Estimates for the Norm of Generalized Maximal Operator on Strong Product of Graphs

Abstract

Let $G=G_1\times G_2\times \cdots \times G_m$ be the strong product of simple, finite connected graphs, and let $\varphi :\mathbb{N}⟶\left(0,\infty \right)$ be an increasing function. We consider the action of generalized maximal operator $M_{\mathbf{G}}^{\varphi }$ on ${\ell }^p$ spaces. We determine the exact value of ${\ell }^p$ -quasi-norm of $M_G^{\varphi }$ for the case when $G$ is strong product of complete graphs, where $0<p\le 1$ . However, lower and upper bounds of ${\ell }^p$ -norm have been determined when $1<p<\infty$ . Finally, we computed the lower and upper bounds of ${‖M_G^{\varphi }‖}_p$ when $G$ is strong product of arbitrary graphs, where $0<p\le 1$ .