Sharp variational inequalities for the Hardy-Littlewood maximal operator on finite undirected graphs

Abstract

We study sharp $p$-variational inequalities for the Hardy-Littlewood maximal operator on complete graphs, answering in the affirmative a question by Feng Liu and Qingying Xue. We also use computational assistance to find sharp constants in $1$-variational inequalities for all connected graphs on at most five vertices and pose a conjecture on the corresponding sharp constants for path graphs. Finally, we construct finite graphs with arbitrarily large $p$-variational constants.

Figures (5)

• Figure 1: Complete graph $K_4$ (left) and star graph $S_4$ (right).
• Figure 2: Cycle graph $C_4$ (left) and path graph $P_4$ (right).
• Figure 3: Paw graph $Y$ (left) and diamond graph $D$ (right).
• Figure 4: Connected graphs on $5$ vertices.
• Figure 5: Examples of graphs with large $\mathbf{C}_{G,p}$.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Theorem 4
• Conjecture 5
• Proposition 6
• proof : Proof of Proposition \ref{['prop:ineq']}
• Lemma 7
• proof : Proof of Lemma \ref{['lm:lemma1']}
• Lemma 8
• proof : Proof of Lemma \ref{['lm:lemma2']}