### Abstract

Let (X,d,μ) be a metric measure space. For ∅≠R⊆(0,∞) consider the Hardy-Littlewood maximal operator. M_{R}f(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}L_{1}(X)→L_{1,∞}(X)sup_{r>}0{double pipe}M_{R∩[r,nr]}{double pipe}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 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∈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}.

Original language | English (US) |
---|---|

Pages (from-to) | 731-779 |

Number of pages | 49 |

Journal | Journal of Functional Analysis |

Volume | 259 |

Issue number | 3 |

DOIs | |

State | Published - Aug 2010 |

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### Keywords

- Ahlfors-David regularity
- Hardy-Littlewood maximal function
- Weak (1,1) norm

### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Functional Analysis*,

*259*(3), 731-779. https://doi.org/10.1016/j.jfa.2009.12.009

**Random martingales and localization of maximal inequalities.** / Naor, Assaf; Tao, Terence.

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 259, no. 3, pp. 731-779. https://doi.org/10.1016/j.jfa.2009.12.009

}

TY - JOUR

T1 - Random martingales and localization of maximal inequalities

AU - Naor, Assaf

AU - Tao, Terence

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

UR - http://www.scopus.com/inward/record.url?scp=77952673611&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77952673611&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2009.12.009

DO - 10.1016/j.jfa.2009.12.009

M3 - Article

VL - 259

SP - 731

EP - 779

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 3

ER -