Restricted testing for positive operators

2018/09/13 by Hytönen, Tuomas P., Li, Kangwei, Sawyer, Eric T.
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1809.04873

Abstract

We prove that for certain positive operators T, such as the Hardy-Littlewood maximal function and fractional integrals, there is a constant D>1, depending only on the dimension n, such that the two weight norm inequality ∫_ℝⁿT( fσ) ²dω≤ C∫_ℝⁿf²dσ holds for all f≥ 0 if and only if the (fractional) A₂ condition holds, and the restricted testing condition ∫_QT( 1_Qσ) ²dω≤ C | Q |_σ holds for all cubes Q satisfying | 2Q |_σ≤ D | Q |_σ. If T is linear, we require as well that the dual restricted testing condition ∫_QT^∗ ( 1_Qω) ²dσ≤ C | Q |_ω holds for all cubes Q satisfying | 2Q |_ω≤ D | Q |_ω.

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