On the conformal gauge of a compact metric space

In this article we study the Ahlfors regular conformal gauge of a compact metric space $(X,d)$, and its conformal dimension $\mathrm{dim}_{AR}(X,d)$. Using a sequence of finite coverings of $(X,d)$, we construct distances in its Ahlfors regular conformal gauge of controlled Hausdorff dimension. We obtain in this way a combinatorial description, up to bi-Lipschitz homeomorphisms, of all the metrics in the gauge. We show how to compute $\mathrm{dim}_{AR}(X,d)$ using the critical exponent $Q_N$ associated to the combinatorial modulus.Comment: 49 pages, 9 figure

Random Martingales and localization of maximal inequalities

Let $(X,d,\mu)$ be a metric measure space. For $\emptyset\neq R\subseteq (0,\infty)$ consider the Hardy-Littlewood maximal operator $$M_R f(x) \stackrel{\mathrm{def}}{=} \sup_{r \in R} \frac{1}{\mu(B(x,r))} \int_{B(x,r)} |f| d\mu.$$ 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: $$\|M_R\|_{L_1(X) \to L_{1,\infty}(X)}\asymp \sup_{r>0} \|M_{R\cap [r,nr]}\|_{L_1(X) \to L_{1,\infty}(X)}.$$ 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 Str\"omberg. 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 $M_R$

Equal area partitions of the sphere with diameter bounds, via optimal transport

We prove existence of equal area partitions of the unit sphere via optimal transport methods, accompanied by diameter bounds written in terms of Monge--Kantorovich distances. This can be used to obtain bounds on the expectation of the maximum diameter of partition sets, when points are uniformly sampled from the sphere. An application to the computation of sliced Monge--Kantorovich distances is also presented.Comment: 12pages. Comments welcome