Analysis & PDE
- Anal. PDE
- Volume 5, Number 1 (2012), 1-60.
A characterization of two weight norm inequalities for maximal singular integrals with one doubling measure
Let and be positive Borel measures on with doubling. Suppose first that . We characterize boundedness of certain maximal truncations of the Hilbert transform from to in terms of the strengthened condition
where , 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,
for all intervals in and all functions . In the case the same result holds if we include an additional necessary condition, the Poisson condition
for all pairwise disjoint decompositions of the dyadic interval into dyadic intervals . We prove that analogues of these conditions are sufficient for boundedness of certain maximal singular integrals in when is doubling and . Finally, we characterize the weak-type two weight inequality for certain maximal singular integrals in when , without the doubling assumption on , in terms of analogues of the second testing condition and the condition.
Anal. PDE, Volume 5, Number 1 (2012), 1-60.
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
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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