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Sharp remainder terms and stability of weighted Hardy-Poincaré and Heisenberg-Pauli-Weyl inequalities related to the Baouendi-Grushin operator

Yerkin Shaimerdenov, Nurgissa Yessirkegenov, Amir Zhangirbayev

Abstract

In this paper, we obtain sharp remainder terms for the Hardy-Poincaré inequalities with general non-radial weights in the setting of Baouendi-Grushin vector fields (see Theorem 2.5). It is worth emphasizing that all of our results are new both in the Baouendi-Grushin and standard Euclidean settings. The method employed allows us to not only unify, but also improve the results of Kombe and Yener [KY18] for any $1<p<\infty$ while holding true for complex-valued functions and providing explicit constants (Corollary 2.7). As a result, we are able to obtain sharp remainder terms to many known weighted Hardy-type inequalities (see Section 3.1). Aside from weighted Hardy-type inequalities, we also recover a sharp remainder formula for the $L^{p}$-Poincaré inequality (Corollary 3.5). In the special case of radial weights, we are naturally able to introduce the notion of Baouendi-Grushin $p$-Bessel pairs (see Definition 2.9). Finally, we apply the technique to establish the sharp remainder term of the Heisenberg-Pauli-Weyl inequality in $L^{p}$ for $1<p<\infty$ (Corollary 3.13), which includes the sharp constant. This makes it possible to obtain the $L^{p}$-analogue for $2\leq p < n$ (Theorem 3.17) of a stability result by Cazacu, Flynn, Lam and Lu [CFLL24].

Sharp remainder terms and stability of weighted Hardy-Poincaré and Heisenberg-Pauli-Weyl inequalities related to the Baouendi-Grushin operator

Abstract

In this paper, we obtain sharp remainder terms for the Hardy-Poincaré inequalities with general non-radial weights in the setting of Baouendi-Grushin vector fields (see Theorem 2.5). It is worth emphasizing that all of our results are new both in the Baouendi-Grushin and standard Euclidean settings. The method employed allows us to not only unify, but also improve the results of Kombe and Yener [KY18] for any while holding true for complex-valued functions and providing explicit constants (Corollary 2.7). As a result, we are able to obtain sharp remainder terms to many known weighted Hardy-type inequalities (see Section 3.1). Aside from weighted Hardy-type inequalities, we also recover a sharp remainder formula for the -Poincaré inequality (Corollary 3.5). In the special case of radial weights, we are naturally able to introduce the notion of Baouendi-Grushin -Bessel pairs (see Definition 2.9). Finally, we apply the technique to establish the sharp remainder term of the Heisenberg-Pauli-Weyl inequality in for (Corollary 3.13), which includes the sharp constant. This makes it possible to obtain the -analogue for (Theorem 3.17) of a stability result by Cazacu, Flynn, Lam and Lu [CFLL24].

Paper Structure

This paper contains 10 sections, 20 theorems, 236 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Applications of Theorem \ref{['main thm']}
  4. Sharp remainders of weighted Hardy-type, Poincaré and Hardy-Poin-caré inequalities
  5. Sharp remainder and stability of the $L^{p}$-Heisenberg-Pauli-Weyl inequality
  6. Sharp Remainder of the Hardy-Poincaré Inequality With General Weights--Proof of Theorem \ref{['main thm']}
  7. Proofs of Corollary \ref{['cor rec KY']} and of Applications of the Main Result
  8. Proofs of sharp remainder formulas of weighted Hardy-type and Poincaré inequalities--Proof of Corollary \ref{['cor nhc']}-\ref{['cor several yener']}
  9. Proof of the sharp remainder formula of the Heisenberg-Pauli-Weyl inequality--Proof of Corollary \ref{['cor hpw sharp']}
  10. Proof of the stability of the Heisenberg-Pauli-Weyl inequality--Proof of Theorem \ref{['stability hpw']}

Key Result

Lemma 2.2

Let $p\geq2$. Then, for $\xi,\eta\in\mathbb{C}^n$, we have where

Theorems & Definitions (41)

  • Definition 2.1
  • Lemma 2.2: cazacu2024hardy
  • Lemma 2.3: CT24
  • Lemma 2.4: CT24
  • Theorem 2.5
  • Corollary 2.6
  • Corollary 2.7
  • Remark 2.8
  • Definition 2.9
  • Corollary 3.1
  • ...and 31 more