Analysis & PDE

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

Michael Lacey, Eric Sawyer, and Ignacio Uriarte-Tuero

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Abstract

Let σ and ω be positive Borel measures on with σ doubling. Suppose first that 1<p2. 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

( s Q ( x ) p d ω ( x ) ) 1 p ( s Q ( x ) p d σ ( x ) ) 1 p C | Q | ,

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

Q T ( χ E σ ) ( x ) p d ω ( x ) C 1 Q d σ ( x )  for all  E Q ,

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 and all functions fLp(σ). In the case p>2 the same result holds if we include an additional necessary condition, the Poisson condition

( r = 1 | I r | σ | I r | p 1 = 0 2 | ( I r ) ( ) | χ ( I r ) ( ) ( y ) ) p d ω ( y ) C r = 1 | I r | σ | I r | p ,

for all pairwise disjoint decompositions Q=r=1Ir 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 n when σ is doubling and 1<p<. Finally, we characterize the weak-type two weight inequality for certain maximal singular integrals T in n when 1<p<, 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
Received: 7 October 2009
Revised: 2 February 2011
Accepted: 2 March 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document
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

Subjects
Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)

Keywords
two weight singular integral maximal function maximal truncation

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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