[Paper Review] Inclusion Properties of Weighted Weak Orlicz Spaces

Al Azhary Masta, Ifronika+1 more|arXiv (Cornell University)|Oct 12, 2017

Advanced Harmonic Analysis Research18 references3 citations

TL;DR

This paper establishes sufficient and necessary conditions for inclusion relations between weighted weak Orlicz spaces and weighted weak Lebesgue spaces on $\mathbb{R}^n$. Using the quasi-norm of characteristic functions of balls and translation invariance, it proves that $wL_{\Phi_1}^{u_1} \subseteq wL_{\Phi_2}^{u_2}$ if and only if $u_1 \preceq u_2$ and $\Phi_1 \prec \Phi_2$, extending known results to the weak and weighted setting with precise functional analytic conditions.

ABSTRACT

In this paper we discuss the structure of weighted weak Lebesgue spaces and weighted weak Orlicz spaces on $\mathbb{R}^n$. First, we present sufficient and necessary conditions for inclusion relation between weighted weak Lebesgue spaces. Next, we also obtain similar results on weighted weak Orlicz spaces. One of the keys to prove our results is to use the norm of the characteristic functions of the balls in $\mathbb{R}^n$.

Motivation & Objective

• To characterize inclusion relations between weighted weak Orlicz spaces $wL_{\Phi}^u(\mathbb{R}^n)$ and weighted weak Lebesgue spaces $wL_p^u(\mathbb{R}^n)$.
• To extend known inclusion results for strong Orlicz and Lebesgue spaces to the weak and weighted setting.
• To identify precise conditions on weights $u_1, u_2$ and Young functions $\Phi_1, \Phi_2$ ensuring $wL_{\Phi_2}^{u_2} \subseteq wL_{\Phi_1}^{u_1}$.
• To generalize previous results on weighted Lebesgue and Orlicz spaces by incorporating weak-type norms and translation invariance.

Proposed method

• Define weighted weak Orlicz and Lebesgue spaces using quasi-norms involving superlevel sets of $|u(x)f(x)|$.
• Use the norm of characteristic functions of balls in $\mathbb{R}^n$ as a key technical tool to analyze inclusion properties.
• Apply translation invariance via $L_xf(y) = f(y - x)$ to relate the norm of $f$ to the weight $u(x)$.
• Employ the inverse Young function $\Phi^{-1}$ and its monotonicity to compare growth conditions between $\Phi_1$ and $\Phi_2$.
• Establish equivalence between $u_1 \preceq u_2$ and inclusion $wL_p^{u_2} \subseteq wL_p^{u_1}$ using scaling and measure estimates.
• Prove inclusion $wL_{\Phi_2}^{u_2} \subseteq wL_{\Phi_1}^{u_1}$ under $\Phi_1 \prec \Phi_2$ and $u_1 \preceq u_2$, using quasi-norm comparison and translation estimates.

Experimental results

Research questions

• RQ1 Under what conditions on weights $u_1, u_2$ and Young functions $\Phi_1, \Phi_2$ does $wL_{\Phi_2}^{u_2}(\mathbb{R}^n) \subseteq wL_{\Phi_1}^{u_1}(\mathbb{R}^n)$ hold?
• RQ2 How do the inclusion properties of weighted weak Orlicz spaces relate to the dominance of Young functions and the comparison of weights?
• RQ3 What is the precise role of the translation operator $L_x$ in characterizing the norm behavior of functions in weighted weak spaces?
• RQ4 Can the inclusion $wL_p^{u_2} \subseteq wL_p^{u_1}$ be characterized solely by the pointwise comparison $u_1 \preceq u_2$?
• RQ5 What conditions ensure that the quasi-norm of a function in $wL_{\Phi}^u$ is controlled by a multiple of its norm in another weighted weak Orlicz space?

Key findings

• The inclusion $wL_p^{u_2}(\mathbb{R}^n) \subseteq wL_p^{u_1}(\mathbb{R}^n)$ holds if and only if $u_1 \preceq u_2$, i.e., $u_1(x) \leq C u_2(x)$ for some $C > 0$ and all $x \in \mathbb{R}^n$.
• For weighted weak Orlicz spaces, $wL_{\Phi_2}^{u_2}(\mathbb{R}^n) \subseteq wL_{\Phi_1}^{u_1}(\mathbb{R}^n)$ holds if and only if $u_1 \preceq u_2$ and $\Phi_1 \prec \Phi_2$, where $\Phi_1(t) \leq \Phi_2(C_1 t)$ for some $C_1 > 0$.
• The norm of the translated function $L_x f$ satisfies $\|L_x f\|_{wL_p^u} \leq u(x) \|f\|_{wL_p^u}$, linking the weight to the translation behavior.
• For non-zero $f$, the norm of $L_x f$ satisfies $\frac{u(x)}{C} \leq \|L_x f\|_{wL_p^u} \leq C u(x)$, showing that the weight $u(x)$ controls the norm up to a constant.
• The inclusion $wL_{\Phi_2}^{u_2} \subseteq wL_{\Phi_1}^{u_1}$ implies $u_1 \preceq u_2$, establishing the necessity of the weight condition.
• The proof relies on estimating superlevel sets of $|u(x)f(x)|$ and using the inverse Young function to compare growth rates of $\Phi_1$ and $\Phi_2$.