A note on the generalized Hausdorff and packing measures of product sets in metric space

Rihab Guedri; Najmeddine Attia

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arXiv · 2110.15774

A note on the generalized Hausdorff and packing measures of product sets in metric space

Rihab Guedri·Najmeddine Attia

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Abstract

Let $μ$ and $ν$ be two Borel probability measures on two separable metric spaces $\X$ and $\Y$ respectively. For $h, g$ be two Hausdorff functions and $q\in \R$, we introduce and investigate the generalized pseudo-packing measure ${\RRR}_μ^{q, h}$ and the weighted generalized packing measure ${\QQQ}_μ^{q, h}$ to give some product inequalities : $${\HHH}_{μ\times ν}^{q, hg}(E\times F) \le {\HHH}_μ^{q, h}(E) \; {\RRR}_ν^{q, g}(F) \le {\RRR}_{μ\times ν}^{q, hg}(E\times F)$$ and $${\PPP}_{μ\times ν}^{q, hg}(E\times F) \le {\QQQ}_μ^{q, h}(E) \; {\PPP}_ν^{q, g}(F) $$ for all $E\subseteq \X$ and $F\subseteq \Y$, where ${\HHH}_μ^{q, h}$ and ${\PPP}_μ^{q, h}$ is the generalized Hausdorff and packing measures respectively. As an application, we prove that under appropriate geometric conditions, there exists a constant $c$ such that $${\HHH}_{μ\times ν}^{q, hg}(E\times F) \le c\, {\HHH}_μ^{q, h}(E) \; {\PPP}_ν^{q, g}(F) $$ $${\HHH}_μ^{q, h}(E) \; {\PPP}_ν^{q, g}(F) \le c\, {\PPP}_μ^{q, hg}(E \times F) $$ $${\PPP}_{μ\times ν}^{q, hg}(E\times F) \le c\, {\PPP}_μ^{q, h}(E) \; {\PPP}_ν^{q, g}(F). $$ These appropriate inequalities are more refined than well know results since we do no assumptions on $μ, ν, h$ and $g$.

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Rihab Guedri, Najmeddine Attia. 2024-01-06. A note on the generalized Hausdorff and packing measures of product sets in metric space. https://arxiv.org/abs/2110.15774

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