Sharp inequalities for maximal operators on finite graphs, II

Let MG be the centered Hardy-Littlewood maximal operator on a finite graph G. We find limp→∞‖MG‖pp when G is the start graph (Sn) and the complete graph (Kn), and we fully describe ‖MSn‖p and the corresponding extremizers for p∈(1,2). We prove that limp→∞‖MSn‖pp=1+n2 when n≥25. Also, we compute the best constant CSn,2 such that for every f:V→R we have Var2MSnf≤CSn,2Var2f. We prove that CSn,2=(n2−n−1)1/2n for all n≥3 and characterize the extremizers. Moreover, when M is the Hardy-Littlewood maximal operator on Z, we compute the best constant Cp such that VarpMf≤Cp‖f‖p for p∈(12,1) and we describe the extremizers.