Random martingales and localization of maximal inequalities

Abstract

Let ( X , d , μ ) be a metric measure space. For ∅ ≠ R ⊆ ( 0 , ∞ ) consider the Hardy–Littlewood maximal operator M R f ( x ) = def sup r ∈ R 1 μ ( B ( x , r ) ) ∫ B ( x , r ) | f | d μ . We show that if there is an n > 1 such that one has the “microdoubling condition” μ ( B ( x , ( 1 + 1 n ) r ) ) ≲ μ ( B ( x , r ) ) for all x ∈ X and r > 0 , then the weak ( 1 , 1 ) norm of M R has the following localization property: ‖ M R ‖ L 1 ( X ) → L 1 , ∞ ( X ) ≍ sup r > 0 ‖ M R ∩ [ r , n r ] ‖ L 1 ( 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 M R is ≲ n log n , 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 ≳ n log n . The localization property of M R is proved by assigning to each f ∈ L 1 ( X ) a distribution over random martingales for which the associated (random) Doob maximal inequality controls the weak ( 1 , 1 ) inequality for M R .