A REPRISE OF THE NTV CONJECTURE FOR THE HILBERT TRANSFORM
ERIC T. SAWYER
Abstract. We give a somewhat di¤erent organization of the proof of the NTV conjecture for the Hilbert
transform that was proved by T. Hytönen, M. Lacey, E.T. Sawyer, C.-Y. Shen and I. Uriartre-Tuero in
[LaSaShUr3], [Lac] and [Hyt2], building on previous work of F. Nazarov, S. Treil and A. Volberg in [NTV3].
After modifying the decomposition of the main bilinear form, the proof alters the bottom-up corona
construction, the size functional, the straddling lemmas, and the use of recursion of admissible collections
of pairs of intervals, from M. Lacey [Lac]. However, the essence of control of the stopping form remains
as in the fundamental work of Lacey. The alternate approach of Sawyer and Wick [SaWi] to controlling
functional energy is also described, which di¤ers from that previously in the literature, [LaSaShUr3] and
[Hyt2].
Contents
1. Introduction
1
2. Reprise of the proof of the N T V conjecture
2
2.1. First steps and outline of the decomposition of forms
3
2.2. Control of the disjoint and comparable forms by testing and o¤set Muckenhoupt characteristics 4
2.3. Control of the neighbour form by the o¤set Muckenhoupt characteristic
4
3. Reduction of the home form by coronas
6
3.1. Control of the far form by way of functional energy
8
4. Reduction of the diagonal form by the NTV reach
12
4.1. Control of the paraproduct form by the testing characteristic
12
4.2. Control of the stopping form by the Poisson-Energy characteristic
13
References
20
1. Introduction
F. Nazarov, S. Treil and A. Volberg formulated the two weight question for the Hilbert transform [Vol],
that in turn led to the famous NTV conjecture:
Conjecture 1. [Vol]Given two positive locally …nite Borel measures and ! on the real line R, the Hilbert
R
transform Hf (x) = pv R fy (y)x dy is bounded from L2 ( ) to L2 (!), i.e. the operator norm
(1.1)
NH ( ; !)
sup
f 2L2 (R;
1
kf
k
)
L2 (
)
kH (f )kL2 (!) ;
is …nite uniformly over appropriate truncations of H, if the two tailed Muckenhoupt characteristic,
Z
Z
1
1
2
2
A2 ( ; !) sup
sI (x) d! (x)
sI (y) d (y) ;
jIj I
jIj I
I2I
is …nite, where sI (x) =
(1.2)
`(I)
`(I)+jx cI j ,
and if the testing characteristic
TH ( ; !)
I2I
is …nite as well as its dual TH (!; ).
1
kH1I kL2 (!) ;
jIj
sup p
1
2
ERIC T. SAW YER
In a groundbreaking series of papers including [NTV1],[NTV2] and [NTV3], Nazarov, Treil and Volberg used weighted Haar decompositions with random grids of (r; ") good grids, introduced their ‘pivotal
condition’, and proved the above conjecture under the side assumption that this pivotal condition held. Subsequently in [LaSaUr2], it was shown that the pivotal condition was not necessary for the norm inequality
to hold in general, a necessary ‘energy condition’was introduced as a substitute, and a hybrid merging of
these two conditions was shown to be su¢ cient for use as a side condition. Eventually, these three authors
with C.-Y. Shen established the NTV conjecture in the absence of common point masses in the measures
and ! in a two part paper; M. Lacey, E. T. Sawyer, C.-Y. Shen and I. Uriarte-Tuero [LaSaShUr3] and M.
Lacey [Lac].
The …nal assumption of no common point masses was removed shortly after by T. Hytönen [Hyt2] who
replaced the two tailed Muckenhoupt characteristic with a variant involving holes,
Ahole
( ; !)
2
P (I; )
sup P I; 1RnI !
I2I
Z
` (I)
R
(` (I) + jy
jIj
jIj
2d
cI j)
+
jIj!
jIj
(y) ;
cI
P I; 1RnI
;
centre of I;
and who also simpli…ed some aspects of the proof. The reason for introducing the holes is that if
share a common point mass, then the classical Muckenhoupt characteristic
A2 ( ; !)
sup
I2I
and !
jIj jIj!
;
jIj jIj
fails to be …nite - simply let I in the sup above shrink to a common point mass. The holed Muckenhoupt
condition trivially implies the more elementary o set Muckenhoupt condition
Z
Z
1
1
Ao2 set ( ; !)
sup
d!
d
;
jQ0 j Q0
jQj Q
(Q;Q0 )2N
where N is the set of neighbouring pairs (Q; Q0 ) of intervals of comparable side length, i.e. (Q; Q0 ) 2 D for
some dyadic grid D where 2 r ` (Q0 ) ` (Q) 2r ` (Q0 ), Q \ Q0 = ;, and Q \ Q0 6= ;. Here r is the goodness
parameter in [NTV3] and [LaSaShUr3].
The purpose of this paper is to give a modi…ed proof of the above results for the Hilbert transform,
and to highlight where the various characteristics enter into the proof. Our proof is somewhat simpli…ed
in that it avoids the recursion of admissible collections of pairs of intervals that was used in Part II [Lac],
and the recent approach to controlling functional energy due to Sawyer and Wick [SaWi], is discussed in the
setting p = 2. However, apart from some reorganization using nonlinear bounds, the essence of control of the
stopping form remains as in the fundamental work of M. Lacey [Lac], and the control of functional energy
is similar in spirit to that in [LaSaShUr3], and can in fact be imported as a black box from [LaSaShUr3] if
one wishes.
Theorem 2. Let and ! be positive locally …nite Borel measures on R. The Hilbert transform H is bounded
from L2 ( ) to L2 (!) if and only if the two tailed Muckenhoupt characteristic with holes, and both of the
testing characteristics, are …nite. More precisely, there are universal positive constants c; C such that for all
pairs ( ; !) of positive locally …nite Borel measures,
p
Ahole
( ; !) + TH ( ; !) + TH (!; )
2
c
C:
NH ( ; !)
Acknowledgement 3. We thank Brett Wick for fruitful discussions on both the far and stopping forms.
2. Reprise of the proof of the N T V conjecture
We assume the Haar supports of all functions f 2 L2 ( ) and g 2 L2 (!) are contained, along with their
"
1 "
two children, in the (r; ")-good grid Dgood , where J 2 Dgood if dist (J; @K) 21 ` (J) ` (K)
for all K J
with ` (K) 2r ` (J). See [NTV3] and [LaSaShUr3] for the reduction of the norm inequality to good functions
with goodness parameters r 2 N and " > 0.
A REPRISE OF THE NTV CONJECTURE FOR THE HILBERT TRANSFORM
3
We will make use of the following bound for sums of Haar coe¢ cients, which is proved e.g. in [LaSaUr2]
and [LaSaShUr3]. For any interval J with center cJ , and any …nite measure , de…ne the Poisson integral,
Z
` (J)
P (J; )
2 d (y) :
cJ j)
R (` (J) + jy
Lemma 4 (Monotonicity and Energy Lemma). Suppose that J
H 1F nI 0 ; 4!
Jg
*
H 1F nI 0 ;
X
aJ 0 h!
J0
J0 J
2
!
+
.
!2
P J; 1F nI 0
` (J)
I0
2
2
. P J; 1F nI 0
!
jJj!
2I 0
F are dyadic intervals. Then
2
2
!
k4!
J xkL2 (!) k4J gkL2 (!) ;
X
2
aJ 0 h!
J0
J0 J
:
L2 (!)
We will also need the following critical Poisson Decay Lemma from [Vol].
Lemma 5 (Poisson Decay Lemma). Suppose J
I
K are dyadic intervals and that d (J; @I) >
"
1 "
2` (J) ` (I)
for some 0 < " < 12 . Then for a positive Borel measure we have
P(J; 1KnI ) .
(2.1)
` (J)
` (I)
1 2"
P(I; 1KnI ):
2.1. First steps and outline of the decomposition of forms. Now we can begin the proof of Theorem
2. Following [NTV3] and [LaSaShUr3], we expand the Hilbert transform bilinear form hH f; gi! in terms of
the Haar decompositions of f and g,
X
hH f; gi! =
hH 4I f; 4!
J gi! ;
I;J2D
and then assuming the Haar supports of f and g lie in an (r; ") good grid Dgood (see [NTV3] for the origin
of goodness, and also [LaSaShUr3] for its use here), we decompose the double sum above as follows,
9
8
>
>
>
>
=
< X
X
X
X
X
+
+
hH 4I f; 4!
+
+
hH f; gi! =
J gi!
>
>
>
>
I;J2D
I;J2D
;
:I;J2D I;J2D I;J2D
J
I
I
J
I\J=;
J I and `(J) 2
`(I)
I J and `(I) 2
`(J)
Bbelow (f; g) + Babove (f; g) + Bdisjoint (f; g) + Bcomparable (f; g) + Bcomparable (f; g) ;
where 2 N is …xed below, larger than the goodness parameter r, and J
I means that J
I and
` (J)
2 ` (I). The …rst two forms are symmetric, and using a CZ-energy corona decomposition with
parameter > 1, we subsequently decompose the below form Bbelow (f; g) into another four forms,
Bbelow (f; g) = Bneigh (f; g) + Bfar (f; g) + Bpara (f; g) + Bstop (f; g) ;
in which there is control of both averages of f and a Poisson-Energy characteristic ( ; !) in each corona.
Altogether, we will eventually have eleven forms in our decomposition of the inner product hH f; gi! . It
turns out that all of the forms, save for the far, paraproduct, and stopping forms, are controlled directly by
the parameter and the testing and holed Muckenhoupt characteristics. Control of the far, paraproduct,
and stopping forms will require …niteness of the energy characteristics introduced in [LaSaUr2],
!2
P Ir ; 1InIr
2 jIr j!
2
E (Ir ; !)
and its dual E2 (!; ) ;
(2.2)
E2 ( ; !)
sup
` (Ir )
jIj
S1
r=1 Ir
I
where the supremum is taken over all pairwise disjoint subdecompositions of an interval I into dyadic
subintervals Ir 2 D [I]. The holed Muckenhoupt characteristic is needed only to control the far form, while
the smaller o¤set Muckenhoupt characteristic su¢ ces elsewhere.
It may help to make a couple of comments on the broad structure of the proof. The testing characteristics are used to control forms in which the measures
and ! ‘see each other’ in inner products
hH 1K 4I f; 4!
K (such as comparable and paraproduct forms), while the holed and o¤set
J gi! with J
Muckenhoupt characteristics are used to control forms in which the measures and ! ‘do not see each other’
in inner products hH 1K 4I f; 4!
J gi! with K \ J = ; (such as disjoint, neighbour, far and stopping forms).
4
ERIC T. SAW YER
In some of these cases, there arise parts of the form in which the coronas associated to the measures are
far apart, and it is then necessary to derive additional geometric decay in the distance between coronas,
much as in the well-known Colar-Stein Lemma.
A main tool for this is the Poisson Decay Lemma, which extracts geometric decay from goodness of
intervals in the Haar supports of the functions, as pioneered in [NTV3]. This decay is used in controlling
the disjoint, outer and stopping forms. Additional sources of needed geometric decay arise from the coronas
themselves as follows. There are three coronas at play in the proof:
(1) a top-down Calderón-Zygmund corona that derives geometric decay from a -Carleson condition,
that holds automatically from the de…nition of the stopping times,
(2) a top-down Poisson-Energy corona that also derives geometric decay from a -Carleson condition,
but that is no longer automatic from the de…nition, and instead requires the energy condition, as
pioneered in [LaSaUr2],
(3) and a bottom-up Energy corona that derives geometric decay from an ‘!-energy Carleson condition’,
that again holds automatically from the de…nition of the stopping times, as pioneered in [Lac].
Since the …rst two corona constructions both yield a -Carleson condition, it is convenient to bundle them
together in a single CZ-PE corona construction, see (3.1) for the associated stopping times. This corona
construction has two additional important properties, control of averages of f that is crucial for controlling
the paraproduct form, and control of a Poisson-Energy characteristic that is crucial for controlling the
stopping form. The -Carleson condition is used in the far, paraproduct and stopping forms while the
‘!-energy Carleson condition’is used in the stopping form.
2.2. Control of the disjoint and comparable forms by testing and o¤set Muckenhoupt characteristics. The disjoint form Bdisjoint (f; g), and the comparable forms Bcomparable (f; g) and Bcomparable (f; g),
are controlled in [SaShUr12, Section 4]1 using only the testing characteristics, the o¤set Muckenhoupt characteristic, and the following weak boundedness characteristic,
Z
(H 1I ) d! :
W ( ; !)
sup
I;J adjacent: 2
`(I) `(J) 2 `(I)
J
Similar results are in [NTV3], [LaSaShUr3], and [Hyt2]. Note that it follows from [Hyt2, Lemma 2.4] that
W ( ; !) Ao2 set ( ; !).
The forms Bbelow (f; g) and Babove (f; g) are symmetric, and so we need only bound the below form
Bbelow (f; g). To acccomplish this we decompose the below form Bbelow further into neighbour, far, paraproduct and stopping forms,
Bbelow = Bneigh + Bfar + Bpara + Bstop ;
which are detailed as we progress through the proof.
Let
Pbelow f(I; J) 2 Dgood
Dgood : J
Ig
be the set of pairs of good dyadic intervals (I; J) with J at least levels below and inside I. We begin by
splitting the below form into home and neighbour forms, where K denotes the dyadic sibling of K 2 D,
X
X
Bbelow (f; g) =
hH (1IJ 4I f ) ; 4!
hH (1 IJ 4I f ) ; 4!
J gi! +
J gi!
(I;J)2Pbelow
(I;J)2Pbelow
Bhome (f; g) + Bneigh (f; g) :
2.3. Control of the neighbour form by the o¤set Muckenhoupt characteristic. The neighbour form
is easily controlled by the Ao2 set condition using Lemma 4 and the fact that the intervals J are good, namely
we claim
q
jBneigh (f; g)j
To see this, momentarily …x an integer s
H
1
(IJ )
Ao2
C"
set
( ; !) kf kL2 (
)
kgkL2 (!) :
. We have
4I f ; 4!
Jg
!
=E
(IJ )
If
H
1
(IJ )
; 4!
Jg
!
;
1 the proof given there is for the T b theorem, and the reader can note that only the o¤set Muckenhoupt condition is used
A REPRISE OF THE NTV CONJECTURE FOR THE HILBERT TRANSFORM
and thus we can write
X
Bneigh (f; g) =
(2.3)
E
I;J2Dgood and J
If
(IJ )
H
1
!
Jg !
;
(IJ )
5
:
I
s
with s
Now we use the Poisson Decay Lemma. Assume that J
I. Let `(J)
`(I) = 2
estimate in Lemma 4 with J I to obtain
q
!
hH 1 (IJ ) ; !
gi
.
k
gk
jJj! P J; 1 (IJ )
J !
J
L2 (!)
q
.k !
jJj! 2 (1 2")s P I; 1 (IJ ) :
J gkL2 (!)
in the pivotal
where we have used (2.1).
(s)
Now we pigeonhole pairs (I; J) of intervals by requiring J 2 CD (I), i.e. J I and ` (J) = 2
we further separate the two cases where IJ = I , the right and left children of I. We have
X
hH 1 (IJ ) I f ; !
A(I; s)
J gi!
s
(s)
J2CD (I)
X
(1 2")s
2
+;
X
.
(1 2")s
2
+;
jEI
I f j P(I; 1I
jEI
I fj
X
)
J : 2s `(J)=`(I): J I
v
q
u
) jI j! u
t
P(I; 1I
X
(s)
J2CD (I
k
!
J gkL2 (!)
q
k
! gk2
J
L2 (!)
;
)
jJj!
where we have used Cauchy-Schwarz in the last line, and we also note that
X X
2
2
k !
kgkL2 (!) :
(2.4)
J gkL2 (!)
I2D J2C(s) (I)
D
Using
(2.5)
E
(IJ )
q
If
E
(IJ )
j
2
I fj
k
I f kL2 ( )
and the o¤set Muckenhoupt condition, we can thus estimate,
v
X
u X
(1 2")s
A(I; s) .
k
2
k I f kL2 ( ) u
t
+;
since P(I; 1I
).
(s)
q
.
Ao2 set
jI j
jIj
! gk2
J
L2 (!)
J2CD (I )
(1 2")s
( ; !)2
k
I f kL2 (
shows that
jI j
1
2
P(I; 1I
v
u
u
)t
X
(s)
J2CD (I)
k
1
2
j (IJ )j
jI j
1
2
;
P(I; 1I
! gk2
J
L2 (!)
p
p
q
q
jI j
jI j!
) jI j! .
. Ao2
jIj
set
;
( ; !):
An application of Cauchy-Schwarz to the sum over I using (2.4) then shows that
v
sX
q
X
uX X
2
(1 2")s
o
set
A(I; s) . A2
( ; !)2
k I f kL2 ( ) u
t
I2D
.
q
Ao2
q
.
Ao2
I2D
set
set
( ; !)2
( ; !)2
(1 2")s
(1 2")s
kf kL2 (
kf kL2 (
v
uX
u
)t
X
I2D J2C(s) (I)
)
D
kgkL2 (!) :
I2D J2C(s) (I)
D
k
! gk2
J
L2 (!)
q
) jI j!
k
! gk2
J
L2 (!)
` (I), and
6
ERIC T. SAW YER
Now we sum in s
to obtain,
X
jBneigh (f; g)j
.
H
I;J2D and J
I
1
q
X
(1 2")s
2
s=
= C
q
Ao2
set
Ao2
1
set
4I f ; 4!
Jg
(IJ )
( ; !) kf kL2 (
( ; !) kf kL2 (
)
)
!
=
1 X
X
A(I; s)
s= I2D
kgkL2 (!)
kgkL2 (!) :
3. Reduction of the home form by coronas
In order to control the home form, we must pigeonhole the pairs of intervals (I; J) 2 Pbelow into a
collection of pairwise disjoint corona boxes in which both averages of f are controlled, and a Poisson-Energy
characteristic is controlled. Then we split the home form into two forms according to this decomposition,
which we call the diagonal and far forms. But …rst we need to construct the corona decomposition.
1
Fix > 1 and a large dyadic interval T . De…ne a sequence of stopping times fFn gn=0 depending on T ,
and ! recursively as follows. Let F0 = fT g. Given Fn , de…ne Fn+1 to consist of the maximal intervals
I 0 2 Dgood for which there is I 2 Fn with I 0 I and
0
2
2 jI j!
either P I 0 ; 1InI 0
E (J; !)
> ;
jI 0 j
Z
Z
1
1
or 0
P
f d >4
P
f d ;
jI j I 0 D[I]
jIj I D[I]
(3.1)
where
2
E (J; !)
Z
1
jJj!
J
1
2
` (J)
Z
zd! (z) d! (x) ;
f ;
for F 2 F:
1
jJj!
x
2
J
P
S1
is the energy functional introduced in [LaSaUr2], and PD[I] = K2D: K I 4K . Set F
n=0 Fn , which
we refer to as the CZ-PE stopping times for T , and !, and which we note consists of good intervals, i.e.
F
Dgood . Denote the associated coronas by CF (F ) and the grandchildren at depth m 2 N of F in the
(m)
(1)
tree F by CF (F ), with CF (F ) abbreviated to CF (F ). We will consistly use calligraphic font C to denote
coronas, and fraktur font C to denote children. Finally, we de…ne
F
(F )
sup
G2F : G F
EG PD[
F G]
S
The point of introducing the corona
decomposition D [T ] = F 2F CF (F ) is that we obtain control of both
R
1
the averages EI PD[F ] f
PD[F ] f d of projections of f , and the Poisson-Energy functional
jIj
I
2
PEF (I)
good
in each good corona CF
(F )
EI jPF f j
EF jPF f j
P I; 1F nI
2
2
E (I; !)
jIj!
;
jIj
CF (F ) \ Dgood , i.e.
2
4 and PEF (I)
;
good
for all I 2 CF
(F ) and F 2 F:
In particular this inequality shows that the Poisson-Energy characteristic
(3.2)
PEF ( ; !)
F
sup PEF
F ( ; !) ; where PEF ( ; !)
F 2F
sup
PEF (I)
good
I2CF
(F )
of and ! with respect to the stopping times F, is dominated by the parameter chosen in (3.1). For
future reference we note that two more re…nements of the characteristic will appear in connection with the
Stopping Child Lemma and the control of the stopping form below.
A REPRISE OF THE NTV CONJECTURE FOR THE HILBERT TRANSFORM
7
Consequences of the energy condition: If we assume the …niteness of the energy characteristic
(2.2) (which is often referred to as the energy condition), and if we take > 4E2 ( ; !) in (3.1), we
obtain a -Carleson, or -sparse, condition for the CZ-energy stopping times F,
R
X
X
jPF f j d
1
2
2
0
F0
min
; P F 0 ; 1F nF 0
E (F 0 ; !) jF 0 j!
jF j
EF jPF f j
0
0
F 2CF (F )
F 2CF (F )
1
E2 ( ; !)
jF j +
jF j
4
(3.3)
1
jF j ;
2
for all F 2 F ;
which can then be iterated to obtain geometric decay in generations,
X
jGj
C 2 m jF j ;
for all m 2 N and F 2 F :
(m)
G2CF
(3.4)
<
(F )
In addition we obtain the quasi-orthogonality inequality, see e.g. [LaSaShUr3] and [SaShUr7], both
of which apply here since EF (jPF f j) = EF (jf EF f j) 2EF (jf j),
Z
X
2
2
jf j d :
jF j F (F )
C
R
F 2F
The …niteness of the energy characteristic E2 ( ; !) will be needed to enforce (3.3), that is in turn
needed to control the far, paraproduct and stopping forms, at which point we can appeal to the
following simple inequality from [LaSaUr2]2 ,
(3.5)
E2 ( ; !)
C TH ( ; !) + Ao2
set
( ; !) ;
that controls E2 ( ; !) by the testing characteristic for the Hilbert transform and the o¤set Muckenhoupt characteristic. Unfortunately this simple inequality fails for most other Calderón-Zygmund
operators in place of the Hilbert transform, including Riesz transforms in higher dimensions, see [Saw]
and [SaShUr11], and this failure limits the current proof to essentially just the Hilbert transform
and similar operators on the real line.
Now we can pigeonhole the pairs of intervals arising in the sum de…ning the below form. Given the corona
decomposition of D according to the Calderón-Zygmund stopping times F constructed above, we de…ne the
analogous decomposition of Pbelow ,
[
Pbelow =
[CF (F ) CF (G)] \ Pbelow
F;G2F : G F
=
(
[
F 2F
Pdiag
[CF (F )
[
CF (F )] \ Pbelow
)
8
[<
Pfar :
:
[
[CF (F )
F;G2F : G$F
CF (G)] \ Pbelow
9
=
;
Then we consider the corresponding decomposition of the home form into diagonal and far forms,
X
X
Bhome (f; g) =
hH (1IJ 4I f ) ; 4!
hH (1IJ 4I f ) ; 4!
J gi! +
J gi!
(I;J)2Pdiag
(I;J)2Pfar
Bdiag (f; g) + Bfar (f; g) :
F;G
We next decompose the far form into corona pieces using Pfar
[CF (F )
CF (G)] \ Pbelow ,
2 In [LaSaUr2], this inequality is stated with a one-tailed Muckenhoupt characteristic A ( ; !) on the right hand side, but as
2
the reader easily observes, the display at the top of page 319 in the proof given in [LaSaUr2] holds for the smaller characteristic
Ao2 set ( ; !).
8
ERIC T. SAW YER
Bfar (f; g)
X
=
F;G2F : G$F
X
=
F;G2F : G$F
where
BF;G
far
X
(f; g)
F;G
(I;J)2Pfar
Now for m >
BF;m
far
*
0
H @
X
(m)
=
hH (1IJ
1
4I f A ;
X
J2CF (G): J
4I f ) ; 4!
J gi!
I
4!
Jg
+
!
X
=
BF;G
far (f; g)
F;G2F : G$F
hH (1IJ 4I f ) ; 4!
J gi! :
(F )
X
1IJ
I2CF (F )
F;G
(I;J)2Pfar
and F 2 F we de…ne
X
BF;G
(f; g)
far (f; g) =
G2CF
X
X
F 0 2CF (F ) G2C(m
1)
F
X
(m)
G2CF
X
(F 0 ) J2CF (G)
X
F;G
(F ) (I;J)2Pfar
0
*
hH (1IJ 4I f ) ; 4!
J gi!
X
H @
1I J
I2CF (F ): J
I
1
4I f A ; 4!
Jg
+
:
!
3.1. Control of the far form by way of functional energy. Here we will control the far form Bfar (f; g)
…rst by the local testing and functional energy characteristic. The far form is given by
0
1
*
+
1 X
X
X
X
X
H @
1IJ 4I f A ; 4!
Bfar (f; g) =
Jg
m=1 F 2F G2C(m) (F ) J2CF (G)
F
=
X
X
G2F J2CF (G)
*
0
H @
I2CF (F ): J
X
I2D: G$I and J
which we write with the dummy variable G replaced by F ,
0
*
X X
X
Bfar (f; g) =
H @
F 2F J2CF (F )
I2(F;T ] and J
I
I
!
1
1IJ 4I f A ; 4!
Jg
1I J
I
1
+
4I f A ; 4!
Jg
;
!
+
:
!
Given any collection H D of intervals, and a dyadic interval J, we de…ne the corresponding Haar projection
!
P!
H and its localization PH;J to J by
X
X
!
(3.6)
P!
4!
4!
H =
H and PH;J =
H :
H2H
H2H: H J
De…nition 6. Given any interval F 2 D, we de…ne the (r; ")-Whitney collection M(r;") deep (F ) of F to
be the set of dyadic subintervals W
F that are maximal with respect to the property that W r;" F . Let
D [W ] fJ 2 D : J W g.
Clearly the intervals in M(r;")
deep
(F ) form a pairwise disjoint decomposition of F .
De…nition 7. Let F ( ; !) be the smallest constant in the ‘functional energy’ inequality below, holding for
all h 2 L2 ( ) and all collections F D, and where x denotes the identity function on R:
2Z
X
X
2
P (W; h1F c )
P!
F ( ; !) khkL2 ( ) :
(3.7)
CF (F )\D[W ] x d!
` (W )
W
F 2F W 2M(r;")
deep (F )
There is a similar de…nition of the dual constant F (!; ). The Intertwining Proposition will control the
following Intertwining form,
E
X X D
BInter (f; g)
H (1IF 4I f ) ; P!
;
CF (F ) g
F 2F I: I%F
!
A REPRISE OF THE NTV CONJECTURE FOR THE HILBERT TRANSFORM
whose di¤erence from Bfar (f; g) is
Bfar (f; g)
BInter (f; g)
=
X
X
X
F 2F I2(F;T ] J2CF (F ) and J
X
X
X
F 2F I2(F;T ] J2CF (F )
=
X
X
F 2F I2(F;T ]
I
9
hH (1IJ 4I f ) ; 4!
J gi!
hH (1IF 4I f ) ; 4!
J gi!
X
J2CF (F )
`(J) `(F )
and J
hH (1IF 4I f ) ; 4!
J gi! :
I
This di¤erence form is easily controlled using the proofs of the disjoint and comparable forms above,
X X
X
jBfar (f; g) BInter (f; g)j
jhH (1IF 4I f ) ; 4!
J gi! j
F 2F I2(F;T ]
.
TH ( ; !) +
Ahole
2
( ; !) kf kL2 (
)
J2CF (F )
`(J) `(F )
and J
I
kgkL2 (!) :
De…nition 8. A collection F of dyadic intervals is -Carleson if
X
jF j
CF jSj ;
S 2 F:
F 2F : F
S
The constant CF is referred to as the Carleson norm of F.
We now show that the functional energy inequality (3.7), together with local interval testing, su¢ ces to
prove the following Intertwining Proposition, [SaShUr7] and [SaShUr12], when F is -Carleson.
Let F be any subset of D. For any J 2 D, we de…ne 0F J to be the smallest F 2 F that contains J. Then
for s 1, we recursively de…ne sF J to be the smallest F 2 F that strictly contains sF 1 J. This de…nition
s t
0 and J 2 D. In particular sF J = sF F where F = 0F J. In the
satis…es s+t
F J = F F J for all s; t
special case F = D we often suppress the subscript F and simply write s for sD . Finally, for F 2 F, we
write CF (F )
F 0 2 F : 1F F 0 = F for the collection of F-children of F .
Proposition 9 (The Intertwining Proposition). Suppose that F is -Carleson. Then
X
X
F 2F I: I%F
D
H (1IF 4I f ) ; P!
CF (F ) g
E
!
. (F ( ; !) + TH ( ; !)) kf kL2 (
Proof. We write the left hand side of the display above as
0
1
*
+
X X
X
X
hH (1IF 4I f ) ; gF i! =
H @
1IF 4I f A ; gF
F 2F I: I%F
F 2F
where
gF = P!
CF (F ) g =
X
J2CF (F )
I: I%F
!
X
4!
J g and fF
I: I%F
)
X
F 2F
kgkL2 (!) :
hH fF ; gF i! ;
1IF 4I f :
Note that gF is supported in F , and that fF is constant on F . The intervals I occurring in this sum are
linearly and consecutively ordered by inclusion, along with the intervals F 0 2 F that contain F . More
precisely, we can write
F F0 $ F1 $ F2 $ ::: $ Fn $ Fn+1 $ :::FN
m
where Fm = F F for all m 1. We can also write
F = F0 $ I1 $ I2 $ ::: $ Ik $ Ik+1 $ ::: $ IK = FN
where Ik =
such that
k
DF
for all k
N
1, and by convention we set I0 = F . There is a (unique) subsequence fkm gm=1
Fm = Ikm ;
Recall that
fF (x) =
1
X
k=1
1
1(Ik )F (x) 4Ik f (x) =
1
X
k=1
m
N:
1Ik nIk
(x)
1
1
X
`=k+1
4I` f (x) :
10
ERIC T. SAW YER
Assume now that km
k < km+1 . Using a telescoping sum, we compute that for
we have
1
X
jfF (x)j =
`=k+2
Now fF is constant on F and
N
X
jfF j
m=0
=
EFm+1 jf j
(EF jf j) 1F +
F
F
=
F
x 2 Ik+1 n Ik
Fm+1 n Fm ;
4I` f (x) = E
Ik+2 f
1Fm+1 nFm = (EF jf j) 1F +
X
E
F 0 2F : F
F 0 2F :
F
F0
F
F0
X
(F ) 1F +
F 0 2F :
F
F
1F c ;
F 2F
FF
0
) 1
(
FF
0
) 1
F
F 00 2F
X
jf j 1
(
X
hH fF ; gF i! =
0
E
m=0
m+1
F
F
jf j
1
m+1
Fn m
F
FF
0 nF 0
FF
0 nF 0
FF
0
1F c
(F 00 ) 1F 00 :
hH (1F fF ) ; gF i! +
F 2F
FF
N
X
for all F 2 F;
where
Now we write
X
FF
F0
X
(F ) 1F +
(F ) 1F +
EIK f . EFm+1 jf j :
X
F 2F
hH (1F c fF ) ; gF i!
I + II:
Then local interval testing and quasi-orthogonality, together with the fact that fF is a constant on F bounded
by F (F ), give
XZ
1F (x) H (1F fF ) (x) gF (x) d! (x)
jIj =
s
R
F 2F
XZ
F 2F
R
j
. Tloc
H ( ; !)
2
F (F ) 1F (x) H (1F ) (x)j d! (x)
s
XZ
F 2F
R
j
s
XZ
F 2F
2
R
jgF (x)j d! (x)
(F )j jF j kgkL2 (!) . Tloc
H ( ; !) kf kL2 (
2
F
)
kgkL2 (!) :
Now 1F c fF is supported outside F , and each J in the Haar support of gF = P!
CF (F ) g is either in Nr (F ) =
fJ 2 D [F ] : ` (J) 2 r ` (F )g, in which case the desired bound for term I is straightforward, or J is (r; ")deeply embedded in F , i.e. J r;" F , and so J r;" W for some W 2 M(r;") deep (F ). Thus with
[
CF
(F ) CF (F ) n Nr (F ) we have
XZ
jIIj =
H (1F c fF ) (x) P!
g (x) d! (x)
C [ (F )
F 2F
=
X
F 2F W 2M(r;")
v
uX
u
. t
F
R
X
deep (F )\CF (F )
F 2F W 2M(r;")
X
Z
deep (F )\CF (F )
v
uX
u
t
R
P!
H (1F c fF ) (x) P!
g (x) d! (x)
C [ (F )\D[W ]
C [ (F )\D[W ]
F
Z
R
F
2
P!
H (1F c fF ) (x) d! (x)
C [ (F )\D[W ]
F
F 2F W 2M(r;")
X
deep (F )\CF (F )
Z
R
2
P!
g (x) d! (x):
C [ (F )\D[W ]
F
A REPRISE OF THE NTV CONJECTURE FOR THE HILBERT TRANSFORM
11
The second factor is bounded by C kgkL2 (!) , and we use the second line in the Energy Lemma 4 on the …rst
factor to bound it by,
v
uX
2Z
X
2
u
P (W; 1F c fF )
t
(3.8)
P!
x d!
C
(F
)\D[W
]
F
` (W )
W
F 2F W 2M(r;")
F ( ; !) k kLp (
deep (F )\CF (F )
)
. F ( ; !) kf kL2 ( ) ;
using the de…nition of the functional energy characteristic F ( ; !), and the dyadic -maximal function
inequality k kLp ( ) . kM f kLp ( ) . kf kLp ( ) . This completes the proof of the Intertwining Proposition
9.
Thus we have the following control of the far form,
jBfar (f; g)j . F ( ; !) + TH ( ; !) + Ahole
( ; !) kf kL2 (
2
)
kgkL2 (!) :
Finally then, to control the quadratic functional energy characteristic F ( ; !), we can of course appeal to
the bound in [LaSaShUr3] as a black box, to conclude that,
2
jBfar (f; g)j . T`H;p ( ; !) + Ahole
( ; !)
2
kf kL2 (
)
kgkL2 (!) ;
thus completing our treatment of the far form.
Alternatively, we can use the following somewhat simpler argument introduced in [SaWi]. Consider the
re…ned functional energy inequality,
2Z
X
X
2
P (W; h1F c )
P!
F ( ; !) khkL2 ( ) ;
CF (F )\D[W ] x d!
` (W )
W
F 2F W 2M(r;")
deep (F )\CF (F )
in which the sum in W 2 M(r;") deep (F ) is restricted to the corona CF (F ). This smaller functional energy
also serves to control the Intertwining Proposition by an inspection of (3.8) above. Using
where
S
denotes that
n the sets M(r;")
o
[
deep
F 2F
M(r;")
deep
(F ) \ CF (F ) ;
(F ) \ CF (F ) are pairwise disjoint, together with the orthogonality
in L2 (!), this re…ned functional energy inequality
R
is equivalent to the boundedness from L ( ) to L (!) of the operator T h (x)
K (x; y) h (y) d (y) with
R
kernel,
Z X
1RnW (y)
!
(3.9)
K (x; y)
2 PCF (F )\D[W ] Z (x) ;
cW j)
R W 2 (` (W ) + jy
of the projections
P!
CF (F )\D[W ] x
W 2M(r;")
2
F 2F
deep (F )\CF (F )
2
where Z denotes the identity map Z (x) = x on R.
Now we enlarge the kernel K (x; y) by removing
the indicator 1RnW (y) from the right hand side and
P
!
Z
(x)
=
replacing the projection P!
J2CF (F )\D[W ] 4J Z (x) by the ‘absolute’projection,
CF (F )\D[W ]
s
X
2
j4!
P!
Z
(x)
CF (F )\D[W ]
J Z (x)j :
J2CF (F )\D[W ]
R
b (x; y) h (y) d (y), where
Then boundedness of the positive linear operator Tbh (x)
K
R
Z X
1
!
K (x; y)
2 PCF (F )\D[W ] Z (x) ;
cW j)
R W 2 (` (W ) + jy
is su¢ cient for the re…ned functional energy inequality.
Now the kernel K is
(1) essentially decreasing in the variable y away from x, i.e.
b (x; y1 )
K
b (x; y2 ) ;
CK
jy1
xj
jy2
xj ;
12
ERIC T. SAW YER
(2) roughly constant in the variable y on intervals away from x, i.e.
b (x; y1 )
K
b (x; y2 )
K
1
C
C;
y1 ; y2 2 I with x 2
= 2I;
(3) and reverse Hölder in the variable x on intervals away from y, i.e. for every 1 < r < 1, there is
Cr < 1 such that,
1
jIj
Z
I
1
r0
r0
b (x; y)
K
d! (x)
Cr
Z
1
jIj
I
b (x; y) d! (x) ;
K
y2
= 2I:
Properties (1) and (2) are easy consequences of the formula for the Poisson kernel, see [SaWi, (7.10) and
(7.11)] for details, and see [SaWi, implicit in the proof of Theorem 51] for property (3). Now as shown
in [SaWi, Section 11, the appendix], this reduces boundedness of Tb to two dual testing conditions, which
can then be controlled as in [SaWi, Subsections 7.3 and 7.4], and we leave these details to the reader (they
simplify considerably in the case p = 2).
4. Reduction of the diagonal form by the NTV reach
We …rst apply the clever ‘NTV reach’of [NTV3], which splits the diagonal form
Bdiag (f; g) =
X
(I;J)2Pdiag
hH (1IJ 4I f ) ; 4!
J gi! =
into a paraproduct and stopping form,
X
Bdiag (f; g) =
X
X
F 2F (I;J)2CF (F ) CF (F )
J
I
X
EIJ 4I f hH 1F ; 4!
J gi!
F 2F (I;J)2CF (F ) CF (F )
J
I
+
X
X
BF
para (f; g) +
F 2F
X
H 1F nIJ ; 4!
Jg
EIJ 4I f
F 2F (I;J)2CF (F ) CF (F )
J
I
X
EIJ 4I f hH 1IJ f; 4!
J gi! ;
!
BF
stop (f; g)
F 2F
Bpara (f; g) + Bstop (f; g) :
4.1. Control of the paraproduct form by the testing characteristic. We control the local paraproduct
form BF
para (f; g) by the testing condition for H. Indeed, from the telescoping identity for Haar projections,
see e.g. [LaSaUr2], we have
BF
para (f; g) =
X
I2CF (F ) and J2CF (F )
J I and `(J) 2 r `(I)
=
X
J2CF (F )
=
X
J2CF (F )
hH 1F ; 4!
J gi!
EIJ 4I f hH 1F ; 4!
J gi!
8
<
X
:
I2CF : J I and `(J) 2
n
hH 1F ; 4!
J gi! EI \ (J) f
EF f
J
where I \ (J) is the smallest I 2 CF (F ) \
satis…es J r I.
f
o
=
(where
r `(I)
*
EIJ 4I f
H 1F ;
f
9
=
;
X
J2CF (F )
n
EI \ (J) f
J
o
EF f 4!
J g
+
;
!
is the Haar support of f ), that contains J and
A REPRISE OF THE NTV CONJECTURE FOR THE HILBERT TRANSFORM
13
Thus from Cauchy-Schwarz and the bound on the coe¢ cients J EI \ (J) f EF f given by j
J
we have
+
*
o
X n
!
F
(4.1)
EI \ (J) f EF f 4J g
Bpara (f; g) = H 1F ;
Jj
.
F
(F ),
J
J2CF (F )
X
k1F H 1F kL2 (!)
TH ( ; !)
F
J
J2CF (F )
(F )
q
jF j
!
.
4!
J g
F
X
(F ) k1F H 1F kL2 (!)
J2CF (F )
L2 (!)
P!
CF (F ) g
L2 (!)
4!
Jg
L2 (!)
:
P
F
Then summing in F we obtain the following bound for the paraproduct form Bpara (f; g)
F 2F Bpara (f; g),
q
X
X
TH ( ; !) F (F ) jF j
BF
jBpara (f; g)j
P!
para (f; g)
CF (F ) g
2
F 2F
TH ( ; !)
sX
F 2F
L (!)
F 2F
2
F
. TH ( ; !) kf kL2 (
)
(F ) jF j
kgkL2 (!) :
s
X
F 2F
P!
CF (F ) g
2
L2 (!)
by quasi-orthogonality in (3.4) and orthogonality in Haar projections P!
CF (F ) g.
4.2. Control of the stopping form by the Poisson-Energy characteristic. To control the stopping
form, it su¢ ces to assume the Haar supports of f and g are contained in a …xed large, but …nite subset z
of the grid D, and to uniformly control over these subsets, each local stopping form
X
BF
EIJ 4I f H 1F nIJ ; 4!
stop (f; g)
Jg ! :
I2CF and J2CF (F )
J I and `(J) 2 `(I)
Indeed, then quasi-orthogonality in (3.4) and orthogonality in Haar projections will be used to …nish control
of the stopping form. In fact we will prove
BF
stop (f; g)
(4.2)
CPEF ( ; !) kf kL2 (
)
kgkL2 (!) ;
for all F 2 F;
where PEF ( ; !) is de…ned in (3.2). The key technical estimate needed is the following Stopping Child
Lemma, which is a reformulation of the ‘straddling’lemmas in M. Lacey [Lac, Lemmas 3.19 and 3.16].
4.2.1. The Stopping Child Lemma. Let F be a collection of good stopping times. Suppose F 2 F is …xed for
the moment and let A CF (F ) be a collection of good stopping times with top interval F . For A 2 A we
de…ne the ‘stopping child’bilinear form,
X
X
BA
EIJ 4I f H 1F nS ; 4!
A (f; g)
Jg !
S2CA (A) (I;J)2(S;A] D[S]
J
I
=
X
X
S2CA (A) J2D[S]: J
f [S]
H 'SJ ; 4!
Jg
!
;
where (I; J]
fK 2 D : J K $ Ig denotes the tower in the grid D with endpoints I (not included)
and J (included), and f [S] is the smallest interval in the Haar support f of f that contains S, and
P
'SJ
EIJ 4I f 1A0 nS . The presence of the indicator 1F nS suggests the name ‘stopping
I2(S;A]: J
f [S]
child’bilinear form. We also de…ne a re…ned Poisson-Energy characteristic by,
s
X
1 P K; 1F nS
2
!
p
(4.3)
PEA;trip
;
;
!
sup
sup
k4!
g
A
J xkL2 (!) ;
` (K)
jKj
S2CA (A) K2W F;trip (S)
!
good
where
!
g
2
fJ 2 D : 4!
J g 6= 0g is the Haar support of g in L (!), and
h
i
F;trip
Wgood
(S)
fSg [ Mtrip
(S)
\ CF (F ) ;
good
J2
g
\D[K]
14
ERIC T. SAW YER
where Mtrip
good (S) is the collection of maximal good subintervals I of S whose triples are contained in S. This
characteristic depends on A, A and !
and !, and is a variation on the crucial size function
g , as well as on
introduced by M. Lacey in [Lac]. Set
PEtrip
A[F ]
!
g;
;!
PEtrip
A[F ]
!
g;
;!
sup PEA;trip
A
A2A[F ]
!
g;
;! ;
and note the trivial inequality
(4.4)
PEF
F ( ; !)
PEF ( ; !) ;
where PEF
F ( ; !) and PEF ( ; !) are de…ned in (3.2).
Lemma 10 (Stopping Child Lemma). (a reformulation of [Lac, Lemmas 3.19 and 3.26]) Let f 2 L2 ( ),
g 2 L2 (!) have good Haar supports, along with their children, let F be a collection of good stopping times,
and let A CF (F ) be a collection of good stopping times with top F . Set
A
I2(
f [S];A
Then for all A 2 A we have the nonlinear bound,
BA
A
CPEA;trip
A
(f; g)
for S 2 CA (A) ; A 2 A:
jEI f j ;
sup
]\Dgood
(S)
!
g;
s X
;!
S2CA (A)
2
jSj
A
(S) kgkL2 (!) :
Proof. By the telescoping property of martingale di¤erences,
i together with the bound A (S) on the averages
of PCA (A) f in the good intervals in the tower
f [S] ; A , and the goodness assumption on f , we have
X
'SJ =
(4.5)
I2(
f [S];A
]:
J
EIJ 4I f 1A0 nIJ .
I
From the Monotonicity Lemma and the fact that
0
!
g
where N (S)
fJ
S : ` (J)
S2CA (A) J2
A
!:
g
X
J
(S)
X
1
K A [ N (S)
f [S]
X
A
trip
straddle
(f; g) + BA
A
(S)
J2
! \N
g
J2
X
J
J K
(S): J
!
!
k4!
J xkL2 (!) k4J gkL2 (!)
!
g:
X
near
straddle
H 'SJ ; 4!
Jg
f [S]
P J; 'SJ
` (J)
X
S2CA (A)
(S) 1F nS :
` (S)g is the set of ‘ -nearby’dyadic intervals in S, we have
F;trip
K2Wgood
(S)
S2CA (A)
BA
A
F
K2Wgood
(S)
S2CA (A) J2D[S]: J
X
+
@
\ D [S]
X
BA
A (f; g)
X
2
[
A
f [S]
f [S]
P J; 1F nS
` (J)
P J; 1F nS
` (J)
!
k4!
J xkL2 (!) k4J gkL2 (!)
(f; g) :
Since
P J; 1F nS
` (J)
.
P K; 1F nS
` (K)
!
k4!
J xkL2 (!) k4J gkL2 (!)
;
A REPRISE OF THE NTV CONJECTURE FOR THE HILBERT TRANSFORM
F;trip
b!
for K 2 Wgood
(S) [ N (S), we have with P
S;K
BA
A
trip
straddle
X
X
(f; g)
A
A
X
(S)
F
(S)
K2Wgood
S2CA (A)
CPEA;trip
A
!
g;
CPEA;trip
A
!
g;
CPEA;trip
A
!
g;
;!
P K; 1F nS
` (K)
X
;!
A
(S)
s X
S2CA (A)
s
X
2
jSj
A
f [S]
X
J
J K
L2 (!)
q
jKj
F
(S)
K2Wgood
S2CA (A)
;!
X
F
(S)
K2Wgood
S2CA (A)
X
!
g:
b! x
P
S;K
A (S)
J
J W
F
(S) J2
K2Wgood
S2CA (A)
!
g:
J2
X
(S)
P
f [S]
4!
J,
P K; 1F nS
` (K)
b! g
P
S;K
b! g
P
S;K
v
u
u
jKj t
15
!
k4!
J xkL2 (!) k4J gkL2 (!)
L2 (!)
L2 (!)
X
F
(S)
K2Wgood
b! g
P
S;K
2
L2 (!)
(S) kgkL2 (!) :
near
The corresponding bound for the form BA
A straddle (f; g) is similar, but easier since there are at most 2
\
N
(S),
and
this
completes
the proof of the Stopping Child Lemma.
intervals J in !
g
+1
4.2.2. Dual tree decomposition. To control the local stopping forms BF
stop (f; g), we need to introduce further
corona decompositions within each corona CF (F ). These coronas will be associated to stopping intervals
U [F ] CF (F ), whose construction uses a dual tree decomposition originating with M. Lacey in [Lac]. For
the sake of generality, we present the decomposition in the setting of trees that are not necessarily dyadic.
De…nition 11. Let T be a tree with root o.
(1) Let P ( ) f 2 T :
g and S ( ) f 2 T :
g denote the predessor and successor sets
of 2 T .
(2) A geodesic g is a maximal linearly ordered subset of T . A …nite geodesic g is an interval g = [ ; ] =
P ( ) n S ( ), and an in…nite geodesic is an interval g = g n P ( ) for some 2 g.
(3) A stopping time3 T for a tree T is a subset T T such that
N
fTn gn=0
S( )\S
0
= ; for all ;
0
2 T with
6=
0
:
(4) A sequence
of stopping times Tn is decreasing if, for every 2 Tn+1 with 0 n < N , there
is 0 2 Tn such that S ( ) S 0 . We think of such a sequence as getting further from the root as
n increases.
(5) For T a stopping time in T and 2 T , we de…ne
[
[T; )
[ ; );
2T
where the interval [ ; ) = ; unless
. In the case [T; ) = ;, we write
T , and in the case
[T; ) 6= ;, we write
T . The set [T; ) can be thought of as the set of points in the tree T that
‘lie between’ T and but are strictly less than .
(6) For any 2 T , we de…ne the set of its children CT ( ) to consist of the maximal elements 2 T
such that
.
P
We de…ne the dual integration operator I on T by I ( )
( ). Here is the dual stopping
2T : 4
time lemma that abstracts that of M. Lacey in [Lac].
Lemma 12. Let T be a tree with root o, and suppose there is a uniform bound on the number of children
of any tree element. Suppose further that : T ! [0; 1) is nontrivial with …nite support, and let T0 be the
stopping time consisting of the minimal tree elements in the support of . Fix > 1. If there is no element
3 This de…nition of stopping time used in the theory of trees is a slight variant of what has been used above, but should
cause no confusion.
16
ERIC T. SAW YER
2 T with I
P
( ) >
increasing sequence
2T :
N +1
fTn gn=0 , with
(4.6)
( ) >
I
I
I
2Tn
1:
2Tn
1:
X
I
X
(o)
is
-irreducible. Otherwise, there is a unique
TN +1 = fog, of stopping times Tn such that for all n 2 N with n
X
I ( );
for all 2 Tn ;
( )
I
( ), we say that
I
( );
for all
2 [ ; Tn
1)
with
N,
2 Tn ;
( ) :
2TN :
Moreover, for 1
n
(4.7)
)j
j(Tn
j(Tn
1;
N +1
N + 1, this unique sequence fTn gn=0 satis…es
X
(
1)
I ( );
for all 2 Tn ;
2Tn
1;
]j
(
1:
X
1)
2Tn
I
( );
for all
2S( )nf gn
1:
[
2Tn
S( ) :
1
P
Proof. If Tn is already de…ned, let Tn+1 consist of all minimal points 2 T satisfying I ( ) >
I
2Tn :
provided at least one such point exists. If not then set N = n and de…ne TN +1 fog. It is easy to see that
N +1
fTn gn=0 is an increasing sequence of stopping times that satis…es (4.6), and is unique with these properties.
Moreover (4.7) also holds since for 2 Tn we have
0
1
X
X
X
@I ( )
I ( )A
j(Tn 1 ; )j
=
j(Tn 1 ; )j =
2CT ( )
X
2CT ( )
(
1)
0
@
2Tn
2CT ( )
X
2Tn
X
1:
I
( )
2Tn
X
2CT ( ) 2Tn
X
I
( )=(
1:
4
1:
1
( )A
I
4
X
1)
2Tn
1:
and the same argument proves the second line in (4.7), since 2 S ( ) n f g n
by the stopping criterion in the …rst line of (4.6), and hence
X
I ( )
I ( ):
2Tn
I
( );
1:
S
2Tn
1
S ( ) was not chosen
1:
4.2.3. Completion of the proof. Momentarily …x > 0 in Lemma 12, which will be chosen at the very
end of the proof. For F 2 F, we consider the dyadic tree T
D, and let U [F ] = U !g [F ] denote the
collection of intervals constructed in Lemma 12 with o = F ,
= 1 + and g : D ! [0; 1) de…ned
(
2
!
!
k4J xkL2 (!) for J 2 g
by g (J) =
for J 2 D. If g is irreducible, the ensuing arguments are
0
for J 2
= !
g
greatly simpli…ed, as the reader can easily check. Note that U [F ] depends only only the !-Haar support !
g
!
of g, and not on g itself. We will sometimes write just U [F ] when !
g is understood. De…ne MIN
g to
good
be the minimal intervals J 2 CF
(F ) for which 4!
J x 6= 0. From (4.7), we have tight control of the good
!-projections of g in the coronas,
2
(4.8)
!;good
P
D no top (K)n
S
P!;good
D(K)\
x
S2CU [F ] [U ]
D(S) \
!
g
L2 (!)
2
!
g
x
L2 (!)
;
( ),
A REPRISE OF THE NTV CONJECTURE FOR THE HILBERT TRANSFORM
17
!
no top
(U ) D (U ) n fU g, and the projections are restricted to good
for K 2 CUgood
g , where D
[F ] (U ) n MIN
intervals. We also have geometric decay in grandchildren from iterating the …rst line in (4.6),
X
(4.9)
(m)
U 0 2CU [F ] [U ]
P!;good
CU [F ] (U 0 )\
!
g
x
2
1
L2 (!)
m
2
P!;good
D(U )\
!
g
x
L2 (!)
;
U 2 U [F ] :
De…nition 13. Fix f 2 L2 ( ) and let F be the stopping times constructed using the criterion (3.1), that
depends only on f , and !. For g 2 L2 (!) let U [F ] = U !g [F ] ; F 2 F be as constructed above, depending
only on g and !. Then for F 2 F, de…ne the form,
X
BF
EIJ 4I f H 1F nIJ ; 4!
stop (f; g)
Jg ! :
I2CF and J2CF (F )
J I and `(J) 2 `(I)
We will prove the bound (4.2) for f and g. Denote the collection of pairs (I; J) arising in the sum de…ning
F
BF
stop (f; g) by P , so that
X
BF
stop (f; g) =
EIJ 4I f
(I;J)2P F
1F nIJ ; 4!
Jg
H
We use the corona decomposition associated with the stopping times U, namely
[
PF =
CU (U ) CU (V )
U;V 2U : V
=
(
[
U 2U
F
Pdiag
U
CU (U )
[
)
CU (U )
8
[<
F
Pfar below ;
[
:
U;V G2U : V $U
CU (U )
:
!
9
=
CU (V )
;
to obtain the decomposition of the stopping form BF
stop (f; g) into ‘diagonal’and ‘far’stopping forms,
BF
stop (f; g)
F
= BF
diag stop (f; g) + Bfar stop (f; g) ;
X
BF
EIJ 4I f H 1F nIJ ; 4!
diag stop (f; g)
Jg
!
;
F
(I;J)2Pdiag
X
BF
far stop (f; g)
F
(I;J)2Pfar below
EIJ 4I f
H
1F nIJ ; 4!
Jg
!
;
where
BF
far stop
(f; g)
=
X
X
U;V 2U : V $U I2CU (U )
=
1 X
X
X
EIJ 4I f
X
t=1 U 2U V 2C(t) (U ) I2CU (U )
U
EI J
*
H
1F nIJ ;
X
J2CU (V )
J I and `(J) 2
D
4I PCU (U ) f H
4!
Jg
+
`(I)
1F nIJ ; P!
CU (V ) g
!
E
!
:
We will control this far stopping form BF
far stop (f; g) using the Stopping Child Lemma, in which we will
!
derive geometric decay from (4.9) and the Poisson-Energy characteristic PEA;trip
gt ; ; ! with
A
A0 replaced by F;
and where gt
S
P!
A replaced by U
(t)
G2C
(U )
U
CU (G) g:
!
g
[F ] , A replaced by U;
18
ERIC T. SAW YER
To reduce notational clutter, we write Ug instead of U
of g. Indeed, we then have for U 2 Ug and t 1,
X
X
CPEU;trip
Ug
!
gt ;
(t)
!
g
, even though it depends only on the !-Haar support
D
H
EIJ 4I PCUg (U ) f
V 2CUg (U ) I2CUg (U )
;!
s
X
S2CUg (U )
jSj
1F nIJ ; P!
CUg (V ) g
E
!
2
F
(S) kgt kL2 (!) ;
where
U
(S)
sup
I2
and gt
P
P
CU (U )
g
(t)
V 2CUg (U )
PEU;trip
Ug
!
gt ;
sup
EI PCUg (U ) f =
#
[S];F \Dgood
f
P!
CU
g (V
;! =
sup
S2CUg (U ) K2W trip (S)
good
P!(t)
)g
CUg (U )
sup
I2(S;U ]\Dgood
g, and where
v
u
u
t
1 P K; 1F nS
p
` (K)
jKj
S2CUg (U ) K2W trip (S)
good
sup
ES M PCUg (U ) f ;
EI PCUg (U ) f
sup
s
(1
1 P K; 1F nS
` (K)
jKj
p
t
")
J2
S
J2 !
g\
X
2
2
(t)
V 2C
(U ):G
Ug
k4!
J xkL2 (!)
! \D[K]
g
X
K
CUg (V )
k4!
J xkL2 (!)
t
") 2 PEU;trip
Ug [F ]
(1
!
g;
;! ;
i.e. we gain geometrically in t when passing from the Haar support of gt to that of g.
We now introduce the size functional of Lacey [Lac], denoted here by
s X
1 P K; 1F nK
2
p
PEF
(
;
;
!)
sup
k4!
F
J xkL2 (!) :
`
(K)
good
jKj
K2C
(F )
J2 \D[K]
F
Recalling the previous characteristics from (3.2) and (4.3), we have
PEtrip
A[F ] ( ; ; !)
(4.10)
PEF
F ( ; ; !)
PEF
F ( ; !)
PEF ( ; !) :
Thus we have from the Stopping Child Lemma applied to each U 2 Ug ,
v
1 X
u X
X
u
U;trip
F
!
Bfar stop (f; g)
CPEUg [F ] gt ; ; ! t
(4.11)
t=1 U 2Ug
1 X
X
C (1
")
t=1 U 2Ug
CPEtrip
Ug [F ]
CPEtrip
Ug [F ]
!
g;
t
2
PEU;trip
Ug [F ]
!
g;
X
v
u
u
;! t
U 2Ug S2CUg (U )
!
g;
; ! kf kL2 (
X
2
jSj
)
M PCU
f
g (U )
2
jSj
g (U )
S2CUg (U )
S2CUg (U )
v
uX
u
;! t
2
jSj ES M PCU
M PCU
g
(U ) f
X
X
(t)
U 2Ug V 2C(t) (U )
Ug
kgt kL2 (!)
P!
CUg (V ) g
V 2CUg (F )
X
f
L2 (!)
P!
CUg (V ) g
L2 (!)
kgkL2 (!) ;
by Cauchy-Schwarz, boundedness of the dyadic maximal function M on L2 ( ), and orthogonality in both
f and g.
Finally it remains to control the diagonal stopping form BF
diag stop (f; g), where
+
*
X
X
X
!
F
EIJ 4I PCUg (U ) f
H 1F nIJ ;
4J g
:
Bdiag stop (f; g) =
U 2Ug I2CUg (U )
J2CUg (U ): J I and `(J) 2
`(I)
!
A REPRISE OF THE NTV CONJECTURE FOR THE HILBERT TRANSFORM
19
no top
Note that the intervals J arising in BF
CUg n fU g by the requirements that
diag stop (f; g) actually lie in CUg
` (J) 2 ` (I) and I 2 CUg (U ). In particular, the minimal stopping intervals U 2 Ug do not contribute to
P
the sum U 2Ug de…ning BF
diag stop (f; g).
Next we note that
*
+
X
X
X
H 1F nIJ ;
4!
EIJ 4I PCUg (U ) f
BF
Jg
diag stop (f; g)
no top
J2CU
(U ): J I and `(J) 2
g
U 2Ug I2CUg (U )
=
X
g
PCUg (U ) f; P!
no top
CU
(U )
g
BF
stop
U 2Ug [F ]
no top
PCUg (U ) f and gU
where fU
=
no top
fU ; gU
BF
stop
!
;
U 2Ug [F ]
P!
g:
C no top (U )
Ug
For F 2 F, set
F
z
X
`(I)
BF
stop (f; g)
sup
PEF
F
f;g6=0
!
f ; g 2z
!;
g
; ! kf kL2 (
)
kgkL2 (!)
;
which is …nite because of our assumption that the Haar supports of f and g are restricted to a …xed …nite
subset z of the grid D. We next use (4.8), (4.10) and orthogonality of the projections PCU (U ) to deduce that
F
z
F;top only
F;no top
BF
far stop (f; g) + Bdiag stop (f; g) + Bdiag stop (f; g)
sup
PEF
F
f;g6=0
!
f ; g 2z
C +C +
P
U 2U
sup
f;g6=0
!
f ; g 2z
C +C +
!;
g
PEF
F
sup
f;g6=0
U 2U
!
g 2z
f;
; ! kf kL2 (
PEF
F
)
!
no top ;
gU
PEF
F
! [F ]
g
)
kgkL2 (!)
no top
BF
stop fU ; gU
! [F ]
g
!;
g
X
; ! kf kL2 (
!;
g
kgkL2 (!)
; ! kfU kL2 (
)
kf kL2 (
;!
no top
gU
)
L2 (!)
kgkL2 (!)
no top
BF
stop fU ; gU
PEF
F
2C +
sup
sup
f;g6=0 U 2U
!
f ; g 2z
! [F ]
g
!
no top ;
gU
; ! kfU kL2 (
!
no top ;
gU
PEF
F
PEF
F
!;
g
;!
)
F
z
;!
no top
gU
2C +
L2 (!)
F
z;
where > 0 is as in (4.8). Note that (4.8) applies here precisely because the tops of the coronas are missing
good
in CUno !top (U ). Indeed, for K 2 CF
(F ), it su¢ ces to show,
g
v
s
X
X
u
2
2
!
u
(4.12)
k4
xk
k4!
2 (!)
J
J xkL2 (!) :
L
t
J2
If
!
no top
gU
J2
! \D[K]
g
\ D [K] is empty, there is nothing to prove, and if K
the remaining case K 2
Since
!
no top \D[K]
g
U
F
z
CUno !top
g
(U ) n MIN
< 1, we conclude that
BF
stop (f; g)
F
F
z PEF
F
z
!
g;
!
g
2C
1
U , the …rst line in (4.8) yields (4.12). In
, the second line in (4.8) yields (4.12).
4C provided 0 <
; ! kf kL2 (
)
kgkL2 (!)
< 12 . Hence we have the inequality,
4CPEF ( ; !) kf kL2 (
)
kgkL2 (!) ;
using (4.10), which when combined with (4.11), completes the required control (4.2) of the stopping form
BF
stop (f; g) when the Haar supports of f and g are in a …xed …nite subset z of the grid D. Since such
functions are dense in L2 ( ) and L2 (!) as z ranges over all …nite subsets z of D, the proof of (4.2) is
complete.
20
ERIC T. SAW YER
Proof of Theorem 2. Collecting all of the above form estimates proves Theorem 2.
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McMaster University
E-mail address : sawyer@mcmaster.ca