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

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1948-206X (online)

ISSN 2157-5045 (print)

Author index

To appear

Other MSP journals

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 $σ$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>σ</mi></math>$ and $ω$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ω</mi></math>$ be positive Borel measures on $R$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ℝ</mi></math>$ with $σ$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>σ</mi></math>$ doubling. Suppose first that $1 < p \leq 2$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1</mn> <mo class="MathClass-rel">&lt;</mo> <mi>p</mi> <mo class="MathClass-rel">≤</mo> <mn>2</mn></math>$ . We characterize boundedness of certain maximal truncations of the Hilbert transform $T_{♮}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>T</mi></mrow><mrow><mi>♮</mi></mrow></msub></math>$ from $L^{p} (σ)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo class="MathClass-open">(</mo><mrow><mi>σ</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ to $L^{p} (ω)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo class="MathClass-open">(</mo><mrow><mi>ω</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ in terms of the strengthened $A_{p}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub></math>$ 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 $s_{Q} (x) = ∣ ∣ Q ∣ ∣ / (∣ ∣ Q ∣ ∣ + ∣ ∣ x - x_{Q} ∣ ∣)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>s</mi></mrow><mrow><mi>Q</mi></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>x</mi></mrow><mo class="MathClass-close">)</mo></mrow> <mo class="MathClass-rel">=</mo> <mo class="MathClass-rel">|</mo><mi>Q</mi><mo class="MathClass-rel">|</mo><mo class="MathClass-bin">∕</mo><mrow><mo class="MathClass-open">(</mo><mrow><mo class="MathClass-rel">|</mo><mi>Q</mi><mo class="MathClass-rel">|</mo> <mo class="MathClass-bin">+</mo> <mo class="MathClass-rel">|</mo><mi>x</mi> <mo class="MathClass-bin">−</mo> <msub><mrow><mi>x</mi></mrow><mrow><mi>Q</mi></mrow></msub><mo class="MathClass-rel">|</mo></mrow><mo class="MathClass-close">)</mo></mrow></math>$ , 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 $Q$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Q</mi></math>$ in $R$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ℝ</mi></math>$ and all functions $f \in L^{p} (σ)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>f</mi> <mo class="MathClass-rel">∈</mo> <msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo class="MathClass-open">(</mo><mrow><mi>σ</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ . In the case $p > 2$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi> <mo class="MathClass-rel">&gt;</mo> <mn>2</mn></math>$ 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 $Q = ⋃_{r = 1}^{\infty} I_{r}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Q</mi> <mo class="MathClass-rel">=</mo><msubsup><mrow><mo class="MathClass-op"> ⋃</mo> </mrow><mrow><mi>r</mi><mo class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></msubsup><msub><mrow><mi>I</mi></mrow><mrow><mi>r</mi></mrow></msub></math>$ of the dyadic interval $Q$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Q</mi></math>$ into dyadic intervals $I_{r}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>I</mi></mrow><mrow><mi>r</mi></mrow></msub></math>$ . We prove that analogues of these conditions are sufficient for boundedness of certain maximal singular integrals in $R^{n}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>n</mi></mrow></msup></math>$ when $σ$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>σ</mi></math>$ is doubling and $1 < p < \infty$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1</mn> <mo class="MathClass-rel">&lt;</mo> <mi>p</mi> <mo class="MathClass-rel">&lt;</mo> <mi>∞</mi></math>$ . Finally, we characterize the weak-type two weight inequality for certain maximal singular integrals $T_{♮}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>T</mi></mrow><mrow><mi>♮</mi></mrow></msub></math>$ in $R^{n}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>n</mi></mrow></msup></math>$ when $1 < p < \infty$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1</mn> <mo class="MathClass-rel">&lt;</mo> <mi>p</mi> <mo class="MathClass-rel">&lt;</mo> <mi>∞</mi></math>$ , without the doubling assumption on $σ$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>σ</mi></math>$ , in terms of analogues of the second testing condition and the $A_{p}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub></math>$ 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