ANALYSIS & PDE Volume 5 No. 1 2012 mathematical sciences publishers Analysis & PDE msp.berkeley.edu/apde EDITORS EDITOR-IN-CHIEF Maciej Zworski University of California Berkeley, USA BOARD OF EDITORS Michael Aizenman Princeton University, USA Nicolas Burq Université Paris-Sud 11, France [email protected] [email protected] Luis A. 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Analysis & PDE, at Mathematical Sciences Publishers, Department of Mathematics, University of California, Berkeley, CA 94720-3840 is published continuously online. Periodical rate postage paid at Berkeley, CA 94704, and additional mailing offices. APDE peer review and production are managed by EditFLOW™ from Mathematical Sciences Publishers. PUBLISHED BY mathematical sciences publishers http://msp.org/ A NON-PROFIT CORPORATION Typeset in LATEX Copyright ©2012 by Mathematical Sciences Publishers ANALYSIS AND PDE Vol. 5, No. 1, 2012 dx.doi.org/10.2140/apde.2012.5.1 msp A CHARACTERIZATION OF TWO WEIGHT NORM INEQUALITIES FOR MAXIMAL SINGULAR INTEGRALS WITH ONE DOUBLING MEASURE MICHAEL LACEY,ERIC T. SAWYER AND IGNACIO URIARTE-TUERO Let σ and ! be positive Borel measures on R with σ doubling. Suppose first that 1 < p ≤ 2. We p characterize boundedness of certain maximal truncations of the Hilbert transform T\ from L (σ / to p L .!/ in terms of the strengthened A p condition Z 1=p Z 1=p0 p p0 sQ.x/ d!.x/ sQ.x/ dσ .x/ ≤ CjQj; R R where sQ.x/ D jQj=.jQj C jx − xQj/, and two testing conditions. The first applies to a restricted class of functions and is a strong-type testing condition, Z Z p T\(χE σ /.x/ d!.x/ ≤ C1 dσ .x/ for all E ⊂ Q; Q Q and the second is a weak-type or dual interval testing condition, Z Z 1=p Z 1=p0 p T\(χQ f σ /.x/ d!.x/ ≤ C2 j f .x/j dσ .x/ d!.x/ Q Q Q for all intervals Q in R and all functions f 2 L p(σ /. In the case p > 2 the same result holds if we include an additional necessary condition, the Poisson condition 1 1 1 Z −` p X p0−1 X 2 X p0 jI j jI j jχ .`/ .y/ d!.y/ ≤ C jI j jI j ; r σ r .`/ .Ir / r σ r j.Ir / R rD1 `D0 rD1 S1 for all pairwise disjoint decompositions Q D rD1 Ir 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 Rn when σ is doubling and 1 < p < 1. Finally, we characterize the weak-type two weight n inequality for certain maximal singular integrals T\ in R when 1 < p < 1, without the doubling assumption on σ, in terms of analogues of the second testing condition and the A p condition. 1. Introduction Sawyer [1984; 1982; 1988] characterized two weight inequalities for maximal functions and other pos- itive operators, in terms of the obviously necessary conditions that the operators be uniformly bounded on a restricted class of functions, namely indicators of intervals and cubes. Thus, these characterizations have a form reminiscent of the T 1 theorem of David and Journé. Lacey is supported in part by the NSF, through grant DMS-0456538. Sawyer is supported in part by NSERC. Uriarte-Tuero is supported in part by the NSF, through grant DMS-0901524. MSC2000: 42B20. Keywords: two weight, singular integral, maximal function, maximal truncation. 1 2 MICHAELLACEY,ERICT.SAWYER AND IGNACIO URIARTE-TUERO Corresponding results for even the Hilbert transform have only recently been obtained [Nazarov et al. 2010; Lacey et al. 2011] and even then only for p D 2; evidently these are much harder to obtain. We comment in more detail on prior results below, including the innovative work of Nazarov, Treil and Volberg [1999; 2008; 2010; 2003]. Our focus is on providing characterizations of the boundedness of certain maximal truncations of a fixed operator of singular integral type. The singular integrals will be of the usual type, for example the Hilbert transform or paraproducts. Only size and smoothness conditions on the kernel are assumed; see (1-9). The characterizations are in terms of certain obviously necessary conditions, in which the class of functions being tested is simplified. For such examples, we prove unconditional characterizations of both strong-type and weak-type two weight inequalities for certain maximal truncations of the Hilbert transform, but with the additional assumption that σ is doubling for the strong-type inequality. A major point of our characterizations is that they hold for all 1 < p < 1. The methods in [Lacey et al. 2011] and those of Nazarov, Treil and Volberg apply only to the case p D 2, where the orthogonality of measure- adapted Haar bases prove critical. The doubling hypothesis on σ may not be needed in our theorems, but is required by the use of Calderón–Zygmund decompositions in our method. As the precise statements of our general results are somewhat complicated, we illustrate them with an important case here. Let Z 1 T f .x/ D lim f .x − y/ dy "!0 Rn.−";"/ y denote the Hilbert transform, let Z D 1 − T[ f .x/ sup f .x y/ dy 0<"<1 Rn.−";"/ y denote the usual maximal singular integral associated with T , and finally let Z 1 T f .x/ D sup f .x − y/ dy \ y 0<"1;"2<1 Rn.−"1;"2/ 1=4<"2="1<4 denote the new strongly (or noncentered) maximal singular integral associated with T that is defined more precisely below. Suppose σ and ! are two locally finite positive Borel measures on R that have no point masses in common. Then we have the following weak and strong-type characterizations, which we emphasize hold for all 1 < p < 1. • The operator T[ is weak type .p; p/ with respect to (σ; !/, that is, kT[. f σ /kL p;1.!/ ≤ Ck f kL p(σ / (1-1) for all f bounded with compact support if and only if the two weight A p condition 1 Z 1 Z p−1 d! dσ ≤ C; jQj Q jQj Q TWOWEIGHTNORMINEQUALITIESFORMAXIMALSINGULARINTEGRALS 3 holds for all intervals Q and the dual T[ interval testing condition Z Z 1=p Z 1=p0 p T[(χQ f σ / d! ≤ C j f j dσ d! ; (1-2) Q Q Q p holds for all intervals Q and f 2 L Q(σ / (part 4 of Theorem 1.8). The same is true for T\. It is easy to see that (1-2) is equivalent to the more familiar dual interval testing condition Z Z ∗ p0 jL (χQ!/j dσ ≤ C d!; (1-3) Q Q for all intervals Q and linearizations L of the maximal singular integral T[ (see (2-10)). • Suppose in addition that σ is doubling and 1 < p < 1. Then the operator T\ is strong-type .p; p/ with respect to (σ; !/, that is, kT\. f σ /kL p.!/ ≤ Ck f kL p(σ / for all f bounded with compact support if and only if these four conditions hold: (1) the strengthened A p condition Z 1=p Z 1=p0 p p0 sQ.x/ d!.x/ sQ.x/ dσ .x/ ≤ CjQj; Q I jQj where sQ.x/ D , holds for all intervals Q; (2) the dual T interval testing condition jQjCjx−xQ j \ Z Z 1=p Z 1=p0 p T\(χQ f σ / d! ≤ C j f j dσ d! ; Q Q Q p holds for all intervals Q and f 2 L Q(σ /; (3) the forward T\ testing condition Z Z p T\(χE σ / d! ≤ C dσ; (1-4) Q Q holds for all intervals Q and all compact subsets E of Q; and (4) the Poisson condition 1 1 1 Z −` p X p0−1 X 2 X p0 jI j jI j χ .`/ .y/ d!.y/ ≤ C jI j jI j ; r σ r .`/ .Ir / r σ r j.Ir / j R rD1 `D0 rD1 S1 for all pairwise disjoint decompositions Q D rD1 Ir of the dyadic interval Q into dyadic inter- vals Ir for any fixed dyadic grid.