On the variation of the Hardy–Littlewood maximal functions on finite graphs

Let G be a connected and finite graph with the set of vertices V and the set of edges E . Let M G be the Hardy–Littlewood maximal function defined on graph G and M α , G ( 0 ≤ α < 1 ) be its fractional version. In this paper, the regularity problems related to M G and M α , G will be studied. We show that M G : BV p ( G ) → BV p ( G ) is bounded and M α , G : ℓ p ( V ) → BV q ( G ) is bounded and continuous for all 0 < p , q ≤ ∞ . Here BV p ( G ) is the set of all functions of bounded p -variation on V . The operator norms of M G and M α , G have also been investigated.