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

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. [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. [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.