, the authors have unified a fragmented field of mathematics, providing a single, powerful tool that is more accurate and versatile than anything that came before.

Technical Summary

Here is a detailed technical summary of the paper "Sharp Remainder Formulae for General Weighted Hardy and Rellich Type Inequalities for $1 < p < \infty$" by Shaimerdenov, Yessirkegenov, and Zhangirbayev.

1. Problem Statement and Motivation

The paper addresses the extension of general weighted $L^p$-Hardy and $L^p$-Rellich type inequalities to the full range of exponents $1 < p < \infty$.

• Context: Previous work by Cossetti and D'Arca [CD25] established general weighted Hardy inequalities and corresponding identities using a unified framework involving distributional solutions to degenerate elliptic $p$-Laplacian equations. However, their proofs relied on specific algebraic identities (Propositions 2.1 and 2.3 in [CD25]) that were only valid for $p \geq 2$.
• Gap: There was a lack of sharp remainder formulae (identities) for the case $1 < p < 2$ in the context of general weighted operators and quasilinear second-order degenerate elliptic operators.
• Objective: The authors aim to:
  1. Extend the results of [CD25] to the entire range $1 < p < \infty$.
  2. Establish a general weighted $L^p$-Rellich type inequality with a sharp remainder term for quasilinear second-order degenerate elliptic operators.
  3. Demonstrate that these identities are new even for the classical Laplacian in the Rellich case.

2. Methodology

The core methodology relies on a specific algebraic identity involving the $C_p$-functional, which serves as the remainder term in the inequalities.

A. The $C_p$-Functional

The authors utilize the functional defined for $\xi, \eta \in \mathbb{C}^\ell$:
$$C_p(\xi, \eta) := |\xi|^p - |\xi - \eta|^p - p|\xi - \eta|^{p-2}\text{Re}((\xi - \eta) \cdot \eta) \geq 0$$
Crucially, unlike the identities used in [CD25], this functional and its associated properties hold for all $1 < p < \infty$. The paper cites recent lemmas (e.g., from [CT25], [CKLL24]) that establish lower and upper bounds for$C_p(\xi, \eta)$depending on whether$p \geq 2$or$1 < p < 2$, ensuring the non-negativity and sharpness of the remainder.

B. General Operator Framework

The authors work within a general setting defined by:

• A matrix $\sigma(x)$ defining vector fields $X_i = \sum \sigma_{ij} \partial_{x_j}$.
• The horizontal gradient $\nabla_L = \sigma \nabla$.
• The linear second-order operator $L = -\nabla_L^* \cdot \nabla_L$.
• The quasilinear degenerate elliptic operator $L_p u = -\nabla_L^* \cdot (|\nabla_L u|^{p-2} \nabla_L u)$.

This framework encompasses standard Euclidean operators ($\Delta, \Delta_p$), Baouendi-Grushin operators, and Heisenberg-Greiner operators.

C. Ground State Representation

The proofs utilize a "ground state representation" technique. By assuming the existence of a non-trivial positive solution $\phi$ (a "ground state") to a specific weighted equation involving the operator, the authors decompose the energy functional of an arbitrary function $u$ into the energy of the ground state plus a non-negative remainder term involving $C_p$.

3. Key Contributions and Results

A. Extension of Hardy Identities ($1 < p < \infty$)

The paper proves two main theorems extending [CD25]:

1. Theorem 3.1 (General Weighted Hardy Identity with Vector Field $Z$):
   For a solution $\phi$ to $-\text{div}_L (V |\nabla_L \phi \cdot Z|^{p-2} (\nabla_L \phi \cdot Z) Z) = \lambda W |\phi|^{p-2}\phi$, the following identity holds for all $u \in C_0^\infty(\Omega)$:
   $$\int_\Omega V |\nabla_L u \cdot Z|^p dx = \lambda \int_\Omega W |u|^p dx + \int_\Omega C_p\left( V^{1/p}\nabla_L u \cdot Z, V^{1/p}\phi \nabla_L\left(\frac{u}{\phi}\right) \cdot Z \right) dx$$
   This extends the validity from $p \geq 2$ to $1 < p < \infty$.

2. Theorem 3.3 (General Weighted Hardy Identity without $Z$):
   Similarly, for the equation $-\text{div}_L (V |\nabla_L \phi|^{p-2} \nabla_L \phi) = \lambda W |\phi|^{p-2}\phi$, the identity:
   $$\int_\Omega V |\nabla_L u|^p dx = \lambda \int_\Omega W |u|^p dx + \int_\Omega C_p\left( V^{1/p}\nabla_L u, V^{1/p}\phi \nabla_L\left(\frac{u}{\phi}\right) \right) dx$$
   is established for all $1 < p < \infty$.

Significance: These identities imply the sharp inequalities by dropping the non-negative remainder term. The "virtual extremizers" are functions of the form $u = c\phi$, which make the remainder vanish (though the integrals may diverge).

B. Sharp $L^p$-Rellich Type Inequality

The paper's most novel contribution is Theorem 3.5, which provides a sharp remainder formula for the Rellich inequality (involving second-order operators).

• Assumption: $\phi$ solves $L(V |L\phi|^{p-2} L\phi) = \lambda W |\phi|^{p-2}\phi$ with $-L\phi/\phi \geq 0$.
• Result: The identity involves the $C_p$ functional of the operator applied to $u$, plus two additional non-negative remainder terms arising from the second-order nature of the operator:
  1. A term involving the difference between the gradient of the modulus and the modulus of the gradient: $|\nabla_L |u| - \frac{|u|}{\phi}\nabla_L \phi|^2$.
  2. A term involving the difference between $|\nabla_L u|^2$ and $|\nabla_L |u||^2$.
• Implication: This yields the general weighted $L^p$-Rellich inequality:
  $$\int_\Omega V |Lu|^p dx \geq \lambda \int_\Omega W |u|^p dx$$
  with a precise characterization of the remainder.

C. Classical and New Applications

The authors apply their general theorems to recover and improve known results:

• Classical $L^p$-Hardy: Recovering the sharp remainder for the standard Euclidean Hardy inequality with weight $|x|^{-p}$.
• Classical $L^p$-Rellich: Deriving a new sharp remainder formula for the classical Rellich inequality ($\Delta u$).
  • Note: The authors explicitly state that even for the classical Laplacian ($p=2$), the specific identity form (4.4) appears to be new.
• Eigenvalue Problems: Applying the results to the Dirichlet $p$-Laplacian eigenvalue problem, recovering Poincaré identities with sharp remainders (Corollary 4.10).
• Generalized Operators: The framework naturally applies to Baouendi-Grushin and Heisenberg-Greiner operators, extending previous results on these manifolds to the full $p$-range.

4. Significance and Impact

1. Unification of $p$-Ranges: The paper successfully unifies the theory of weighted Hardy and Rellich inequalities for the entire range $1 < p < \infty$, removing the restriction to$p \geq 2$ that plagued previous approaches relying on convexity inequalities.
2. Sharpness: By providing exact identities rather than just inequalities, the authors characterize the "distance" to the optimal constant, which is crucial for stability analysis and understanding the behavior of extremal functions.
3. Generality: The results apply to a vast class of operators (degenerate, sub-elliptic, non-Euclidean) via the abstract $L$ and $L_p$ framework, making the results widely applicable in geometric analysis and PDE theory.
4. Novelty in Classical Cases: The discovery of new identities for the classical Laplacian (even for $p=2$) in the Rellich context highlights that the general framework yields insights even in well-studied classical settings.

In summary, this paper advances the field of functional inequalities by providing a robust, unified algebraic framework that extends sharp remainder formulae to the full range of $p$, covering both first-order (Hardy) and second-order (Rellich) operators in general geometric settings.