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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
We also prove some general results for $0<p\leq 1$.
📌 3 Key Takeaways
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Objective & Hypothesis:Let $G{m}_{n}$ be a simple, connected and finite graph.
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Key Findings & Data:We consider the action of generalized maximal operator $M^{\phi}_{G^{m}_{n}}$ on $\ell^{p}$ spaces and find optimal bound for the quasi norm $\|M^{\phi}_{G^{m}_{n}}\|_{p}$ for the case $0<p\leq 1$.
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Conclusion & Impact:We also prove some general results for $0<p\leq 1$.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:generalized maximal operator · $\ell^{p}$-estimates · $G^{m}_{n}$ graph
Abstract
Let $G{m}_{n}$ be a simple, connected and finite graph. Suppose $\phi: \mathbb{N}\to \mathbb{R^{+}}$ is a positive and increasing function. We consider the action of generalized maximal operator $M^{\phi}_{G^{m}_{n}}$ on $\ell^{p}$ spaces and find optimal bound for the quasi norm $\|M^{\phi}_{G^{m}_{n}}\|_{p}$ for the case $0<p\leq 1$. In addition we find bounds for the norm $\|M^{\phi}_{G^{m}_{n}}\|_{p}$ for the case $1<p<\infty$. We also prove some general results for $0<p\leq 1$.
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Cite This Article
Hussain, Z., Talib, S. (2020). A note on the paper “Best constants for the Hardy−Litllewood maximal operator on finite graphs”. Hacettepe Journal of Mathematics and Statistics, Volume: 49 Issue: 2. https://doi.org/10.15672/hujms.471101