Sharp Inequalities for maximal operators on finite graphs

2020/05/06 by González-Riquelme, Cristian, Madrid, José
#05C12 #26A45 #39A12 #42B25 #46E35 #46E39 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2005.03146

Abstract

Let G=(V,E) be a finite graph and M_G be the centered Hardy-Littlewood maximal operator defined there. We find the optimal value \bfC_G,p such that the inequality Var_p(M_Gf)≤ C_G,pVar_p(f) holds for every f:V→ ℝ, where Var_p stands for the p-variation, when: (i) G=K_n (complete graph) and p∈ [(log(4))/(log(6)),∞) or G=K₄ and p∈ (0,∞); (ii) G=S_n (star graph) and 1≥ p≥ (1)/(2); p∈ (0,(1)/(2)) and n≥ C(p) or G=S₃ and p∈ (1,∞). We also find the value of the norm ‖M_G‖₂ when: (i) G=K_n and n≥ 3; (ii) G=S_n and n≥ 3.

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