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Weighted Orlicz-Poincaré inequalities in product spaces

Lucas Yong

TL;DR

The work addresses the problem of establishing weighted Orlicz-Poincaré inequalities on product spaces, providing necessary and sufficient conditions in terms of one-dimensional constants $K_{p_i,Φ}(μ_i,ν_i,w_i)$ and a two-dimensional gauge-norm framework. The approach generalizes the Chuawheeden framework to Orlicz spaces, employing repeated-norm tools and submultiplicativity of $Φ$ with convex $Γ_i(t)=Φ(t^{1/p_i})$ to derive a two-dimensional bound on $\| f - f_{\nu,av} \|_{L_μ^Φ(I\times J)}$ via the directional derivatives $∂_{x_1}f$ and $∂_{x_2}f$. The main results show that finiteness of the 1D constants yields the two-variable inequality, and, under invertibility of $Φ$, a necessity direction as well, thereby linking product-space regularity to constituent one-dimensional conditions. This provides a framework for regularity theory of degenerate PDEs on product domains using Orlicz-Sobolev bounds.

Abstract

This article is a follow-up to arXiv:2304.04373. We establish necessary and sufficient conditions for weighted Orlicz-Poincaré inequalities in product spaces. These results follow the work of Chua and Wheeden, who established similar results for weighted Poincaré inequalities in product spaces.

Weighted Orlicz-Poincaré inequalities in product spaces

TL;DR

The work addresses the problem of establishing weighted Orlicz-Poincaré inequalities on product spaces, providing necessary and sufficient conditions in terms of one-dimensional constants and a two-dimensional gauge-norm framework. The approach generalizes the Chuawheeden framework to Orlicz spaces, employing repeated-norm tools and submultiplicativity of with convex to derive a two-dimensional bound on via the directional derivatives and . The main results show that finiteness of the 1D constants yields the two-variable inequality, and, under invertibility of , a necessity direction as well, thereby linking product-space regularity to constituent one-dimensional conditions. This provides a framework for regularity theory of degenerate PDEs on product domains using Orlicz-Sobolev bounds.

Abstract

This article is a follow-up to arXiv:2304.04373. We establish necessary and sufficient conditions for weighted Orlicz-Poincaré inequalities in product spaces. These results follow the work of Chua and Wheeden, who established similar results for weighted Poincaré inequalities in product spaces.
Paper Structure (3 sections, 6 theorems, 35 equations)

This paper contains 3 sections, 6 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs of the main theorems

Key Result

Theorem 1

Suppose that $\Phi$ is submultiplicative and invertible on $[0, \infty)$, and that $\Gamma_i(t) := \Phi\left(t^{1/{p_i}}\right)$ is convex for $i = 1, 2$. If $K_{p_i,\Phi}(\mu_i, \nu_i, w_i) < \infty$ for $i = 1,2$, then for all Lipschitz continuous functions $f \colon I \times J \to \mathbb{R}$. Moreover, if $C_i$ are the best possible constants for $i = 1,2$, then Here, $C_0(\Phi) := 2 \left[

Theorems & Definitions (19)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Definition 4
  • Remark 2
  • Definition 5
  • ...and 9 more