BOUNDEDNESS, AND A TWO WEIGHT ACCRETIVE GLOBAL T b THEOREM
ERIC T. SAWYER, CHUN-YEN SHEN, AND IGNACIO URIARTE-TUERO
Abstract. Let and!be locally …nite positive Borel measures onRn, letT be a standard -fractional Calderón-Zygmund operator onRn with 0 < n, and assume as side conditions theA2 conditions, puncturedA2 conditions, and certain -energy conditions. Then the weak boundedness property associated with the operatorT and the weight pair( ; !), is ‘good- ’ controlled by the testing conditions and the Muckenhoupt and energy conditions. As a consequence, assuming the side conditions, we can eliminate the weak boundedness property from Theorem 1 of [SaShUr9] to obtain thatT is bounded from L2( ) toL2(!) if and only if the testing conditions hold forT and its dual. As a corollary we give a simple derivation of a two weight accretive globalT btheorem from a relatedT1theorem. The role of two di¤erent parameterizations of the family of dyadic grids, by scale and by translation, is highlighted in simultaneously exploiting both goodness and NTV surgery with families of grids that arecommon to both measures.
Contents
1. Introduction 1
1.1. Quasicubes 3
1.2. Standard fractional singular integrals and the norm inequality 4
1.3. Quasicube testing conditions 5
1.4. Quasiweak boundedness and indicator/touching property 5
1.5. Poisson integrals andA2 6
1.6. A weighted quasiHaar basis 7
1.7. The strong quasienergy conditions 7
2. The Good- Lemma 9
2.1. Corollaries 9
3. Proof of the Good- Lemma 11
3.1. Good/bad technology 12
3.2. Control of the indicator/touching property 14
4. Appendix 25
References 26
Dedication: This paper is dedicated to Dick Wheeden on the occasion of his retirement from Rutgers University, and for all of his fundamental contributions to the theory of weighted inequalities, in particular for the beautiful paper of Hunt, Muckenhoupt and Wheeden that started it all back in 1973.
1. Introduction
The theory of weighted norm inequalities burst into the general mathematical consciousness with the celebrated theorem of Hunt, Muckenhoupt and Wheeden [HuMuWh] that extended boundedness of the
Date: September 12, 2016.
Research supported in part by NSERC.
C.-Y. Shen supported in part by the NSC, through grant NSC 104-2628-M-008 -003 -MY4.
I. Uriarte-Tuero has been partially supported by grants DMS-1056965 (US NSF), MTM2010-16232, MTM2015-65792-P (MINECO, Spain), and a Sloan Foundation Fellowship.
1
Hilbert transform to measures more general than Lebesgue’s, namely showing that H was bounded on the weighted spaceL2(Rn;w)if and only if theA2 condition of Muckenhoupt,
1 jQj
Z
Q
w(x)dx 1 jQj
Z
Q
1
w(x)dx .1 ;
holds when taken uniformly over all cubes Q in Rn. The ensuing thread of investigation culminated in the theorem of Coifman and Fe¤erman [CoFe] that characterized those nonnegative weights w on Rn for which all of the ‘nicest’of theL2(Rn)bounded singular integrals T above are bounded on weighted spaces L2(Rn;w), and does so in terms of the aboveA2 condition of Muckenhoupt.
Attention then turned to the corresponding two weight inequalities for singular integrals, which turned out to be considerably more complicated. For example, Cotlar and Sadosky gave a beautiful function theoretic characterization of the weight pairs ( ; !) for which H is bounded from L2(R; ) to L2(R;!), namely a two-weight extension of the Helson-Szegö theorem, which illuminated a deep connection between two quite di¤erent function theoretic conditions, but failed to shed much light on when either of them held1. On the other hand, the two weight inequality for positive fractional integrals, Poisson integrals and maximal functions were characterized using testing conditions by one of us in [Saw] (see also [Hyt2] for the Poisson inequality with ‘holes’) and [Saw1], but relying in a very strong way on the positivity of the kernel, something the Hilbert kernel lacks. In a groundbreaking series of papers including [NTV1],[NTV2]
and [NTV4], Nazarov, Treil and Volberg used weighted Haar decompositions with random grids, introduced their ‘pivotal’condition, and proved that the Hilbert transform is bounded fromL2(R; )toL2(R;!)if and only if a variant of theA2condition ‘on steroids’held, and the norm inequality and its dual held when tested locally over indicators of cubes - butonlyunder the side assumption that their pivotal conditions held.
The last dozen years have seen a resurgence in the investigation of two weight inequalities for singular integrals, beginning with the aforementioned work of NTV, and due in part to applications of the two weight T1 theorem in operator theory, such as in [LaSaShUrWi], where embedding measures are characterized for model spacesK , where is an inner function on the disk, and where norms of composition operators are characterized that mapK into Hardy and Bergman spaces. AT1theorem could also have implications for a number of problems that are higher dimensional analogues of those connected to the Hilbert transform (rank one perturbations [Vol], [NiTr]); products of two densely de…ned Toeplitz operators; subspaces of the Hardy space invariant under the inverse shift operator [Vol], [NaVo]; orthogonal polynomials [VoYu], [PeVoYu]; and quasiconformal theory [IwMa], [LaSaUr], [AsGo], [AsZi]), and we refer the reader to [SaShUr10] for more detail on these applications.
Following the groundbreaking work of Nazarov, Treil and Volberg, two of us, Sawyer and Uriarte-Tuero, together with Lacey in [LaSaUr2], showed that the pivotal conditions were not necessary in general, and introduced instead a necessary ‘energy’condition as a substitute, along with a hybrid merging of these two conditions that was shown to be su¢ cient for use as a side condition. The resurgence was then capped along the way with a resolution - involving the work of Nazarov, Treil and Volberg in [NTV4], the authors and M. Lacey in the two part paper [LaSaShUr3], [Lac] and T. Hytönen in [Hyt2] - of the two weight Hilbert transform conjecture of Nazarov, Treil and Volberg ([Vol]):
Theorem 1. The Hilbert transform is bounded fromL2(R; )toL2(R;!), i.e.
(1) kH(f )kL2(R;!).kfkL2(R; ); f 2L2(R; ); if and only if the two weightA2 condition with holes holds,
jQj jQj
1 jQj
Z
RnQ
s2Qd!(x)
!
+ 1
jQj Z
RnQ
s2Qd (x)
!jQj!
jQj .1 ; uniformly over all cubesQ, and the two testing conditions hold,
k1QH(1Q )kL2(R;!) . k1QkL2(R; )= q
jQj ; k1QH (1Q!)kL2(R; ) . k1QkL2(R;!)=
q jQj! ; uniformly over all cubesQ.
1However, the testing conditions in Theorem 1 are sub ject to the same criticism due to the highly unstable nature of singular integrals acting on measures.
HereHf(x) =R
R f(y)
y xdyis the Hilbert transform on the real lineR, and and! are locally …nite positive Borel measures on R. The two weight A2 condition with holes is also a testing condition in disguise, in particular it follows from
kH(sQ )kL2(R;!).ksQkL2(R; ) ; tested over all ‘indicators with tails’sQ(x) = `(Q)+`(Q)jx c
Qj of intervalsQin R. Below we discuss the precise interpretation of the above inequalities involving the singular integralH.
At this juncture, attention naturally turned to the analogous two weight inequalities forhigher dimensional singular integrals, as well as -fractional singular integrals such as the Cauchy transform in the plane. A variety of results were obtained, e.g. [SaShUr8], [LaSaShUrWi], [LaWi] and [SaShUr9], in which aT1theorem was proved under certain side conditions that implied the energy conditions. However, in [SaShUr10], the authors have recently shown that the energy conditions are not in general necessary for elliptic singular integrals.
The aforementioned higher dimensional results require re…nements of the various one-dimensional con- ditions associated with the norm inequalities, namely the A2 conditions, the testing conditions, the weak boundedness property and energy conditions. The purpose of this paper is to prove in higher dimensions that the weak boundedness constant WBPT ( ; !) that is associated with an -fractional singular inte- gral T and a weight pair ( ; !) in Rn, is ‘good- ’ controlled by the usual testing conditions TT ( ; !), TT ( ; !)and two side conditions on weight pairs, namely the Muckenhoupt conditionsA2( ; !)and the energy conditions Estrong( ; !), Estrong; ( ; !): more precisely, for every 0 < < 12, we have the Good- Lemma:
WBPT ( ; !) C 1p
A2+TT +TT +Estrong+Estrong; +p4
NT :
The …rst instance of this type of conclusion appears in Lacey and Wick in [LaWi]) - see Remark 1 in Subsection 2.1 below.
Applications of the Good- Lemma are then given to obtain bothT1 and T btheorems for two weights.
We now turn to a description of the higher dimensional conditions appearing in the above display. As the Good- Lemma, along with its corollaries, hold in the more general setting of quasicubes, we describe them
…rst. But the reader interested only in cubes can safely ignore this largely cosmetic generalization (but crucial for our ‘measure on a curve’T1theorem in [SaShUr8]) by simply deleting the pre…x ‘quasi’wherever it appears.
1.1. Quasicubes. We begin by recalling the notion of quasicube used in [SaShUr9] - a special case of the classical notion used in quasiconformal theory.
De…nition 1. We say that a homeomorphism :Rn!Rn is a globally biLipschitz map if
(2) k kLip sup
x;y2Rn
k (x) (y)k kx yk <1; and 1 Lip<1.
Notation 1. We de…ne Pn to be the collection of half open, half closed cubes in Rn with sides parallel to the coordinate axes. A half open, half closed cubeQ inRn has the formQ=Q(c; `)
Yn k=1
ck 2`; ck+2` for some` >0 andc= (c1; :::; cn)2Rn. The cubeQ(c; `) is described as having centerc and sidelength`.
De…nition 2. Suppose that :Rn!Rn is a globally biLipschitz map.
(1) If E is a measurable subset of Rn, we de…ne E f (x) :x2Eg to be the image of E under the homeomorphism .
(a) In the special case thatE=Qis a cube inRn, we will refer to Qas a quasicube (or -quasicube if is not clear from the context).
(b) We de…ne the centerc Q=c( Q)of the quasicube Qto be the point cQ wherecQ=c(Q) is the center ofQ.
(c) We de…ne the side length`( Q)of the quasicube Q to be the sidelength`(Q)of the cube Q.
(d) For r > 0 we de…ne the ‘dilation’r Q of a quasicube Q to be rQ where rQ is the usual
‘dilation’ of a cube inRn that is concentric with Qand having side length r`(Q).
(2) If K is a collection of cubes inRn, we de…ne K f Q:Q2 Kg to be the collection of quasicubes QasQranges over K.
(3) If F is a grid of cubes inRn, we de…ne the inherited quasigrid structure on F by declaring that Qis a child of Q0 in F if Qis a child ofQ0 in the gridF.
Note that if Qis a quasicube, thenj Qjn1 jQjn1 =`(Q) =`( Q). For a quasicubeJ = Q, we will generally use the expressionjJjn1 in the various estimates arising in the proofs below, but will often use`(J) when de…ning collections of quasicubes. Moreover, there are constantsRbig and Rsmall such that we have the comparability containments
Q+ xQ Rbig QandRsmall Q Q+ xQ :
Example 1. Quasicubes can be wildly shaped, as illustrated by the standard example of a logarithmic spiral in the plane f"(z) = zjzj2"i =zei"ln(zz). Indeed, f" : C!C is a globally biLipschitz map with Lipschitz constant 1 +C"since f"1(w) =wjwj 2"i and
rf"= @f"
@z;@f"
@z = jzj2"i+i"jzj2"i; i"z
zjzj2"i :
On the other hand,f" behaves wildly at the origin since the image of the closed unit interval on the real line under f" is an in…nite logarithmic spiral.
1.2. Standard fractional singular integrals and the norm inequality. Let 0 < n. We de…ne a standard -fractional CZ kernelK (x; y)to be a real-valued function de…ned on Rn Rn satisfying the following fractional size and smoothness conditions of order1 + for some >0: Forx6=y,
jK (x; y)j CCZjx yj n and jrK (x; y)j CCZjx yj n 1; (3)
jrK (x; y) rK (x0; y)j CCZ jx x0j
jx yj jx yj n 1; jx x0j jx yj
1 2;
and the last inequality also holds for the adjoint kernel in whichxandy are interchanged. We note that a more general de…nition of kernel has only order of smoothness >0, rather than1 + , but the use of the Monotonicity and Energy Lemmas in arguments below, which involve …rst order Taylor approximations to the kernel functionsK (; y), requires order of smoothness more than1 to handle remainder terms.
1.2.1. De…ning the norm inequality. We now turn to a precise de…nition of the weighted norm inequality (4) kT fkL2(!) NT kfkL2( ); f 2L2( ):
For this we introduce a family n
;R
o
0< <R<1 of nonnegative functions on [0;1) so that the truncated kernelsK ;R(x; y) = ;R(jx yj)K (x; y)are bounded with compact support for …xedxor y. Then the truncated operators
T ; ;Rf(x) Z
Rn
K;R(x; y)f(y)d (y); x2Rn; are pointwise well-de…ned, and we will refer to the pair K ;n
;R
o
0< <R<1 as an -fractional singular integral operator, which we typically denote byT , suppressing the dependence on the truncations.
De…nition 3. We say that an -fractional singular integral operatorT = K ;n
;R
o
0< <R<1 satis…es the norm inequality (4) provided
T ; ;Rf
L2(!) NT kfkL2( ); f 2L2( );0< < R <1:
It turns out that, in the presence of Muckenhoupt conditions, the norm inequality (4) is essentially independent of the choice of truncations used, and we now explain this in some detail. Asmooth truncation of T has kernel ;R(jx yj)K (x; y) for a smooth function ;R compactly supported in ( ; R), 0 <
< R <1, and satisfying standard CZ estimates. A typical example of an -fractional transform is the -fractionalRiesz vector of operators
R ;n=fR`;n: 1 ` ng; 0 < n:
The Riesz transformsRn;` are convolution fractional singular integralsRn;` f K`n; f with odd kernel de…ned by
K`;n(w) w` jwjn+1
`(w)
jwjn ; w= w1; :::; wn :
However, in dealing with energy considerations, and in particular in the Monotonicity Lemma below where …rst order Taylor approximations are made on the truncated kernels, it is necessary to use thetangent line truncation of the Riesz transform R`;n whose kernel is de…ned to be `(w) ;R(jwj) where ;R is continuously di¤erentiable on an interval(0; S)with0< < R < S, and where ;R(r) =r nif r R, and has constant derivative on both(0; )and(R; S)where ;R(S) = 0. HereS is uniquely determined by Rand . Finally we set ;R(S) = 0as well, so that the kernel vanishes on the diagonal and common point masses do not ‘see’ each other. Note also that the tangent line extension of aC1; function on the line is againC1; with no increase in theC1; norm.
It was shown in the one dimensional case with no common point masses in [LaSaShUr3], that boundedness of the Hilbert transformH with one set of appropriate truncations together with theA2 condition without holes, is equivalent to boundedness ofH with any other set of appropriate truncations, and this was extended to R ;n and more general operators in higher dimensions, permitting common point masses as well. Thus we are free to use the tangent line truncations throughout the proofs of our results.
1.3. Quasicube testing conditions. The following ‘dual’ quasicube testing conditions are necessary for the boundedness ofT fromL2( )toL2(!),
T2T sup
Q2 Pn
1 jQj
Z
QjT (1Q )j2! <1; (TT )2 sup
Q2 Pn
1 jQj!
Z
Q
(T ) (1Q!)2 <1;
and where we interpret the right sides as holding uniformly over all tangent line truncations ofT . Equally necessary are the following ‘full’testing conditions where the integrations are taken over the entire spaceRn:
FT2T sup
Q2 Pn
1 jQj
Z
RnjT (1Q )j2! <1; (FTT )2 sup
Q2 Pn
1 jQj!
Z
Rn
(T ) (1Q!)2 <1;
1.4. Quasiweak boundedness and indicator/touching property. The quasiweak boundedness prop- erty forT with constantC is given by
Z
Q
T (1Q0 )d! WBPT q
jQj!jQ0j ; (5)
for all quasicubesQ; Q0 with 1 C
`(Q)
`(Q0) C;
and eitherQ 3Q0nQ0 orQ0 3QnQ;
and where we interpret the left side above as holding uniformly over all tangent line trucations ofT . This condition is used in ourT1theorem with an energy side condition in [SaShUr9], but will be removed in our T1theorem with an energy side condition obtained here as a corollary of the Good- Lemma.
We say that two quasicubesQandQ0 in Pn aretouching quasicubes if the intersection of their closures is nonempty and contained in the boundary of the larger quasicube. Finally, let IT =IT ( ; !) be the best constant in theindicator/touching inequality
jT (1Q;1Q0)j IT ( ; !)k1QkL2( )k1Q0kL2(!) ; (6)
for all touching quasicubesQ; Q02 Pn; with 1
C
`(Q)
`(Q0) C;
and eitherQ 3Q0nQ0 orQ0 3QnQ:
1.5. Poisson integrals and A2. Let be a locally …nite positive Borel measure on Rn, and supposeQis an -quasicube inRn. Recall thatjQj1n `(Q)for a quasicubeQ. The two -fractional Poisson integrals of on a quasicubeQare given by:
P (Q; ) Z
Rn
jQjn1 jQjn1 +jx xQj n+1
d (x);
P (Q; ) Z
Rn
0
B@ jQjn1 jQjn1 +jx xQj 2
1 CA
n
d (x);
where we emphasize thatjx xQjdenotes Euclidean distance betweenxandxQandjQjdenotes the Lebesgue measure of the quasicubeQ. We refer toP as the standard Poisson integral and toP as thereproducing Poisson integral.
We say that the pair K; K0 in Pn are neighbours if K and K0 live in a common dyadic grid and both K 3K0 nK0 and K0 3KnK, and we denote by Nn the set of pairs (K; K0) in Pn Pn that are neighbours. Let
Nn=f( K; K0) : (K; K0)2 Nng
be the corresponding collection of neighbour pairs of quasicubes. Let and!be locally …nite positive Borel measures onRn, and suppose0 < n. Then we de…ne the classicalo¤ set A2 constants by
(7) A2( ; !) sup
(Q;Q0)2 Nn
jQj jQj1 n
jQ0j!
jQj1 n:
Since the cubes in Pn are products of half open, half closed intervals[a; b), the neighbouring quasicubes (Q; Q0)2 Nnare disjoint, and any common point masses of and!do not simultaneously appear in each factor.
We now de…ne theone-tailed A2 constant usingP . The energy constantsEstrong introduced below will use the standard Poisson integralP .
De…nition 4. The one-tailed constants A2 andA2; for the weight pair ( ; !)are given by
A2 sup
Q2 PnP (Q;1Qc ) jQj!
jQj1 n <1; A2; sup
Q2 PnP (Q;1Qc!) jQj
jQj1 n <1:
Note that these de…nitions are the analogues of the corresponding conditions with ‘holes’introduced by Hytönen [Hyt2] in dimension n= 1 - the supports of the measures 1Qc and 1Q! in the de…nition of A2
are disjoint, and so the common point masses of and! do not appear simultaneously in each factor. Note also that, unlike in [SaShUr5], where common point masses were not permitted, we can no longer assert the equivalence ofA2 with holes taken overquasicubes withA2 with holes taken overcubes.
1.5.1. Punctured A2 conditions. The classical A2 characteristic supQ2 Qn jQj!
jQj1 n jQj
jQj1 n fails to be …nite when the measures and! have a common point mass - simply letQin thesupabove shrink to a common mass point. But there is a substitute that is quite similar in character that is motivated by the fact that for large quasicubesQ, thesupabove is problematic only if justone of the measures ismostly a point mass when restricted toQ.
Given an at most countable set P=fpkg1k=1 in Rn, a quasicubeQ2 Pn, and a locally …nite positive Borel measure , de…ne as in [SaShUr9],
(Q;P) jQj supf (pk) :pk 2Q\Pg; where the supremum is actually achieved since P
pk2Q\P (pk) <1 as is locally …nite. The quantity (Q;P) is simply the e measure of Q where e is the measure with its largest point mass from P in Q removed. Given a locally …nite measure pair ( ; !), let P( ;!)=fpkg1k=1 be the at most countable set of
common point masses of and!. Then the weighted norm inequality (4) typically implies …niteness of the followingpunctured Muckenhoupt conditions (see [SaShUr9]):
A2;punct( ; !) sup
Q2 Pn
! Q;P( ;!) jQj1 n
jQj jQj1 n; A2; ;punct( ; !) sup
Q2 Pn
jQj!
jQj1 n
Q;P( ;!) jQj1 n : Now we turn to the de…nition of a quasiHaar basis ofL2( ).
1.6. A weighted quasiHaar basis. We will use a construction of a quasiHaar basis inRn that is adapted to a measure (c.f. [NTV2] for the nonquasi case). Given a dyadic quasicubeQ2 D, whereDis a dyadic grid of cubes fromPn, let4Q denote orthogonal projection onto the …nite dimensional subspaceL2Q( )of L2( )that consists of linear combinations of the indicators of the childrenC(Q)ofQthat have -mean zero overQ:
L2Q( ) 8<
:f = X
Q02C(Q)
aQ01Q0 :aQ0 2R; Z
Q
f d = 0 9=
;:
Then we have the important telescoping property for dyadic quasicubes Q1 Q2 that arises from the martingale di¤erences associated with the projections4Q:
(8) 1Q0(x) 0
@ X
Q2[Q1;Q2]
4Qf(x) 1
A=1Q0(x) EQ0f EQ2f ; Q02C(Q1); f 2L2( ):
We will at times …nd it convenient to use a …xed orthonormal basis n hQ;ao
a2 n
of L2Q( ) where n f0;1gnn f1g is a convenient index set with1= (1;1; :::;1). Then n
hQ;ao
a2 nand Q2 D is an orthonormal basis for L2( ), with the understanding that we add the constant function 1if is a …nite measure. In particular we have
kfk2L2( )= X
Q2 D
4Qf 2
L2( )= X
Q2 D
X
a2 n
fb(Q)2; fb(Q)2 X
a2 n
D
f; hQ;aE 2
;
where the measure is suppressed in the notationfb. Indeed, this follows from (8) and Lebesgue’s di¤erentiation theorem for quasicubes. We also record the following useful estimate. IfI0 is any of the2n D-children of I, anda2 n, then
(9) jEI0hI;aj
q
EI0(hI;a)2 1 qjI0j
:
1.7. The strong quasienergy conditions. Given a dyadic quasicubeK2 D and a positive measure we de…ne the quasiHaar projectionPK P
J2 D:J K4J onK by PKf = X
J2 D:J K
X
a2 n
hf; hJ;ai hJ;a so that kPKfk2L2( )= X
J2 D:J K
X
a2 n
hf; hJ;ai 2;
and where a quasiHaar basisfhJ;aga2 nandJ2 D adapted to the measure was de…ned in the subsection on a weighted quasiHaar basis above.
Now we de…ne various notions for quasicubes which are inherited from the same notions for cubes. The main objective here is to use the familiar notation that one uses for cubes, but now extended to -quasicubes.
We have already introduced the notions of quasigrids D, and center, sidelength and dyadic associated to quasicubes Q 2 D, as well as quasiHaar functions, and we will continue to extend to quasicubes the additional familiar notions related to cubes as we come across them. We begin with the notion of deeply
embedded. Fix a quasigrid D. We say that a dyadic quasicube J is (r; ")-deeply embedded in a (not necessarily dyadic) quasicubeK, which we write as J br;"K, whenJ Kand both
`(J) 2 r`(K); (10)
qdist (J; @K) 1
2`(J)"`(K)1 ";
where we de…ne the quasidistance qdist (E; F) between two sets E and F to be the Euclidean distance dist 1E; 1F between the preimages 1E and 1F of E and F under the map , and where we recall that`(J) jJjn1. For the most part we will consider J br;" K when J andK belong to a common quasigrid D, but an exception is made when de…ning the strong energy constants below.
Recall that in dimensionn= 1, and for = 0, the energy condition constant was de…ned by E2 sup
I= _[Ir
1 jIj
X1 r=1
P (Ir;1I ) jIrj
2
P!Irx 2L2(!) ;
where I, Ir andJ are intervals in the real line. The extension to higher dimensions we use here is that of
‘strong quasienergy condition’de…ned in [SaShUr9] and recalled below.
We de…ne a quasicube K (not necessarily in D) to be an alternate D-quasicube if it is a union of2n D-quasicubesK0 with side length `(K0) = 12`(K)(such quasicubes were called shifted in [SaShUr5], but that terminology con‡icts with the more familiar notion of shifted quasigrid). Thus for any D-quasicube Lthere are exactly2n alternate D-quasicubes of twice the side length that containL, and one of them is of course the D-parent ofL. We denote the collection of alternate D-quasicubes byA D.
The extension of the energy conditions to higher dimensions in [SaShUr5] used the collection Mr;" deep(K) fmaximalJ br;"Kg
of maximal (r; ")-deeply embedded dyadic subquasicubes of a quasicube K (a subquasicube J of K is a dyadicsubquasicube ofKifJ 2 Dwhen Dis a dyadic quasigrid containingK). This collection of dyadic subquasicubes ofK is of course a pairwise disjoint decomposition ofK. We also de…ned there a re…nement and extension of the collection M(r;") deep(K) for certain K and each ` 1. For an alternate quasicube K2 A D, de…neM(r;") deep; D(K)to consist of themaximalr-deeply embedded D-dyadic subquasicubes J ofK. (In the special case thatKitself belongs to D, thenM(r;") deep; D(K) =M(r;") deep(K).) Then in [SaShUr5] for` 1we de…ned the re…nement
M`(r;") deep; D(K) J 2 M(r;") deep; D `K0 for some K0 2C D(K) :
J L for someL2 M(r;") deep(K) ;
where C D(K) is the obvious extension to alternate quasicubes of the set of D-dyadic children. Thus M`(r;") deep; D(K)is the union, over all quasichildrenK0 of K, of those quasicubes inM(r;") deep `K0 that happen to be contained in some L 2 M(r;") deep; D(K). We then de…ne the strong quasienergy condition as follows.
De…nition 5. Let0 < nand …x ‘goodness’parameters(r; "). Suppose and! are locally …nite positive Borel measures onRn. Then the strongquasienergy constantEstrongis de…ned by
Estrong 2 sup
I= _[Ir
1 jIj
X1 r=1
X
J2Mr;" deep(Ir)
P (J;1I ) jJj1n
!2
kP!Jxk2L2(!)
+ sup
D
sup
I2A D
sup
` 0
1 jIj
X
J2M`(r;") deep; D(I)
P (J;1I ) jJjn1
!2
kP!Jxk2L2(!) :
Similarly we have a dual version ofEstrongdenoted Estrong; , and both depend onrand"as well as onn and . An important point in this de…nition is that the quasicube I in the second line is permitted to lie outside the quasigrid D, but only as an alternate dyadic quasicubeI2 A D. In the setting of quasicubes we continue to use the linear functionxin the …nal factorkP!Jxk2L2(!)of each line, and not the pushforward ofx by . The reason of course is that this condition is used to capture the …rst order information in the Taylor expansion of a singular kernel.
2. The Good- Lemma
The basic new result of this paper is the following ‘Good- Lemma’ whose utility will become evident when we pursue its corollaries below. Setfraktur A2 to be the sum of the fourA2 conditions:
A2 =A2 +A2; +A2;punct+A2; ;punct:
Lemma 1 (The Good- Lemma). Suppose that T is a standard -fractional singular integral in Rn, and that and! are locally …nite positive Borel measures onRn. For every 2 0;12 , we have
WBPT ( ; !) (11)
C 1q
A2( ; !) + (TT +TT ) ( ; !) + Estrong+Estrong; ( ; !) +p4
NT ( ; !) : Thus the e¤ect of the Good- Lemma is to ‘good- replace’ the quasiweak boundedness property with just the usual testing conditions in the presence of the side conditions of Muckenhoupt and energy on the weight pair. However, in dimension n= 1a much stronger inequality can be proved (see e.g. [NTV4] and [LaSaUr2]):
WBPT C p
A2 +TT +TT :
2.1. Corollaries. Now we come to the corollaries of the Good- Lemma. We …rst remove the hypothesis of the quasiweak boundedness property from the conclusion of part (1) of Theorem 1 in [SaShUr9].
Remark 1. In[LaWi], Lacey and Wick have removed the weak boundedness property from theirT1theorem by using NTV surgery with two independent grids, one for each functionf andg inhT f; gi, in the course of their argument. The use of independent grids for each off andggreatly simpli…es the NTV surgery, but does not accommodate our control of functional energy by Muckenhoupt and energy conditions.
Theorem 2. Suppose0 < n, that T is a standard -fractional singular integral operator on Rn, and that! and are locally …nite positive Borel measures onRn. SetT f =T (f )for any smooth truncation of T . Let : Rn ! Rn be a globally biLipschitz map. Then the operator T is bounded from L2( ) to L2(!), i.e.
kT fkL2(!) NT kfkL2( ); uniformly in smooth truncations ofT , and moreover
NT C p
A2 +TT +TT +Estrong+Estrong; ;
provided that the two dualA2 conditions and the two dual punctured Muckenhoupt conditions all hold, and the two dual quasitesting conditions forT hold, and provided that the two dual strong quasienergy conditions hold uniformly over all dyadic quasigrids D Pn, i.e. Estrong+Estrong; <1, and where the goodness parameters r and " implicit in the de…nition of the collections M(r;") deep(K) and M`(r;") deep; D(K) appearing in the strong energy conditions, are …xed su¢ ciently large and small respectively depending only onn and .
Proof. Let T ;R be a tangent line approximation to T as introduced above. Then NT
;R < 1, indeed NT
;R Cn; ; ;Rp
A2 by an easy argument, and by part (1) of Theorem 1 in [SaShUr9] applied to the -fractional singular integralT ;Rwe have
NT
;R C p
A2 +TT
;R+TT
;R+Estrong+Estrong; +WBPT;R ;
withC independent of andR. We obtain from the Good- Lemma applied toT ; ;R in place ofT , WBPT ;R C 1p
A2 +TT
;R+TT
;R+Estrong+Estrong; +p4 NT
;R ; and then combining inequalities gives
NT;R C0 1p
A2 +TT;R+TT
;R+Estrong+Estrong; +p4
NT;R ;
with C0 independent of and R. Since NT
;R < 1, we can absorb the term C0p NT
;R on the right hand side above into the left hand side for >0 su¢ ciently small. Since T;R is an arbitrary tangent line approximation toT , the proof of Theorem 2 is complete.
The …rst case of the followingT1 theorem was proved in [SaShUr8], and the second case is a corollary of Theorem 2 above and Theorem 2 in [SaShUr9].
Theorem 3. Suppose0 < n, that T is a standard -fractional singular integral operator on Rn, and that! and are locally …nite positive Borel measures onRn. SetT f =T (f )for any smooth truncation of T . Let :Rn!Rn be a globally biLipschitz map. Then
NT p
A2 +TT +TT ; in the following two cases:
(1) whenT is a strongly elliptic standard -fractional singular integral operator onRn, and one of the weights or !is supported on a compact C1; curve in Rn,
(2) whenT =R is the vector of -fractional Riesz transforms, and both weights and!arek-energy dispersed where0 k n 1 satis…es
n k < < n; 6=n 1 if 1 k n 2 0 < n; 6= 1; n 1 if k=n 1 :
There is a further corollary that can be easily obtained, namely atwo weightaccretive globalT btheorem whenever a two weightT1theorem holds for strictly comparable weight pairs. We say that two weight pairs ( ; !) and (e;!)e are strictly comparable if e = h1 and !e = h2! where each hi is a function bounded between two positive constants. The simple proof of the following accretive globalT btheorem uses only the statement of a related T1theorem. We say that a complex-valued functionb isaccretive onRn if
0< cb Reb(x) jb(x)j Cb<1; x2Rn :
Theorem 4. Suppose0 < n, that T is a standard -fractional singular integral operator on Rn, and that! and are locally …nite positive Borel measures onRn for which we have the ‘T1 theorem’for strictly comparable weight pairs, i.e.
(12) NT (e;!)e
q
A2(e;e!) +TT (e;!) +e TT (e;!)e ;
whenever ( ; !) and(e;!)e are strictly comparable. Finally, let b andb be two accretive functions onRn. Then the best constant NT =NT ( ; !) in the two weight norm inequality
kT fkL2(!) NT kfkL2( ); taken uniformly over tangent line truncations of T , satis…es
(13) NT p
A2 +TbT +Tb ;T ; where the two dualb-testing conditions forT are given by
Z
QjT (1Qb)j2d! TbT jQj ; for all cubes Q;
Z
QjT!; (1Qb )j2d Tb ;T jQj! ; for all cubesQ;
and where we interpret the left sides above as holding uniformly over all tangent line truncations ofT . Note that Theorem 4 applies in particular to both cases (1) and (2) of Theorem 3.
Proof. We …rst note that since the kernelK is real-valued, Z
QjT (1QReb)j2d! = Z
QjReT (1Qb)j2d!
Z
QjT (1Qb)j2d! TbT jQj ; Z
QjT!; (1QReb )j2d = Z
QjReT!; (1Qb )j2d Z
QjT!; (1Qb )j2d Tb ;T jQj! ; and if we now de…ne measures
e
! (Reb )! ande (Reb) ;
we see that the operatorT and the weight pair(e;!)e satisfy (12). But it follows thatTT (e;!)e TbT ( ; !) andTT (e;!)e Tb ;T ( ; !), and since the MuckenhouptA2 conditions are clearly comparable for strictly comparable weight pairs, we have the equivalence
NT (e;!)e q
A2( ; !) +TbT ( ; !) +Tb ;T ( ; !):
Finally, since0< c Reb;Reb C, we see thatNT (e;!)e NT ( ; !), and this completes the proof of (13).
Note that the presence of a (b; b )-variant of the weak boundedness property here would complicate matters, since in general,
Re Z
Q
T (1Q0b )b d!6= Z
Q
T (1Q0Reb ) Reb d!:
To remind the reader of the versatility of even aglobal T btheorem, we reproduce a proof of the boundedness of the Cauchy integral onC1; curves.
2.1.1. Boundedness of the Cauchy integral onC1; curves. Here we point out how the aboveT btheorem can apply to obtain the boundedness of the Cauchy integral onC1; curves in the plane (which can be obtained in many other easy ways as well, see e.g. [Ste, Section 4 of Chapter VII]). Recall that the problem reduces to boundedness onL2(R)of the singular integral operatorCA with kernel
KA(x; y) 1
x y+i(A(x) A(y));
where the curve has graphfx+iA(x) :x2Rg. Nowb(x) 1+iA0(x)is accretive and we have theb-testing condition
Z
IjCA(1Ib) (x)j2dx TbHjIj; and its dual. Indeed, ifI= [ ; ], then
CA(1Ib) (x) =
Z 1 +iA0(y)
x y+i(A(x) A(y))dy
= log (x y+i(A(x) A(y)))j
= log x +i(A(x) A( )) x +i(A(x) A( )) ; gives
jCA(1Ib) (x)j2 lnx
x; x2I= [ ; ]; and it follows that
Z
IjCA(1Ib) (x)j2dx Z
I
lnx x
2
dx Z
0
ln x 2
dx= ( ) Z 1
0 jlnwj2dw=CjIj: Since the kernelKA isC1; , theT btheorem above applies withT =CAand =!=dxLebesgue measure, to show that CA is bounded onL2(R). Of course this proof just misses the case of Lipschitz curves since our two weightT btheorem does not apply to kernels that fail to beC1; .
3. Proof of the Good- Lemma
We will prove the Good- Lemma by …rst replacing the quasiweak boundedness constant on the left hand side of (11) with the indicator/touching constant introduced in (6) above. To control the indicator/touching constant, we will need to tweak the usual good/bad technology of NTV a bit in the following subsection.
3.1. Good/bad technology. First we recall the good/bad cube technology of Nazarov, Treil and Volberg [Vol] as in [SaShUr7], but with a small simpli…cation introduced in the real line by Hytönen in [Hyt2]. This simpli…cation does not impact the validity of the arguments in [SaShUr6], but will facilitate the use of NTV surgery in later subsections.
Following [Hyt2], we momentarily …x a large positive integer M 2 N, and consider the tiling ofRn by the family of cubes DM IM
2Zn having side length 2 M and given by IM I0M + 2 M where I0M = 0;2 M n. Adyadic grid Dbuilt on DM is de…ned to be a family of cubesDsatisfying:
(1) EachI2 Dhas side length2 `for some `2Zwith` M, andI is a union of2n(M `) cubes from the tilingDM,
(2) For ` M, the collection D`of cubes inDhaving side length2 ` forms a pairwise disjoint decom- position of the spaceRn,
(3) GivenI2 Di andJ 2 Dj withj i M, it is the case that eitherI\J =;or I J.
We now momentarily …x anegative integerN 2 N, and restrict the above grids to cubes of side length at most2 N:
DN I2 D:side length ofI is at most2 N .
We refer to such grids DN as a (truncated) dyadic grid D built on DM of size2 N. There are now two traditional means of constructing probability measures on collections of such dyadic grids.
Construction #1: Consider …rst the special case of dimensionn= 1. Then for any
=f igi2NM 2!NM f0;1gZNM;
whereZNM f`2Z:N ` Mg, de…ne the dyadic gridD built onDM of size2 N by D =
8<
:2 ` 0
@[0;1) +k+ X
i:`<i M
2 i+` i 1 A
9=
;
N ` M; k2Z
:
Place the uniform probability measure NM on the …nite index space !NM = f0;1gZNM, namely that which charges each 2!NM equally. This construction is then extended to Euclidean spaceRn by taking products in the usual way and using the product index space NM !NM n and the uniform product probability measure NM = NM ::: NM.
Construction #2: Momentarily …x a (truncated) dyadic gridDbuilt onDM of size 2 N. For any
= ( 1; :::; n)2 NM 2 MZn+:j ij<2 N ; whereZ+=N[ f0g, de…ne the dyadic gridD built on DM of size2 N by
D D+ :
Place the uniform probability measure NM on the …nite index set NM, namely that which charges each multiindex in NM equally.
The two probability spaces fD g 2 NM; NM and fD g 2 NM; NM are isomorphic since both collections fD g 2 NM andfD g 2 NM describe the setANM ofall(truncated) dyadic gridsD built onDM of size2 N, and since both measures NM and NM are the uniform measure on this space. Indeed, it su¢ ces to verify this in the case n = 1. The …rst construction may be thought of as beingparameterized by scales - each component i in =f igi2NM 2!NM amounting to a choice of the two possible tilings at level ithat respect the choice of tiling at the level below - and since any grid inANM is determined by a choice of scales , we see that fD g 2 NM =ANM. The second construction may be thought of as beingparameterized by translation - each 2 NM amounting to a choice of translation of the grid D …xed in construction #2 - and since any grid inANM is determined by any of the cubes at the top level, i.e. with side length2 N, we see that fD g 2 NM =ANM as well, since every cube at the top level inANM has the formQ+ for some 2 NM and Q2 Dat the top level inANM (i.e. every cube at the top level inANM is a union of small cubes inDM, and so must be a translate of some Q2 D by an amount 2 M times an element of Zn+). Note also that in all dimensions,# NM = # NM = 2n(M N). We will useE NM to denote expectation with respect to this common probability measure onANM.
The usual NTV probabilistic reduction to ‘good’cubes will be implemented below for each positive integer M and each negative integer N assuming that the functionsf andg are supported in a large cubeL with R
Lf d = 0 =R
Lgd!, and moreover assuming that N is su¢ ciently large compared to`(L)that the small probability estimates claimed below hold ( N > `(L) +rwill work whereris the goodness constant), and
…nally assuming that f and g are constant on each cube Q in the tilingDM. Recall that we can always reduce to the caseR
Lf d = 0 =R
Lgd!by simply subtracting o¤ averages and controlling the resulting error terms by the testing conditions (see e.g. [Vol]).
Notation 2. For purposes of notation and clarity, we often suppress all reference toM andN in our families of grids, and in the notations and for the parameter sets, and we will useP andE to denote probability and expectation, and instead proceed as if all grids considered are unrestricted. The careful reader can supply the modi…cations necessary to handle the assumptions made above on the gridsDand the functionsf andg regarding M andN. In fact, we will exploit the integersM andN explicitly in the subsubsections on NTV surgery below.
In the case of one independent family of grids, as is the case here, the main result is the following conditional probability estimate: for everyI2 Pn,
(14) P fD:Iis a bad cube inD jI2 Dg C2 "r:
Provided we obtain estimates independent ofM andN, this will be su¢ cient for our proof - this follows the procedure withtwo independent grids initiated by Hytönen for the Hilbert transform inequality in [Hyt2].
The key point of introducing the two di¤erent parameterizations above of the same probability space, is that construction #1 is well-adapted to the reduction to good cubes in asingle independent family of grids, as used in the proof of the main theorem in [SaShUr6], which is in turn needed below, while construction #2 facilitates the use of NTV surgery below when combined with the construction ofQ-goodgrids,to which we next turn.
3.1.1. Q-good quasicubes and Q-good quasigrids. We …rst introduce these notions for usual cubes, and later pass to quasicubes. LetQ2 Pn be an arbitrary cube inRn with sides parallel to the coordinate axes. For technical reasons associated to our application below, we also want to consider the ‘siblings’ofQ, i.e. the
‘triadic children’of3Q.
De…nition 6. We say that a cubeI2 Pn isQ-goodif either`(I)>2 `(Q), or for every siblingQ0 of Q, we have
dist (I; @Q0) 1
2`(I)"`(Q0)1 "
when`(I) 2 `(Q). We sayI2 Pn isQ-badif I isnot Q-good.
Note that for a …xed cubeQ2 Pn, we donothave a conditional probability estimateP fD:I2 DandI isQ-badg C2 "r since the property of a cubeI beingQ-bad is independent of which gridsDit belongs to. To rectify
this complication we will introduce below asecond independent family of grids - but this second family will also be used to simultaneously Haar-decompose bothf 2L2( )andg2L2(!)2.
We next wish to capture the idea of a gridD being ‘Q-good’with respect to this …xed cubeQ, and the idea will be to require thatQisI-good for all su¢ ciently larger cubesI in the gridD. Here wewill obtain a ‘goodness’estimate in Lemma 2 below.
De…nition 7. Let r and" be goodness constants as in [SaShUr7]. For Q2 Pn we declare a grid Dto be Q-goodif for every siblingQ0 ofQand for everyI2 Dwith`(I) 2r`(Q), the following holds: the distance from the cubeQ0 to the boundary of the cubeI satis…es the ‘deeply embedded’ inequality,
dist (Q0; @I) 1
2`(Q0)"`(I)1 ": We say the gridDisQ-bad if it is notQ-good.
2Traditionally, two independent grids are applied tofandgseparately, something weavoidsince the treatment of functional energy in the arguments of [SaShUr9], [SaShUr6] (which we use here) relies on using acommongrid forf andg.
Note that Q is …xed in this de…nition and it is easy to see, using the translation parameterization in construction #2 above, that the collection of gridsDthat areQ-bad occur with small probability. Indeed, ifI Qhas side length at least2rtimes that ofQ, then the translates ofIsatisfyQbrI with probability near1.
Lemma 2. Fix a cube Q2 Pn. Then P fD:DisQ-badg C2 "r.
The following is our tweaking of the good/bad technology of NTV [Vol]. Fix a cubeQ2 Pn and let D be randomly selected. De…ne linear operators (depending on the gridD),
PQ;goodf
P
I2D: Iisr-go o d inD4If if DisQ-good
0 if DisQ-bad ;
PQ;badf f PQ;goodf ; and likewise forP!Q;goodg andP!Q;badg.
Proposition 1. Fix a cube Q2 Pn. Then we have the estimates E PQ;badf
L2( ) C2 "r2 kfkL2( ); E P!Q;badg
L2(!) C2 "r2 kgkL2(!): Proof. We have from (14) and Lemma 2 that
E kPbadfk2L2( )=E 1fDisQ-go o dg X
I2Dis bad
k4Ifk2L2( )
!
+E 1fDisQ-badgX
I2D
k4Ifk2L2( )
!
C2 "rX
I2D
k4Ifk2L2( )+E 1fDisQ-badg
X
I2D
k4Ifk2L2( ).C2 "rkfk2L2( ):
From this we conclude that there is an absolute choice ofrdepending on0< " <1 so that the following holds. LetT : L2( )!L2(!)be a bounded linear operator, and letQ2 Pnbe a …xed cube. We then have (15) kTkL2( )!L2(!) 2 sup
kfkL2 ( )=1
sup
kgkL2 (!)=1E j TPQ;goodf;P!Q;goodg !j: Indeed, we can choosef 2L2( )of norm one, andg2L2(!)of norm one so that
kTkL2( )!L2(!)=hT f; gi!
E j TPQ;goodf;P!Q;goodg
!j+E j TPQ;badf;P!Q;goodg
!j +E j TPQ;goodf;P!Q;badg
!j+E j TPQ;badf;P!Q;badg
!j E j TPQ;goodf;P!Q;goodg
!j+ 3C 2 r"16kTkL2( )!L2(!) ; And this proves (15) forrsu¢ ciently large depending on" >0.
Clearly, all of this extends automatically to the quasiworld.
Implication: Given a quasicube Q2 Pn,it su¢ ces to consider only Q-good quasigrids and Q-good quasicubes in these quasigrids, and to prove an estimate forkT kL2( )!L2(!) that is independent of these assumptions.
3.2. Control of the indicator/touching property. Recall the indicator/touching constantIT de…ned in (6) above. Here we will prove that
(16) IT C 1p
A2 +TT +TT +Estrong+Estrong; +p4
NT ;
from which it easily follows that we have the same inequality for the weak boundedness property constant WBPT de…ned in (5) above,
(17) WBPT C 1p
A2 +TT +TT +Estrong+Estrong; +p4
NT :
