Random martingales and localization of maximal inequalities

Let (X,d,μ) be a metric measure space. For ∅≠R⊆(0,∞) consider the Hardy-Littlewood maximal operator. M_Rf(x)=defsup1/r∈R1^μ(B(x,r))∫_B(x,r){pipe}f{pipe}dμ. We show that if there is an n>1 such that one has the " microdoubling condition" μ(B(x,(1+1n)r))≲μ(B(x,r)) for all x∈X and r>0, then the weak (1,1) norm of MR has the following localization property:. {double pipe}M_R{double pipe}L₁(X)→L_1,∞(X)sup_r>0{double pipe}M_R∩[r,nr]{double pipe}L₁(X)→L_1,∞(X). An immediate consequence is that if (X,d,μ) is Ahlfors-David n-regular then the weak (1,1) norm of MR is ≲nlogn, generalizing a result of Stein and Strömberg (1983) [47]. We show that this bound is sharp, by constructing a metric measure space (X,d,μ) that is Ahlfors-David n-regular, for which the weak (1,1) norm of M(0,∞) is ≳nlogn. The localization property of MR is proved by assigning to each f∈L₁(X) a distribution over random martingales for which the associated (random) Doob maximal inequality controls the weak (1,1) inequality for M_R.