Two Weight Inequalities for Positive Operators: Doubling Cubes

2018/12/12 by Wei Chen, Michael T. Lacey, Chen, Wei +1
Mathematics · #Advanced Harmonic Analysis Research #Nonlinear Partial Differential Equations #Mathematical Approximation and Integration

paper · pdf · doi:10.48550/arxiv.1812.04952

Abstract

For the maximal operator M on \mathbb R ^d, and 1< p , ρ< ∞ , there is a finite constant D = D _{p, ρ} so that this holds. For all weights w, σ on \mathbb R ^d, the operator M (σ⋅ ) is bounded from L ^p (σ) → L ^p (w) if and only the pair of weights (w, σ) satisfy the two weight A _p condition, and this testing inequality holds: ∫ _Q M (σ\mathbf 1_Q ) ^p d w \lesssim σ( Q), for all cubes Q for which there is a cube P ⊃ Q satisfying σ(P) < D σ(Q), and ℓ P = ρℓ Q. This was recently proved by Kangwei Li and Eric Sawyer. We give a short proof, which is easily seen to hold for several closely related operators.

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