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

Zaryab Hussain,Toqeer Mahmood,Jia-Bao Liu,Ghulam Murtaza,Sakander Hayat

doi:10.1155/2021/9338269

Abstract

Let G = G 1 × G 2 × ⋯ × G m be the strong product of simple, finite connected graphs, and let ϕ : ℕ ⟶ 0 , ∞ be an increasing function. We consider the action of generalized maximal operator M G ϕ on ℓ p spaces. We determine the exact value of ℓ p -quasi-norm of M G ϕ for the case when G is strong product of complete graphs, where 0 < p ≤ 1 . However, lower and upper bounds of ℓ p -norm have been determined when 1 < p < ∞ . Finally, we computed the lower and upper bounds of M G ϕ p when G is strong product of arbitrary graphs, where 0 < p ≤ 1 .

Highlights

We review some of the standard facts on graphs and metric on the graphs
Hardy–Littlewood maximal operator MxG: lp ⟶ lp [4,5,6,7] is defined as 1 |B(q, r)|
We have considered the action of generalized maximal operator on lp spaces and calculated the quasinorm ‖MφK‖p for 0 < p ≤ 1

Summary

Introduction

We review some of the standard facts on graphs and metric on the graphs. All the graphs considered in this paper are simple, finite, and connected. Let G(V(G), E(G)) be a graph, where V(G) is the set of vertices and E(G) is the set of edges of G. e vertices which are at distance one from any vertex x ∈ V(G) are called neighbors of x. E fractional maximal operator [8] on graphs is defined as Mathematical Problems in Engineering. For 0 < p < ∞, the lp norm of the Hardy–Littlewood maximal operator is defined as. For every function f: V(G) ⟶ R, the generalized maximal operator MφG: lp ⟶ lp [9, 10] is defined as.

Preliminaries

Conclusion