TY - JOUR

T1 - Random martingales and localization of maximal inequalities

AU - Naor, Assaf

AU - Tao, Terence

N1 - Funding Information:
We thank Raanan Schul for pointing out that the Lindenstrauss maximal inequality implies the Hardy–Littlewood maximal inequality under strong microdoubling, and Zubin Guatam for explaining the proof of the Lindenstrauss maximal inequality. A.N. was supported in part by NSF grants CCF-0635078 and CCF-0832795, BSF grant 2006009, and the Packard Foundation. T.T. was supported by a grant from the MacArthur foundation, by NSF grant DMS-0649473, and by the NSF Waterman award.

PY - 2010/8

Y1 - 2010/8

N2 - Let (X,d,μ) be a metric measure space. For ∅≠R⊆(0,∞) consider the Hardy-Littlewood maximal operator. MRf(x)=defsup1/r∈R1μ(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}MR{double pipe}L1(X)→L1,∞(X)supr>0{double pipe}MR∩[r,nr]{double pipe}L1(X)→L1,∞(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 MR.

AB - Let (X,d,μ) be a metric measure space. For ∅≠R⊆(0,∞) consider the Hardy-Littlewood maximal operator. MRf(x)=defsup1/r∈R1μ(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}MR{double pipe}L1(X)→L1,∞(X)supr>0{double pipe}MR∩[r,nr]{double pipe}L1(X)→L1,∞(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 MR.

KW - Ahlfors-David regularity

KW - Hardy-Littlewood maximal function

KW - Weak (1,1) norm

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U2 - 10.1016/j.jfa.2009.12.009

DO - 10.1016/j.jfa.2009.12.009

M3 - Article

AN - SCOPUS:77952673611

VL - 259

SP - 731

EP - 779

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 3

ER -