## Analysis & PDE

### A characterization of two weight norm inequalities for maximal singular integrals with one doubling measure

#### Abstract

Let $σ$ and $ω$ be positive Borel measures on $ℝ$ with $σ$ doubling. Suppose first that $1. We characterize boundedness of certain maximal truncations of the Hilbert transform $T♮$ from $Lp(σ)$ to $Lp(ω)$ in terms of the strengthened $Ap$ condition

$( ∫ ℝ s Q ( x ) p d ω ( x ) ) 1 ∕ p ( ∫ ℝ s Q ( x ) p ′ d σ ( x ) ) 1 ∕ p ′ ≤ C | Q | ,$

where $sQ(x)=|Q|∕(|Q|+|x−xQ|)$, and two testing conditions. The first applies to a restricted class of functions and is a strong-type testing condition,

and the second is a weak-type or dual interval testing condition,

$∫ Q T ♮ ( χ Q f σ ) ( x ) d ω ( x ) ≤ C 2 ( ∫ Q | f ( x ) | p d σ ( x ) ) 1 ∕ p ( ∫ Q d ω ( x ) ) 1 ∕ p ′$

for all intervals $Q$ in $ℝ$ and all functions $f∈Lp(σ)$. In the case $p>2$ the same result holds if we include an additional necessary condition, the Poisson condition

$∫ ℝ ( ∑ r = 1 ∞ | I r | σ | I r | p ′ − 1 ∑ ℓ = 0 ∞ 2 − ℓ | ( I r ) ( ℓ ) | χ ( I r ) ( ℓ ) ( y ) ) p d ω ( y ) ≤ C ∑ r = 1 ∞ | I r | σ | I r | p ′ ,$

for all pairwise disjoint decompositions $Q= ⋃r=1∞Ir$ of the dyadic interval $Q$ into dyadic intervals $Ir$. We prove that analogues of these conditions are sufficient for boundedness of certain maximal singular integrals in $ℝn$ when $σ$ is doubling and $1. Finally, we characterize the weak-type two weight inequality for certain maximal singular integrals $T♮$ in $ℝn$ when $1, without the doubling assumption on $σ$, in terms of analogues of the second testing condition and the $Ap$ condition.

#### Article information

Source
Anal. PDE, Volume 5, Number 1 (2012), 1-60.

Dates
Revised: 2 February 2011
Accepted: 2 March 2011
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513731196

Digital Object Identifier
doi:10.2140/apde.2012.5.1

Mathematical Reviews number (MathSciNet)
MR2957550

Zentralblatt MATH identifier
1279.42016

#### Citation

Lacey, Michael; Sawyer, Eric; Uriarte-Tuero, Ignacio. A characterization of two weight norm inequalities for maximal singular integrals with one doubling measure. Anal. PDE 5 (2012), no. 1, 1--60. doi:10.2140/apde.2012.5.1. https://projecteuclid.org/euclid.apde/1513731196

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