Vol. 5, No. 1, 2012

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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

Let σ and ω be positive Borel measures on with σ doubling. Suppose first that 1 < p 2. 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

(sQ(x)pdω(x))1p(sQ(x)p dσ(x))1p 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,

QT(χEσ)(x)pdω(x) C 1Qdσ(x) for all E Q,

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

QT(χQfσ)(x)dω(x) C2(Q|f(x)|pdσ(x))1p(Qdω(x))1p

for all intervals Q in 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=1|I r|σ|Ir|p1 =0 2 |(Ir)()|χ(Ir)()(y))pdω(y) C r=1|I r|σ|Ir|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.

two weight, singular integral, maximal function, maximal truncation
Mathematical Subject Classification 2000
Primary: 42B20
Received: 7 October 2009
Revised: 2 February 2011
Accepted: 2 March 2011
Published: 25 June 2012
Michael Lacey
School of Mathematics
Georgia Institute of Technology
686 Cherry Street NW
Atlanta, GA 30332-0160
United States
Eric T. Sawyer
Department of Mathematics and Statistics
McMaster University
1280 Main St. West
Hamilton, ON L8S 4K1
Ignacio Uriarte-Tuero
Department of Mathematics
Michigan State University
East Lansing, MI 48824
United States