Random Martingales and localization of maximal inequalities

Abstract

Let $(X,d,\mu )$ be a metric measure space. For $\varnothing \ne R\subseteq (0,\infty )$ consider the Hardy-Littlewood maximal operator $$M_{R}f(x)\stackrel{\mathrm{def}}{=}\underset{r\in R}{\sup }\frac{1}{\mu (B(x,r))}{\int }_{B(x,r)}|f|d\mu .$$ We show that if there is an $n>1$ such that one has the "microdoubling condition" $\mu (B(x,(1+\frac{1}{n})r))\lesssim \mu (B(x,r))$ for all $x\in X$ and $r>0$, then the weak $(1,1)$ norm of $M_R$ has the following localization property: $$\Vert M_R{\Vert }_{L_1(X)\to L_{1,\infty }(X)}\asymp \underset{r>0}{\sup }\Vert M_{R\cap [r,nr]}{\Vert }_{L_1(X)\to L_{1,\infty }(X)}.$$ An immediate consequence is that if $(X,d,\mu )$ is Ahlfors-David $n$-regular then the weak $(1,1)$ norm of $M_R$ is $\lesssim n\log n$, generalizing a result of Stein and Strömberg. We show that this bound is sharp, by constructing a metric measure space $(X,d,\mu )$ that is Ahlfors-David $n$-regular, for which the weak $(1,1)$ norm of $M_{(0,\infty )}$ is $\gtrsim n\log n$. The localization property of $M_R$ is proved by assigning to each $f\in L_1(X)$ a distribution over {\em random} martingales for which the associated (random) Doob maximal inequality controls the weak $(1,1)$ inequality for $M_R$.

References in corpus (2)

• Ramsey partitions and proximity data structures
• Maximum gradient embeddings and monotone clustering