A note on the paper “Best constants for the Hardy−Litllewood maximal operator on finite graphs”

Year 2020, Volume: 49 Issue: 2, 498 - 504, 02.04.2020

Zaryab Hussain , Sadia Talib

https://doi.org/10.15672/hujms.471101

Cited By: 2

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 $0<p\leq 1$.

Keywords

generalized maximal operator, $\ell^{p}$-estimates, $G^{m}_{n}$ graph

References

• [1] I. Ahmad and W. Nazeer, Optimal Couples of Rearrangement Invariant Spaces for Generalized Maximal Operators, J. Funct. Spaces 2014, Article ID 647123, 5 pages, 2014.
• [2] J. Soria and P. Tradacete, Best constants for the Hardy-Littlewood maximal operator on finite graphs, J. Math. Anal. Appl. 436 (2), 661–682, 2016.