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

Lacey, Michael; Sawyer, Eric; Uriarte-Tuero, Ignacio

doi:10.2140/apde.2012.5.1

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

Michael Lacey, Eric T. Sawyer and Ignacio Uriarte-Tuero

Vol. 5 (2012), No. 1, 1–60

DOI: 10.2140/apde.2012.5.1

Abstract

Let $σ$ and $ω$ be positive Borel measures on $ℝ$ with $σ$ doubling. Suppose first that $1 < p \leq 2$ . We characterize boundedness of certain maximal truncations of the Hilbert transform $T_{♮}$ from $L^{p} (σ)$ to $L^{p} (ω)$ in terms of the strengthened $A_{p}$ condition

{(\int_{ℝ} s_{Q} {(x)}^{p} d ω (x))}^{1 ∕ p} {(\int_{ℝ} s_{Q} {(x)}^{p^{'}} d σ (x))}^{1 ∕ p^{'}} \leq C | Q |,

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

\int_{Q} T_{♮} (χ_{E} σ) {(x)}^{p} d ω (x) \leq C_{1} \int_{Q} d σ (x) for all E \subset Q,

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

\int_{Q} T_{♮} (χ_{Q} f σ) (x) d ω (x) \leq C_{2} {(\int_{Q} | f (x) |^{p} d σ (x))}^{1 ∕ p} {(\int_{Q} d ω (x))}^{1 ∕ p^{'}}

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

\int_{ℝ} {(\sum_{r = 1}^{\infty} | I_{r} |_{σ} | I_{r} |^{p^{'} - 1} \sum_{ℓ = 0}^{\infty} \frac{2^{- ℓ}}{| {(I_{r})}^{(ℓ)}} | χ_{{(I_{r})}^{(ℓ)}} (y))}^{p} d ω (y) \leq C \sum_{r = 1}^{\infty} | I_{r} |_{σ} | I_{r} |^{p^{'}},

for all pairwise disjoint decompositions $Q = ⋃_{r = 1}^{\infty} I_{r}$ of the dyadic interval $Q$ into dyadic intervals $I_{r}$ . We prove that analogues of these conditions are sufficient for boundedness of certain maximal singular integrals in $ℝ^{n}$ when $σ$ is doubling and $1 < p < \infty$ . Finally, we characterize the weak-type two weight inequality for certain maximal singular integrals $T_{♮}$ in $ℝ^{n}$ when $1 < p < \infty$ , without the doubling assumption on $σ$ , in terms of analogues of the second testing condition and the $A_{p}$ condition.

Keywords

two weight, singular integral, maximal function, maximal truncation

Mathematical Subject Classification 2000

Primary: 42B20

Milestones

Received: 7 October 2009

Revised: 2 February 2011

Accepted: 2 March 2011

Published: 25 June 2012

Authors

Michael Lacey
School of Mathematics Georgia Institute of Technology 686 Cherry Street NW Atlanta, GA 30332-0160 United States
http://www.math.gatech.edu/~lacey
Eric T. Sawyer
Department of Mathematics and Statistics McMaster University 1280 Main St. West Hamilton, ON L8S 4K1 Canada
http://www.math.mcmaster.ca/~sawyer/
Ignacio Uriarte-Tuero
Department of Mathematics Michigan State University East Lansing, MI 48824 United States

Vol. 5, No. 1, 2012