TRANSFORM: A REAL VARIABLE CHARACTERIZATION, I

TRANSFORM: A REAL VARIABLE CHARACTERIZATION, I

MICHAEL T. LACEY, ERIC T. SAWYER,

CHUN-YEN SHEN, and IGNACIO URIARTE-TUERO

Abstract

Let andw be locally finite positive Borel measures on R which do not share a common point mass. Assume that the pair of weights satisfy a PoissonA2condition, and satisfy the testing conditions below, for the Hilbert transformH,

Z

I

H.1I/²dw .I /;

Z

I

H.w1I/²dw.I /;

with constants independent of the choice of intervalI. Then H./maps L². /to L².w/, verifying a conjecture of Nazarov, Treil, and Volberg. The proof has two com- ponents, aglobal-to-localreduction, carried out in this article, and an analysis of the localproblem, to be elaborated in a future Part II version of this article.

1. Introduction

Define a truncated Hilbert transform of a locally bounded signed measureby H;ı.x/WD

Z

<jyxj<ı

d.y/

yx; 0 < < ı:

Given weights (i.e., locally bounded positive Borel measures) andwon the real line R, we consider the followingtwo-weight norm inequalityfor the Hilbert transform,

sup

0<<ı

Z

R

ˇˇH_;ı.f /ˇˇ²dwN² Z

R

jfj²d; f 2L². /; (1.1)

DUKE MATHEMATICAL JOURNAL

Vol. 163, No. 15, © 2014 DOI10.1215/00127094-2826690 Received 13 March 2012. Revision received 23 February 2014.

2010Mathematics Subject Classification. Primary 42A50; Secondary 47B38, 42B20.

Lacey’s work partially supported by National Science Foundation grant DMS-0968499, Simons Foundation grant 229596, and Australian Research Council grant ARC-DP120100399.

Sawyer’s work partially supported by the Natural Sciences and Engineering Research Council of Canada.

Uriarte-Tuero’s work partially supported by National Science Foundation grants DMS-0901524 and DMS- 1056965, Ministerio de Educación y Ciencias (Spain) grant MTM2010-16232 and MTM2009-14694-C02- 01, and by a Sloan Foundation Fellowship.

2795

where N is the best constant in the inequality, uniform over all truncations of the Hilbert transform kernel. Below, will write the inequality above askH.f /k_L2.w/

Nkfk_L2.w/, that is, the uniformity over the truncation parameters is suppressed.

The primary question is to find a real variable characterization of this inequality, and the theorem below is an answer to the beautiful conjecture of Nazarov, Treil, and Volberg (see [29, p. 127]). Set

P .; I /WD Z

R

jIj

jIj²Cdist.x; I /² .dx/;

which is approximately the Poisson extension of to the upper half-plane, evaluated at.xI;jIj/, wherexI is the center ofI.

THEOREM1.2

Let and w be locally finite positive Borel measures on the real line R with no common point masses. Then, the two-weight inequality (1.1) holds if and only if these three conditions hold uniformly over all intervalsI:

P .; I /P .w; I /A2; (1.3)

Z

I

ˇˇH.1_I /ˇˇ²dwT² .I /;

Z

I

ˇˇH.1_Iw/ˇˇ²dT²w.I /: (1.4)

We have

N A¹⁼²₂ CT DWH; (1.5)

whereA2andT are the best constants in the inequalities above.

It is well known (see [29]) that theA2condition is necessary for the norm inequal- ity, and the inequalities (1.4) are obviously necessary, thus the content of the theorem is the sufficiency of theA2and testing inequalities. In the present article we will carry out aglobal-to-localreduction in the proof of sufficiency, while the analysis of the localproblem will be made in a future article (Part II of this series; see [5]).

The Nazarov–Treil–Volberg conjecture has only been verified before under addi- tional hypotheses on the pair of weights, hypotheses which are not necessary for the two-weight inequality. The so-called pivotal condition of [29] is not necessary, as was proved in [9]. The pivotal condition is still an interesting condition: It is all that is needed to characterize the boundedness of the Hilbert transform, together with the maximal function in both directions. But, the boundedness of this triple of operators is decoupled in the two-weight setting (see [24]).

Our argument has these attributes. Certain degeneracies of the pair of weights must be addressed, such as the contribution made by Nazarov, Treil, and Volberg in their innovative 2004 paper [18] (see also [29]), which was further sharpened with the property ofenergyin [9], a crucial property of the Hilbert transform. This theme is fur- ther developed in the following pages, with the notion of functional energy explained in Section5.

The proof should proceed through the analysis of the bilinear formhH.f /; gwi, as one expects certain paraproducts to appear. Still, the paraproducts have no canoni- cal form, suggesting that the proof is highly nonlinear inf andg. The nonlinear point of view was initiated in [7] and is central to our work here. A particular feature of our arguments is a repeated appeal to certainquasiorthogonality arguments, providing (many) simplifications over prior arguments. For instance, we never find ourselves constructing auxiliary measures, and verifying that they are Carleson, a frequent step in many related arguments.

One can phrase a two-weight inequality question for any operator T, a ques- tion that became apparent with the foundational paper of Muckenhoupt [12] onAp

weights for the maximal function. Indeed, the case of Hardy’s inequality was quickly resolved by Muckenhoupt [11]. The maximal function was resolved by the second author in [26], as well as the fractional integrals, and, essential for this paper, Poisson integrals (see [27]). The latter paper established a result which closely paralleled the contemporaneousT 1theorem of David and Journé [1]. This connection, fundamental in nature, was not fully appreciated until the innovative work of Nazarov, Treil, and Volberg [14]–[16] in developing a nonhomogeneous theory of singular integrals. The two-weight problem for dyadic singular integrals was only resolved recently in [17].

Partial information about the two-weight problem for singular integrals in [21] was basic to the resolution of theA₂conjecture in [3], and several related results (see [4], [6], [21], [22]). Our result is the first real variable characterization of a two-weight inequality for a continuous singular integral.

Interest in the two-weight problem for the Hilbert transform arises from its natu- ral occurrence in questions related to operator theory (see [20], [25]), spectral theory (see [20]), model spaces (see [23]), and analytic function spaces (see [10]). In the context of operator theory, Sarason posed the conjecture (see [2]) that the Hilbert transform would be bounded if the pair of weights satisfied the (full) PoissonA2con- dition. This was disproved by Nazarov [13]. Advances on these questions have been linked to a finer understanding of the two-weight question (see, e.g., [19], [20]) built upon Nazarov’s counterexample.

2. Dyadic grids and Haar functions 2.1. Choice of truncation

We have stated the main theorem withhard cutoffs in the truncation of the Hilbert transform. There are many possible variants in the choice of truncation; moreover, the proof of sufficiency requires a different choice of truncation.

Consider a truncation given by HQ˛;ˇ.f /.x/WD

Z

f .y/K˛;ˇ.yx/ .dy/;

whereK˛;ˇ.y/is chosen to minimize the technicalities associated with off-diagonal considerations. Specifically, setK˛;ˇ.0/D0, and otherwiseK˛;ˇ.y/is odd and for y > 0,

K˛;ˇ.y/WD 8ˆ ˆˆ ˆ<

ˆˆ ˆˆ :

_˛^y₂C²_˛ 0 < y < ˛;

1

y ˛yˇ;

^y

ˇ²C_ˇ² ˇ < y < 2ˇ;

0 2ˇy:

This is aC¹function on.0; 2ˇ/, and is Lipschitz, convex, and monotone on.0;1/.

We now argue that we can use these truncations in the proof of the sufficiency bound of our main theorem.

PROPOSITION2.1

If the pair of weights; wsatisfy theA2bound (1.3), then one has the uniform norm estimate with thehardtruncations (1.1) if and only if one has uniform norm estimate for thesmoothtruncations,

sup

0<˛<ˇ

HQ_˛;ˇ.f /

_wNkfk:

Indeed,jH˛;ˇ.f / QH˛;ˇ.f /jA˛.jfj/CAˇ.jfj/, where these last two operators aresingle-scaleaverages, namely,

A˛. /.x/D˛¹ Z

.x3˛;xC3˛/

.y/ .dy/:

But, the (simple)A2 bound is all that is needed to provide a uniform bound on the operatorsA˛. /. So the proposition follows.

Henceforth we use the truncationsHQ_˛;ˇ, and we suppress the tilde in the notation.

The particular choice of truncation is motivated by this off-diagonal estimate on the kernels.

PROPOSITION2.2

Suppose that2jxx⁰j<jxyj; then

K_˛;ˇ.yx⁰/K_˛;ˇ.yx/DCx;x⁰;y

x⁰x .yx/.yx⁰/; whereCx;x⁰;yD1; 2˛ <jxyj<1

2ˇ; (2.3)

and is otherwise positive and never more than4.

Proof

The assumptions imply thatyx⁰ andyx have the same sign. Assume without loss of generality that 0 < yx⁰< yx. If2˛ <jxyj< ˇ=2, it follows that

˛ <jx⁰yj< ˇ, and so by the definition, K_˛;ˇ.yx⁰/K_˛;ˇ.yx/D 1

yx⁰ 1

yx D x⁰x .yx/.yx⁰/: And, in the general case, we havej_dt^dK_˛;ˇ.t /j 4t², so that

0K_˛;ˇ.yx⁰/K_˛;ˇ.yx/

Z yx yx⁰

4

t²dtD4 x⁰x .yx/.yx⁰/: 2.2. Dyadic grids

A collection of intervalsG is agridif for allG; G⁰2G, we haveG\G⁰2 ¹;; G; G⁰º.

By a dyadic grid we mean a gridD of intervals of R such that for each interval I 2D, the subcollection¹I⁰2DW jI⁰j D jIjºpartitionsR, aside from endpoints of the intervals. In addition, the left and right halves ofI, denoted byI˙, are also inD. ForI2D, the left and right halvesI˙are referred to as thechildrenofI. We denote by_DI the unique interval inD havingI as a child, and we refer to_DI as theD-parentofI.

We will work with subsets F D. We say that I has F-parentFI DF if F 2F is the minimal element ofF that containsI.

2.3. Haar functions

Let be a weight onR, one that does not assign positive mass to any endpoint of a dyadic gridD. IfI 2D is such that assigns nonzero weight to both children ofI, the associated Haar function is

h_IWD s

.I/ .IC/ .I /

I

.I/C IC

.IC/ :

In this definition, we are identifying an interval with its indicator function, and we will do so throughout the remainder of the paper. This is anL². /-normalized function, and has-integral zero. For any dyadic interval I₀, we have that ¹ .I₀/¹⁼²I₀º [

¹h_IWI 2D; II0ºis an orthogonal basis forL².I0; /.

We will use the notationf .I /O D hf; h_Ii, as well as

_If D hf; h_Iih_IDICEI_Cf CIEIf IEIf:

The second equality is the familiar martingale difference equality, and so we will refer to_If as amartingale difference. It implies the familiar telescoping identity EJf DP

IWIJEJ_If.

For any function theHaar supportoff is the collection¹I2DW Of .I /¤0º.

2.4. Good-bad decomposition

With a choice of dyadic gridD understood, we say thatJ2D is.; r/-goodif and only if for all intervalsI2DwithjIj 2^r1jJj, the distance fromJto the boundary ofeither childofI is at leastjJjjIj¹.

For f 2L². /, we set P_good f DP

I2D Iis.; r/-good

_If. The projection P_good^w g is defined similarly. To make the two reductions below, one must make a random selection of grids, as is detailed in [9] and [29]. The use of random dyadic grids has been a basic tool since the foundational work of [14]–[16]. Important elements of the suppressed construction of random grids are the following.

(1) It suffices to consider a single dyadic gridD.

(2) For any fixed0 < < 1=2, we can choose an integerr sufficiently large so that it suffices to consider f such that f DP_good f, and likewise for g2 L².w/. Namely, it suffices to estimate the constant below, for an arbitrary dyadic gridD,

ˇˇhHf; giw

ˇˇN_goodkfkkgkw;

where it is required thatf DP_good 2L². /andgDP_good^w 2L².w/.

That the functions are good is, at some moments, an essential property. We sup- press it in notation, however, taking care to emphasize in the text those places in which we appeal to the property of being good.

A reduction, using randomized dyadic grids, allows one the extraordinarily useful reduction in the next lemma. This is a well-known reduction, due to Nazarov, Treil, and Volberg, explained in full detail in the current setting in [18, Section 4]. Below, H is as in (1.5), the normalized sum of theA2and testing constants.

LEMMA2.4

For all sufficiently small, and sufficiently larger, this holds. Suppose that for any

dyadic gridD, such that no endpoint of an intervalI 2D is a point mass for or w,we have

ˇˇhHP_good f; P_good^w giw

ˇˇHkfkkgkw: (2.5)

Then, the same inequality holds without the projectionsP_good , andP_good^w .

Inequality (2.5) should be understood as an inequality, uniform over the class of smooth truncations of the Hilbert transform. But, we can suppress this in the notation without causing confusion. The bilinear form only needs to be controlled for.; r/- good functionsf andg, goodness being defined with respect to a fixed dyadic grid.

Suppressing the notation, we writegoodfor.; r/-good, and it is always assumed that the dyadic gridD is fixed, and only good intervals are in the Haar support off and g, though is also suppressed in the notation.

3. The global-to-local reduction

The goal of this section is to reduce the analysis of the bilinear form in (2.5) to the local estimate, (3.4). It is sufficient to assume thatf andgare supported on an interval I⁰; by trivial use of the interval testing condition, we can further assume thatf andg are of integral zero in their respective spaces. Thus,f is in the linear span of (good) Haar functionsh_I forI I⁰, and similarly forg, and

hHf; giwD X

I;JWI;JI⁰

hH_If; ^w_Jgiw:

The argument is independent of the choice of truncation that implicitly appears in the inner product above.

The double sum is broken into different summands. Many of the resulting cases are elementary, and we summarize these estimates as follows. Define the bilinear form

B^above.f; g/WD X

IWII⁰

X

JWJI_J

EI_JIf hHIJ; ^w_Jgiw;

where here and throughout, J I means J I and2^rjJj jIj. In addition, the argument of the Hilbert transform,I_J, is the child ofI that containsJ, so that_If is constant onIJ. DefineB^below.f; g/in the dual fashion.

This set of dyadic grids that fail this condition have probability zero in standard constructions of the random dyadic grids.

LEMMA3.1

We have, with the notation of (1.5),

ˇˇhHf; giwB^above.f; g/B^below.f; g/ˇ

ˇHkfkkgkw:

This is a common reduction in a proof of aT 1theorem, and in the current context, it only requires goodness of intervals and the A2 condition. For a proof, one can consult [29] and [18]. The lemma is specifically phrased and proved in this way in [7, Section 8].

These definitions are needed to phrase the global-to-local reduction. The follow- ing definition depends upon the essential energy inequality (4.7) in Section 4.

Definition 3.2

Given any intervalF0, defineF_energy.F0/to be the maximal subintervalsFF0such that

P .F₀; F /²E.w; F /²w.F / > 10C₀H² .F /;

whereE.w; F / is defined in (4.6) and whereC0 is the constant in Proposition4.8.

We have .S

¹F WF 2F.F0/º/.=10/.F0/.

Definition 3.3

LetI0 be an interval, and letS be a collection of disjoint intervals contained inI0. A functionf 2L²₀.I₀; /is said to beuniform (with respect toS)if the following conditions are met.

(1) Each energy stopping intervalF 2F_energy.I0/is contained in someS2S.

(2) The functionf is constant on each intervalS2S.

(3) For any intervalII0which is not contained in anyS2S,EIjfj 1.

We will say that gisweakly adapted to a functionf uniform with respect toS if JS for some intervalS 2S implies thathg; h^w_Ji_wD0. We will also say thatgis weakly adapted toS.

The constantLis defined as the best constant in thelocal estimate ˇˇB^above.f; g/ˇˇL®

.I0/¹⁼²C kfk¯

kgkw; (3.4)

wheref; g are of mean zero on their respective spaces, supported on an intervalI0. Moreover,f is uniform andgis weakly adapted tof. The inequality above is homo- geneous ing, but notf, since the term .I0/¹⁼²is motivated by the bounded averages property off.

THEOREM3.5 (Global-to-local reduction) We have

ˇˇB^above.f; g/ˇ

ˇ¹HCLºkfkkgkw: The same inequality holds for the dual formB^below.f; g/.

A reduction of this type is a familiar aspect of many proofs of aT 1 theorem, proved by exploiting standard off-diagonal estimates for Calderón–Zygmund ker- nels, but in the current setting, it is a much deeper fact, a consequence of thefunc- tional energy inequality of Section 5. We make the following construction for an f 2L².I⁰; /, of-integral zero. AddI⁰ toF, and set˛f.I⁰/WDE_I0jfj. In the inductive stage, ifF 2F is minimal, add toF those maximal descendantsF⁰ofF such thatF⁰2F_energy.F /orE_F⁰jfj 10˛f.F /. Then define

˛_f.F⁰/WD

´˛_f.F / EF⁰jfj< 2˛_f.F /;

EF⁰jfj otherwise:

If there are no such intervalsF⁰, the construction stops. We refer toF and˛f./as Calderón–Zygmund stopping data forf, following the terminology of [7, Definition 3.5]. Their key properties are collected here.

LEMMA3.6

ForF and˛f./as defined above, the following hold.

(1) The dyadic intervalI0is the maximal element ofF. (2) For allI 2D,II⁰, we haveE_Ijfj 10˛f.FI /.

(3) The datum˛f is monotonic: IfF; F⁰2F andFF⁰, then˛f.F /˛f.F⁰/.

(4) The collectionF is-Carleson in that X

F2FWFS

.F /2 .S /; S2D: (3.7) (5) We have the inequality

X

F2F

˛f.F /F

kfk: (3.8)

Proof

The first three properties are immediate from the construction. The fourth, the -Carleson property is seen this way. It suffices to check the property forS 2F.

Now, theF-children can be inF_energy.S /, which satisfy X

F⁰2F_energy.S /

.F⁰/ 1 10 .S /:

Otherwise, note that by choice of˛_f./, we haveESjfj 2˛_f.S /. These intervals F⁰ satisfyEF⁰jfj 10˛_f.S /5ESjfj. These intervals satisfy the display above with1=10replaced by1=5. Hence, (3.7) holds.

For the final property, let G F be the subset at which the stopping values change. IfF 2F G, andGis theG-parent ofF, then˛f.F /D˛f.G/. Set

ˆGWD X

F2FW_GFDG

F:

DefineG_kWD ¹ˆG2^kº, forkD0; 1; : : :. The-Carleson property implies integra- bility of all orders in-measure ofˆ_G. Using the third moment, we have .G_k/ 2^3k .G/. Then, estimate

X

F2F

˛f.F /F

2 D

X

G2G

˛f.G/ˆG

2

X1 kD0

.kC1/^C11X

G2G

˛f.G/2^k1G_k

²

X1 kD0

.kC1/²X

G2G

˛_f.G/2^k1G_k.x/

2

X1 kD0

.kC1/² X

G2G

˛f.G/²2^2k .Gk/

X

G2G

˛_f.G/² .G/kMfk²kfk²:

Note that we have used Cauchy–Schwarz in k at the step marked by an . In the step marked with, for each pointx, the nonzero summands are a (super)geometric sequence of scalars, so the square can be moved inside the sum. Finally, we use the estimate on the-measure ofG_k, and compare to the maximal functionMf to com- plete the estimate.

We will use the notation

P_Ff WD X

I2DW_FIDF

_If; F2F;

and similarly forQ^w_F, but rather than useFJin the definition, we usePFJ, defined to be the minimalF 2F withJF. Without this alternate definition, some delicate case analysis would be forced upon us. The inequality (3.8) allows us to estimate

X

F2F

®˛f.F / .F /¹⁼²C kP_Ffk

¯kQ^w_Fgkw

h X

F2F

®˛f.F /² .F /C kP_Ffk²¯ X

F2F

kQ_F^wgk²_wi1=2

kfkkgkw: (3.9)

We will refer to this as thequasiorthogonalityargument, and we remark that it only requires orthogonality of the projectionsQ^w_Fg. It is very useful.

LEMMA3.10 We have

ˇˇB^above.f; g/B_F^above.f; g/ˇ

ˇHkfkkgkw; whereB_F^above.f; g/WD X

F2F

B^above.P_Ff; Q_F^wg/:

Proof

We apply the functional energy inequality of Section5. Observe thatf DP

F2FP_Ff and that

X

JWJI₀

^w_JgD X

F2F

Q^w_Fg:

From the definition of B^above.f; g/, we can assume that g equals the sum above.

Therefore,

B^above.f; g/D X

F⁰2F

X

F2F

B^above.P_F0f; Q^w_Fg/:

In the sum above, we can also add the restriction that F⁰\F ¤ ;, for otherwise B^above.P_F0f; Q^w_Fg/D0. For a pair of intervalsJ IJ, note that this implies that JFI; that is,PFJFI. Therefore, we can add the restrictionF F⁰. The case ofF⁰DF is the definition ofB_F^above.f; g/, so that it suffices to estimate

X

F ;F⁰2F F⁰F

B^above.P_F0f; Q^w_Fg/: (3.11)

Observe that the functionsgF WDQ^w_Fg areF-adapted in the sense of Defini- tion5.1, and by constructionF satisfies the Carleson measure condition (3.7). We take these steps to apply the functional energy inequality. The argument of the Hilbert transform isIF, the child of I that contains F. WriteIF DF C.IF F /, and use linearity ofH. Note that by the standard martingale difference identity and the construction of stopping data,

ˇˇ ˇ X

IWIF

EI_FIfˇˇˇ˛_f.F /; F 2F:

Hence, invoking interval testing, ˇˇ

ˇX

F2F

X

IWIF

EI_FIf hHF; gFiw

ˇˇ ˇ X

F2F

˛_f.F /ˇˇhHF; gFiw

ˇˇ

H X

F2F

˛_f.F / .F /¹⁼²kgFkw: Quasiorthogonality bounds this last expression.

For the second expression, when the argument of the Hilbert transform isIFF, first note that

ˇˇ ˇ X

IWIF

EI_FIf .IF F / ˇˇ

ˇˆWD X

F⁰2F

˛f.F⁰/F⁰; F2F:

Therefore, by the definition ofF-adapted, the monotonicity property (4.3) applies, and yields

ˇˇ ˇ X

IWIF

EI_FIf˝

H.IFF /; gF˛

w

ˇˇ

ˇ X

J2J.F /

P .ˆ; J /D x

jJj; J g_FE

w; F 2F: Here,J.F /are the maximal good intervalsJF, andg_F WDP

J2J.F /WJFjbg.J /j h^w_J, so that every term has a positive inner product withx. The sum overF 2F of this last expression is controlled by functional energy, and by the property that kˆkkfk. This completes the bound for (3.11).

Proof of Theorem3.5

By Lemma3.10, it remains to controlB_F^above.f; g/. Keeping the quasiorthogonality argument in mind, we see that appropriate control on the individual summands is enough to control it. For eachF 2F, letSF be theF-children ofF. Observe that the function

C ˛f.F /1

P_Ff (3.12)

is uniform onFwith respect toSF, for an appropriate absolute constantC. Moreover, the functionQ_F^wgdoes not have any intervalJ in its Haar support strongly contained in an intervalS2S_F. That is, it is weakly adapted to the function in (3.12). Therefore, by assumption,

ˇˇB^above.P_Ff; Q^w_Fg/ˇ ˇL®

˛F.F / .F /¹⁼²C kP_Ffk¯

kQ_F^wgkw:

The sum over F 2F of the right-hand side is bounded by the quasiorthogonality argument of (3.9).

4. Energy, monotonicity, and Poisson

Our theorem (Theorem 1.2) is particular to the Hilbert transform, and so depends upon special properties of it. They largely extend from the fact that the derivative of1=yis positive. The following monotonicity property for the Hilbert transform was observed in [7, Lemma 5.8], and is basic to the analysis of the functional energy inequality.

LEMMA4.1 (Monotonicity property)

LetKI be two intervals, and assume that does not have point masses at the end point ofI. Then, for any functiong2L².I; w/, withw-integral zero, andˇ > 2jKj,

P

.KI /; ID x jIj; gE

wlim inf

˛#0

˝H˛;ˇ

.KI /

; g˛

w: (4.2)

Here,gDP

J⁰jbg.J⁰/jh^w_J0 is a Haar multiplier applied tog. IfJ is a good interval, JI, then, for functiong2L².J; w/, withw-integral zero, and signed measures andsupported onKI, withjj , we have

sup

0<˛<ˇ

ˇˇhH_˛;ˇ; gi_wˇ

ˇP .; J /D x jJj; gE

w: (4.3)

The truncations enter into the formulation of the lemma, since they play a notable role here. We need this preparation.

LEMMA4.4

LetI andJ be two intervals which share an endpointa, at which neither norw have a point mass. Then,

sup

0<˛<ˇ

ˇˇhH_˛;ˇI; Jiw

ˇˇA¹⁼²₂ p

.I /w.J /: (4.5) Proof

IfjIj ' jJj, this inequality is the weak boundedness principle of [9, Section 2.2]. So, let us assume that10jIj<jJj. Then, it remains to bound

ˇˇ˝H_˛;ˇI; .J n10I /˛

w

ˇˇ X1 nD11

.I /w.J\..nC1/InnI //

njIj .I /

jIj¹⁼²P .w; I /¹⁼²w.J /¹⁼²A¹⁼²₂ p

.I /w.J /:

This depends upon obvious kernel bounds, and an application of Cauchy–Schwarz to derive the Poisson term above.

Proof of Lemma4.1

By linearity, it suffices to prove (4.2) in the case ofgDh^w_I. The point is to separate the supports of the functions involved. SinceI does not have a point mass at the end point ofI, we have . I nI /#0as #1. It follows that we can fix a > 1sufficiently small so thatP . .KI /; I /'P . .K I /; I /, and one more condition that we will come back to. Then, for0 < ˛ < .1=2/. 1/jIj, we estimate as below, where x_I is the center ofI,

˝H˛;ˇ

.K I /

; h^w_I˛

w

D Z

KI

Z

I

®K˛;ˇ.yx/K˛;ˇ.yxI/¯

h^w_I.x/w.dx/ .dy/

D Z

KI

Z

I

xxI

.yx/.yxJ/h^w_I.x/w.dx/ .dy/

P

.KI /; IDxxI

jIj ; h^w_IE

w:

We have subtracted the term, sinceh^w_I has integral zero, then we applied (2.3) with Cx;x_J;yD1, as follows from our choices of˛andˇ. Then, note that.xxJ/h^w_I 0, so that we can pull out the Poisson term. The last line follows by our selection of sufficiently close to1. Then, the last condition needed is to select sufficiently close to1that, in view of (4.5),

sup

˛;ˇ

ˇˇ˝H_˛;ˇ. InI /; h^w_I˛

w

ˇˇA¹⁼²₂ p

. I nI / < cP

.KI /; IDxxI

jIj ; h^w_IE

w:

In the last line,c > 0is an absolute constant. This completes the proof of (4.2).

Turn to (4.3). The estimate (2.3) applies:

ˇˇhH_˛;ˇ; giw

ˇˇDˇˇˇ Z

KI

Z

J

®K˛;ˇ.yx/K˛;ˇ.yxJ/¯

h^w_J.x/w.dx/.dy/ˇˇˇ D

ˇˇ ˇ Z

KI

Z

J

Cx;x_J;y

.xxJ/

.yx/.yxJ/h^w_J.x/w.dx/.dy/

ˇˇ ˇ:

But recall that0Cx;x_J;y4and that it equals1for˛sufficiently small. Moreover, yxandyxJ have the same sign, and.xxJ/h^w_J.x/0. So an upper bound is obtained by passing fromto:

ˇˇhH_˛;ˇ; giw

ˇˇ Z

KI

Z

J

.xxJ/

.yx/.yxJ/h^w_J.x/w.dx/.dy/

'P .; J /D x jJj; h^w_JE

w:

The concept ofenergyis fundamental to the subject. For intervalI, define E.w; I /²WDE^w.dx/I E^w.dxI ^0/

.xx⁰/² jIj² D 2

w.I / X

JI

D x jIj; h^w_JE2

w: (4.6)

Now, consider theenergy constant, the smallest constant E such that this condition holds, as presented or in its dual formulation. For all dyadic intervalsI0, all partitions P ofI0 into dyadic intervals, we have

X

I2P

P .I0; I /²E.w; I /²w.I /E² .I0/: (4.7) This was shown in [9, Proposition 2.11].

PROPOSITION4.8 For a finite constantC0,

E²C₀¹A¹⁼²₂ CTº²DC₀H²: We will always estimateE byH. The proof is recalled here.

Proof

It suffices to consider the case of finite partitionsP ofI. We first prove a version of the energy inequality withholesin the argument of the Poisson. It follows from (4.2) that we can fix0 < ˛ < ˇsuch that

P

.I0I /; I2

E.w; I /²w.I /H_˛;ˇ

.I0I /²_{L2.I; /}; I2P: Then, using linearity and interval testing, we have

X

I2P

H_˛;ˇ.I₀/²_{L2.I; /}H_˛;ˇ.I₀/²_{L2.I; /}H² .I₀/

and

X

I2P

H_˛;ˇ.I /²_{L2.I; /}H²X

I2P

.I /H² .I0/:

Then, by theA2 bound, we haveP .I; I /²E.w; I /²w.I / .I /, which we can sum over the partition. This completes the proof.

One should keep in mind that the concept of energy is related to the tails of the Hilbert transform. The energy inequality, and its multiscale extension to the functional energy inequality, show that the control of the tails is very subtle in this problem.

We also need the following elementary Poisson estimate from [29]; used occa- sionally in this argument, it is crucial to the proof of Lemma3.1.

LEMMA4.9

Suppose thatJII0, and thatJ is good. Then jJj²¹P

.I0I /; J

jIj²¹P

.I0I /; I

: (4.10)

Proof

We have dist.J; I0I / jJjjIj¹, so that for anyx2I0I, we have jJj²

.jJj Cdist.x; J //² jIj² .jIj Cdist.x; I //²: Integrating this last expression, it follows that

jJj²¹P

.I0I /; J

D jJj²¹ Z

I₀I

jJj

.jJj Cdist.x; J //²d jIj²

Z

I₀I

1

.jJj Cdist.x; J //²d:

And this proves the inequality.

5. The functional energy inequality

We state an important multiscale extension of the energy inequality (4.7).

Definition 5.1

LetF be a collection of dyadic intervals. A collection of (good) functions¹gFºF2F

inL².w/is said to beF-adaptedif, for allF 2F, the Haar support of the function gF is contained in¹JW P_FJDFº.

Definition 5.2

LetF be the smallest constant in the inequality below, or its dual form. The inequality holds for all nonnegativeh2L². /, all-Carleson collectionsF, and allF-adapted collections¹gFºF2F:

X

F2F

X

J2J.F /

P .h; J/ ˇˇ ˇD x

jJj; gFJE

w

ˇˇ

ˇFkhk

h X

F2F

kgFk²_wi1=2

:

HereJ.F /consists of themaximalgood intervalsJF. Note that the estimate is universal inhandF, separately.

This constant was identified in [7], and is herein shown to be necessary from the A2and interval testing inequalities. Recall the definition ofH in (1.5).

THEOREM5.3

Assume thatF satisfies (3.7); then,F H.

The first step in the proof is the domination of the constantF by the best con- stant in a certain two-weight inequality for the Poisson operator, with the weights being determined bywandin a particular way. This is the decisive step, since there is a two-weight inequality for the Poisson operator proved by one of us in [27, Theo- rem 2]. It reduces the full norm inequality to simpler testing conditions, which are in turn controlled by theA2and Hilbert transform testing conditions.

5.1. The two-weight Poisson inequality Consider the weight

X

F2F

X

J2J.F /

P_{F ;J}^w x

jJj ²

wı_.xJ;jJj/: Here,P_{F ;J}^w WDP

J⁰WJ⁰J;P_FJDF^w_J0. We can replacexbyxcfor any choice ofc we wish; the projection is unchanged. Andıqdenotes a Dirac unit mass at a pointqin the upper half-planeR²C. We prove the two-weight inequality for the Poisson integral

P.h /_L2.R2

C;/Hkhk;

for all nonnegativeh. Above,P./denotes the Poisson extension operator to the upper half-plane, so that in particular

P.h / ²_L2.R2

C;/D X

F2F

X

J2J.F /

P.h /

x_J;jJj2P_{F ;J}^w x jJj

²

w;

wherex_J is the center of the intervalJ. The proof of Theorem5.3follows by duality.

Phrasing things in this way brings a significant advantage. The characterization of the two-weight inequality for the Poisson operator (see [27]) reduces the full norm inequality above to these testing inequalities. For any dyadic intervalI2D we have

Z

R²_CP.I /²d.x; t /H² .I /; (5.4) Z

RP.tI /b ² .dx/A₂ Z

b I

t²d.x; t /; (5.5) wherebI DI Œ0;jIjis the box overI in the upper half-plane and wherePis the dual Poisson operator

P.tbI /D Z

b I

t²

t²C jxyj².dy; dt /:

One should keep in mind that the intervalsI are restricted to be in our fixed dyadic grid, a reduction allowed as the integrations on the left-hand side in (5.4) and (5.5) are done over the entire space, eitherR²CorR. (Goodness of the intervalsI above is not needed.) This reduction is critical to the analysis below.

Remark 5.6

A gap in the proof of the Poisson inequality in [27, p. 542] can be fixed as in [28]

or [8].

5.2. The Poisson testing inequality: The core

This section is concerned with a part of inequality (5.4). Restrict the integral on the left-hand side to the setbIR²C:

Z

b

IP.I /²d.x; t /H² .I /:

Since.xJ;jJj/2Ibif and only ifJI, we have Z

b

IP.I /.x; t /²d.x; t /D X

F2F

X

J2J.F /WJI

P.I /

xJ;jJj2P_{F ;J}^w x jJj

²_w:

For eachJ,

P_{F ;J}^w x jJj

2 w

Z

J

ˇˇ

ˇxE^w_Jx jJj

ˇˇ ˇ

2

dw.x/D2E.w; J /²w.J /2w.J /: (5.7) LetF0 be the maximalF 2F which are strictly contained inI, and letJ^]be those dyadic J such that.x_J;jJj/is in the support of, but has no parent inF₀.

These intervals are necessarily disjoint. Observe that by (5.7) and the energy inequal- ity,

X

J2J^]

P.F /

xJ;jJj2

xJ;jJj

X

J2J^]

P .F; J /²E.w; J /²w.J /

H² .F /: (5.8)

We claim that X

F2F₀

Z

FOP

.InF /

.x; t /²d.x; t /H .I /: (5.9) This is sufficient, since

Z

b

IP.I /.x; t /²d.x; t /

LHS(5.8) C LHS(5.9) C X

F2F₀

Z

FOP.F /.x; t /²d.x; t / H² .I /C X

F2F₀

Z

FOP.F /.x; t /²d.x; t /:

The individual terms in the last sum are set up for a recursive application of this inequality. Due to the Carleson condition (3.7), this recursion will finish the proof.

It remains to prove (5.9), which is another instance of the energy inequality. For an intervalF02F0, andF 2F strictly contained inF0, each intervalJ2J.F /is contained in someJ₀2J.F₀/. Then, the intervalsF 2F are not good, butJ and J0are good, hence

P

.InF0/

xJ;jJj2

xJ;jJj DhZ

InF₀

jJj jJj²C jxxJj²

i2P_{F ;J}^w x jJj

²

w

DhZ

InF₀

1 jJj²C jxxJj²

i2

kP_{F ;J}^w xk²_w

hZ

InF₀

jJ0j jJ0j²C jxxJ₀j²

i2 P_{F ;J}^w x

jJ0j

2 w: This follows from goodness. Forx2InF0,

jJj²C jxxJj² jxxJj² jxxJ₀j² jJ0jjF0j¹:

But then, we can add the projectionsP_{F ;J}^w , due to orthogonality, and use (5.7) again to see that

X

F2F FF0

X

J2J.F / JJ₀

P

.InF0/

xJ;jJj2

xJ;jJj

P.I /

xJ₀;jJ0j2 X

F2F FF₀

X

J2J.F / JJ₀

P_{F ;J}^w x

jJ0j

2 w

P.I /

xJ₀;jJ0j2

E.w; J0/²w.J0/:

The sum overF02F0, andJ02J.F0/is controlled by the energy inequality. This completes the proof of (5.9).

5.3. The Poisson testing inequality: The remainder

Now we turn to proving the following estimate for the global part of the first testing condition (5.4):

Z

R²CbIP.I /²dA2 .I /:

Decompose the integral on the left-hand side into four terms. WithFJ the unique F 2F withJ2J.F /, and using (5.7),

Z

R²_CbIP.I /²d

D X

JW.x_J;jJj/2R²_CbI

P.I /

xJ;jJj2P_F^w

J;J

x jJj

2 w

° X

JWJ\3ID;

jJjjIj

C X

JWJ3II

C X

JWJ\ID;

jJj>jIj

C X

JWJI

±P.I /

xJ;jJj2

w.J /

DACBCCCD:

Decompose termAaccording to the length ofJ and its distance fromI, to obtain A

X1 nD0

X1 kD1

X

JWJ3^kC1I3^kI jJjD2ⁿjIj

2ⁿjIj

dist.J; I /² .I /2

w.J /

X1 nD0

2²ⁿ X1 kD1

jIj² .I /w.3^kC1I3^kI / j3^kIj⁴ .I /

X1 nD0

2²ⁿ X1 kD1

3^2k° .3^kC1I /w.3^kC1I / j3^kIj²

± .I /A₂ .I /:

Decompose termBaccording to the length ofJand then use the Poisson inequal- ity (4.10), available to use because of goodness of intervalsJ. We then obtain

B X1 nD0

X

JWJ3II jJjD2ⁿ

2^n.24/ .I /² jIj² w.J /

X1 nD0

2^n.24/ .3I /w.3I /

j3Ij² .I /A2 .I /:

For termC, fornD1; 2; : : :, setJn to be those good dyadic intervalsJ with jJj>jIj,J\I D ;and

.n1/jJj dist.I; J / < njJj:

These intervals have bounded overlaps. Indeed, suppose that J1 Jr are all members forJ₁. Then, by goodness,

dist.J1; I /dist.Jr; I /.n1/2^rjJ1j Cdist.J1; @Jr/ ®

.n1/2^rC2^r.1/¯ jJ1j;

which is a contradiction to membership inJn. Restricting the sum to intervals inJn, we have

X

J2Jn

P.I /

xJ;jJj2

w.J / .I /² X

J2Jn

w.J / n⁴jJj² .I /²

jIj X

J2Jn

w.J / jIj n⁴jJj² .I /

n² .I /

jIj P .w; I /A2

.I / n² : And this is summable inn2N.

In the last termD, all the intervalsJ containI. Note that X

JWJI

P.I /

xJ;jJj2

w.J / .I /² X

JWJI

w.J / jJj² .I / .I /

jIj X

JWJI

w.J / jIj jJj² .I / .I /

jIj P .w; I /A2 .I /:

5.4. The dual Poisson testing inequality

We are considering (5.5). Note that there is a power oft on both sides and that the expressions on the two sides of this inequality are

Z

b I

t².dx; dt /D X

F2F

X

J2J.F / JI

kP_{F ;J}^w xk²_w;

P.tbI /.x/D X

F2F

X

J2J.F / JI

kP_{F ;J}^w xk²_w jJj²C jxxJj²:

We are to dominatekP.tI /kb ²by the first expression above. The squared norm will be the sum over integerssofTs below, in which the relative lengths ofJ andJ⁰are fixed bys. Suppressing the requirement thatJ; J⁰I,

TsWD X

F2F

X

J2J.F /

X

F⁰2F

X

J⁰2J.F / jJ⁰jD2^sjJj

Z kP_{F ;J}^w xk²_w

jJj²C jxxJj² kP_F^w0;J⁰xk²_w jJ⁰j²C jxxJ⁰j²d

Ms

X

F2F

X

J2J.F /

kP_{F ;J}^w xk²_w;

where Ms sup

F2F

sup

J2J.F /

X

F⁰2F

X

J⁰2J.F / jJ⁰jD2^sjJj

Z 1

jJj²C jxxJj² w.J⁰/ jJ⁰j² jJ⁰j²C jxxJ⁰j²d:

The estimate (5.7) has been used in the definition ofMs. We claim the termMs is at most a constant timesA22^s, and it is here that the full PoissonA2condition is used.

FixJ, and letn2Nbe the integer chosen so that.n1/jJj dist.J; J⁰/njJj.

Estimate the integral in the definition ofMs by w.J⁰/

jJ⁰j

Z jJ⁰j²

jJj²C jxx_Jj² jJ⁰j

jJ⁰j²C jxx_J⁰j²dA22^2s: This estimate is adequate fornD0; 1; 2. Then estimate the sum overJ⁰as

X

F⁰2F

X

J⁰2J.F⁰/WjJ⁰jD2^sjJj .n1/jJjdist.J;J⁰/njJj

2^2s2^s;

because the relative lengths of J and J⁰ are fixed, and each J⁰ is in at most one J.F /.

For the case ofn3, restrictJ⁰to be to the right ofJ, and lettnDxJCxJ⁰=2, so thatjxJtnj;jxJ⁰tnj 'njJj. First, estimate the integral in the definition ofMs

on the intervalŒt_n;1/:

w.J⁰/ jJ⁰j

Z 1 tn

jJ⁰j²

jJj²C jxxJj² jJ⁰j

jJ⁰j²C jxxJ⁰j²dA2

2^2s n² :

Then estimate the sum overJ⁰as follows:

X

F⁰2F

X

J⁰2J.F⁰/WjJ⁰jD2^sjJj .n1/jJjdist.J;J⁰/njJj

2^2s n² 2^s

n² :

This is clearly summable inn4.

Now, estimate on the integral on the interval.1; tn/:

w.J⁰/ jJ⁰j

Z t_n 1

jJ⁰j²

jJj²C jxxJj² jJ⁰j

jJ⁰j²C jxxJ⁰j²d D2^2sw.J⁰/

jJj Z t_n

1

jJj

jJj²C jxxJj² jJj²

jJ⁰j²C jxxJ⁰j²d 2^2sw.J⁰/

n²jJjP .; J /:

Drop the term with the geometric decay ins, and sum overnandJ⁰to see that X1

nD4

X

F⁰2F

X

J⁰2J.F⁰/WjJ⁰jD2^sjJj .n1/jJjdist.J;J⁰/njJj

w.J⁰/

n²jJjP .; I /P .w; J /P .; J /A2:

Here, we have appealed to the full PoissonA2condition. This completes the control of the dual Poisson testing condition.

Acknowledgments.The authors benefited from a stimulating conference on two-weight inequalities in October 2011 at the American Institute of Mathematics in Palo Alto, California, for which they express their appreciation. The reviewing process led to many improvements in this paper, including a streamlining of the main contribution, for which we thank the referees.

References

[1] G. DAVIDandJ.-L. JOURNÉ,A boundedness criterion for generalized Calderón-Zygmund operators, Ann. of Math. (2)120(1984), 371–397.

MR 0763911.DOI 10.2307/2006946. (2797)