Abstract
LetG=G1×G2×⋯×Gmbe the strong product of simple, finite connected graphs, and letϕ:ℕ⟶0,∞be an increasing function. We consider the action of generalized maximal operatorMGϕonℓpspaces. We determine the exact value ofℓp-quasi-norm ofMGϕfor the case whenGis strong product of complete graphs, where0<p≤1. However, lower and upper bounds ofℓp-norm have been determined when1<p<∞. Finally, we computed the lower and upper bounds ofMGϕpwhenGis strong product of arbitrary graphs, where0<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
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.
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