PERFECT DYADIC OPERATORS: WEIGHTED T (1) THEOREM AND
TWO WEIGHT ESTIMATES.
OLEKSANDRA BEZNOSOVA
Abstract. Perfect dyadic operators were first introduced in [1], where a local T (b) theorem
was proved for such operators. In [2] it was shown that for every singular integral operator
T with locally bounded kernel on Rn × Rn there exists a perfect dyadic operator T such that
T − T is bounded on Lp (dx) for all 1 < p < ∞.
In this paper we show a decomposition of perfect dyadic operators on real line into
four well known operators: two selfadjoint operators, paraproduct and its adjoint. Based
on this decomposition we prove a sharp weighted version of the T (1) theorem for such
operators, which implies A2 conjecture for such operators with constant which only depends
on kT (1)kBM Od , kT ∗ (1)kBM Od and the constant in testing conditions for T . Moreover, the
constant depends on these parameters at most linearly. In this paper we also obtain sufficient
conditions for the two weight boundedness for a perfect dyadic operator and simplify these
conditions under additional assumptions that weights are in the Muckenhoupt class Ad∞ .
1. Introduction
A perfect dyadic operator T is defined by:
Z
T f :=
K(x, y) f (y) dy for x ∈
/ supp f,
where kernel K satisfies the following conditions:
standard size condition:
(1.1)
|K(x, y)| ≤
1
|x − y|
and perfect cancellation condition:
K(x, y) − K(x, y 0 ) + K(x, y) − K(x0 , y) = 0
(1.2)
whenever x, x0 ∈ I ∈ D, y, y 0 ∈ J ∈ D, I ∩ J = ∅. Where D is a set of dyadic intervals on
the real line D := {I = [2j k, 2j (k + 1)) : k, j ∈ Z}.
Perfect dyadic operators appeared in the context of the local T (b) theorems (see [1]) and
later were extended to spaces of homogeneous type (see [3], [25]). Perfect dyadic operators
are essentially Calderón–Zygmund singular integrals with singularity adapted to the fixed
dyadic grid. Their main property is that any function supported on a dyadic cube with zero
mean is mapped into the function supported on the same cube. In [2] it was shown that
in L2 (dx) perfect dyadic operators are a good approximation of Calderón–Zygmund singular
integral operators.
2010 Mathematics Subject Classification. Primary 42B20, 42B25 ; Secondary 47B38.
Key words and phrases. Weighted norm estimate, Dyadic operators, Joint A2 -weights, Perfect dyadic operators, T(1) theorem.
The author was supported by the University of Alabama RGC grant.
1
2
O. BEZNOSOVA
In this paper we stick to the real line as a model case. In Section 2 we derive a very useful
decomposition of the perfect dyadic operator into four well known operators: two selfadjoint
operators, dyadic paraproduct and its adjoint. In particular, we show that a perfect dyadic
operator T and its adjoint formally can be written as
T f (x) =
(1.3)
p
1X
1X
KI+ + KI− |I| mI f χI −
KI+ − KI−
|I| hf ; hI i χI
4
4
I∈D
I∈D
1X
1X
+
KI+ − KI− |I|3/2 mI f hI −
KI+ + KI− |I| hf ; hI i hI ,
4
4
I∈D
T ∗ f (x) =
(1.4)
I∈D
p
1X
1X
|I| hf ; hI i χI (x)
KI+ + KI− |I| mI f χI (x) +
KI+ − KI−
4
4
I∈D
I∈D
1X
1X
−
KI+ − KI− |I|3/2 mI f hI (x) −
KI+ + KI− |I| hf ; hI i hI (x).
4
4
I∈D
I∈D
For a given dyadic interval I, I + and I − are its left and right halves. Coefficients KI+ and
are defined to be values of the kernel K of the perfect dyadic operator T on the dyadic
cubes I + × I − and I − × I + respectively.R The notation mI f stands for the average of the
1
function f over the interval I, mI f := |I|
I f.
In Section 3 using our decomposition we prove the T (1) theorem and show several useful
estimates on the kernel. We prove the following theorem.
KI−
Theorem 1. Let T be a perfect dyadic operator that satisfies:
(i) BMO conditions: kT (1)kBM Od and kT ∗ (1)kBM Od are bounded by a finite numerical
constant Q;
(ii) Testing conditions: hT hI ; hI i are uniformly bounded by Q for every dyadic interval
I ∈ D.
Then T is bounded on L2 .
Moreover, T accepts decomposition (1.3) with coefficients KI± satisfying
size conditions:
+
K + K − |I| ≤ 2Q,
(1.5)
∀J ∈ D
J
J
T (1) conditions:
(1.6)
∀J ∈ D
1 X
|J|
KI+ − KI−
2
|I|3 ≤ 16Q
I∈D(J)
and testing conditions:
(1.7)
∀J ∈ D
X
1
2
+
−
KI + KI |I| ≤ 8Q.
|J|
I∈D(J)
In Section 4 we ‘lift’ the above T (1) theorem to the one weight case, L2 (w), and provide
an elementary proof of the A2 conjecture for such operators; in particular we trace how the
weighted norm of perfect dyadic operator depends on the dyadic BM Od and testing constants
of the perfect dyadic operator. We prove the following theorem.
PERFECT DYADIC OPERATORS: WEIGHTED T (1) THEOREM AND TWO WEIGHT ESTIMATES.
3
Theorem 2. Let T be a perfect dyadic operator on R such that
kT (1)kBM Od 6 Q and kT ∗ (1)kBM Od 6 Q
and
∀I ∈ D hT hI , hI i 6 Q
. Then T is bounded on
L2 (w)
and
kT f kL2 (w) 6 CQ[w]Ad kf kL2 (w) ,
(1.8)
2
with some constant C independent of the operator T .
In particular, by [14], this implies that under assumptions of the theorem a perfect dyadic
max{1, p1 }
operator T is also bounded in Lp (w) for all w ∈ Adp , its norms are bounded by C[w]Ad
,
p
and dependence C(Q) can be traced as well.
To the best of our knowledge, such sharp weighted version of T (1) theorem is new. In
particular it is interesting that the constant in (1.8) depends only on the BM Od norms of
T (1) and T ∗ (1) and on the constant in testing conditions. Moreover, the dependence is at
most linear.
In Section 5 we go even further and give sufficient conditions for the two weight boundedness for a single perfect dyadic operator. Currently there are two schools of thought regarding
the two weight problem. First, given one operator T find necessary and sufficient conditions
on the weights to ensure boundedness of the operator on the appropriate spaces. Second,
given a family of operators find necessary and sufficient conditions on the weights to ensure
boundedness of the family of operators. In both cases we are mostly interested in Calderón–
Zygmund singular integral operators. In the first case, the conditions are usually “testing
conditions” obtained from checking boundedness of the given operator on a collection of test
functions. In the second case, the conditions are more “geometric”, meaning they seem to only
involve the weights and not the operators, such as Carleson conditions or bilinear embedding
conditions, Muckenhoupt A2 type conditions or bumped conditions. Operators of interest are
the maximal function [31, 26, 30, 35], fractional and Poisson integrals [32, 10], the Hilbert
transform [8, 9, 19, 28, 23, 20] and general Calderón–Zygmund singular integral operators and
their commutators [13, 12, 11, 17, 27], the square functions [21, 22], paraproducts and their
dyadic counterparts. Necessary and sufficient conditions are only known for the maximal
function, fractional and Poisson integrals [31], square functions [21] and the Hilbert transform [20, 23], and among the dyadic operators for the martingale transform, the dyadic square
functions, positive and well localized dyadic operators [37, 28, 29, 34, 15, 16, 18, 24, 33, 36].
We prove the following sufficient conditions for the two weight boundedness of the perfect
dyadic operators.
Theorem 3. Let (v, u) be a pair of measurable functions, such that u and v −1 , the reciprocal
of v, are weights on R such that
(1.9)
(1.10)
(v, u) ∈ Ad2 , i.e. [v, u]Ad := sup mI (v −1 )mI u < ∞,
2
∀J ∈ D
I∈D
1 X
(∆I u)2 mI (v −1 )|I| 6 CmI u,
|J|
I∈D(J)
4
(1.11)
O. BEZNOSOVA
1 X
(∆I (v −1 ))2 mI u|I| 6 CmI (v −1 ),
|J|
∀J ∈ D
I∈D(J)
and operator T0 is bounded from L2 (v) in L2 (u), i.e.
2

Z
X ∆I u∆I (v −1 ) 1
χI (x) u(x)dx 6 CmJ (v −1 )

(1.12)
∀J ∈ D
|J|
mI u
J
I∈D(J)
and
2
X ∆I u∆I (v −1 ) 
 −1
mI (v −1 ) χI (x) v (x)dx 6 CmJ u.

(1.13)
1
∀J ∈ D
|J|
Z
J
I∈D(J)
Let T be a perfect dyadic operator with kT (1)kBM Od , kT ∗ (1)kBM Od 6 Q and for every I ∈ D
we have hT hI ; hI i 6 Q.
Then T is bounded from L2 (v) to L2 (u) whenever for any dyadic interval J ∈ D
1 X
(KI+ + KI− )|I|2 mI u 6 CmJ u,
|J|
(1.14)
I∈D(J)
(1.15)
1 X
(KI+ + KI− )|I|2 mI (v −1 ) 6 CmJ (v −1 ),
|J|
I∈D(J)
1 X (KI+ − KI− )2 |I|3
≤ CmJ (v −1 )
|J|
mI u
(1.16)
I∈D(J)
and
1 X (KI+ − KI− )2 |I|3
≤ CmJ u.
|J|
mI (v −1 )
(1.17)
I∈D(J)
When both weights are in the Muckenhoupt class Ad∞ we simplify conditions of Theorem
3 to the following.
Theorem 4. Let (v, u) be a pair of measurable functions, such that u and v −1 , the reciprocal
of v, are Ad∞ weights on R such that
(1.18)
(v, u) ∈ Ad2 , i.e. [v, u]Ad := sup mI (v −1 )mI u < ∞,
2
L2 (v)
and operator T0 is bounded from

Z
X
1

(1.19)
∀J ∈ D
|J| J
I∈D(J)
in
I∈D
L2 (u),
i.e.
2
∆I u∆I (v −1 ) χI (x) u(x)dx 6 CmJ (v −1 )
mI u
and
2
X ∆I u∆I (v −1 ) 
 −1
mI (v −1 ) χI (x) v (x)dx 6 CmJ u.

(1.20)
1
∀J ∈ D
|J|
Z
J
I∈D(J)
PERFECT DYADIC OPERATORS: WEIGHTED T (1) THEOREM AND TWO WEIGHT ESTIMATES.
5
Let T be a perfect dyadic operator with kT (1)kBM Od , kT ∗ (1)kBM Od 6 Q and for every I ∈ D
we have hT hI ; hI i 6 Q.
Then T is bounded from L2 (v) to L2 (u) whenever for any dyadic interval J ∈ D
1 X (KI+ − KI− )2 |I|3
≤ CmJ (v −1 )
|J|
mI u
(1.21)
I∈D(J)
and
1 X (KI+ − KI− )2 |I|3
≤ CmJ u.
|J|
mI (v −1 )
(1.22)
I∈D(J)
The main novelty of this paper is the decomposition, very strong quantitative estimates in
the A2 conjecture and sufficient conditions for the two weight boundedness of perfect dyadic
operators.
2. Perfect dyadic operator on R: definition and formal decomposition.
Let T be a perfect dyadic singular integral operator, i.e an operator defined by:
Z
(2.23)
T f :=
K(x, y) f (y) dy for x ∈
/ supp f,
where kernel K satisfies the following conditions:
standard size condition:
|K(x, y)| ≤
(2.24)
1
|x − y|
and perfect cancelation condition:
K(x, y) − K(x, y 0 ) + K(x, y) − K(x0 , y) = 0
(2.25)
whenever x, x0 ∈ I ∈ D, y, y 0 ∈ J ∈ D, I ∩ J = ∅.
Note that, by perfect dyadic cancellation condition, K(x, y) is constant on I + × I − and
−
I × I + for any dyadic interval I ∈ D. We define
KI+ := K(x, y),
x ∈ I +, y ∈ I −
KI− := K(x, y),
x ∈ I −, y ∈ I +.
and
Then
K(x, y) =
X
KI+ χI + ×I − + KI− χI − ×I + .
I∈D
Then perfect dyadic singular operator can be written as
X
(2.26)
T f (x) =
KI+ fI − χI + (x) + KI− fI + χI − (x),
I∈D
R
where fI := I f (x)dx.
Now we are going to rewrite T in a more convenient form. It is easy to see that
KI+ fI − χI + (x) + KI− fI + χI − (x)
=
KI+ fI − + KI− fI +
K + f − − KI− fI +
(χI + (x) + χI − (x)) + I I
(χI + (x) − χI − (x))
2
2
6
O. BEZNOSOVA
(2.27) =
p
1
1
|I| hI (x),
KI+ fI − + KI− fI + χI (x) +
KI+ fI − − KI− fI +
2
2
where {hI }I∈D is the Haar system of functions, normalized in L2 , hI := |I|−1/2 (χI + − χI − ).
Thus, T f (x) can be written as:
X1
p
1
(2.28)
T f (x) =
KI+ fI − + KI− fI + χI +
KI+ fI − − KI− fI + hI |I|.
2
2
I∈D
Now let us handle the coefficients KI+ fI − + KI− fI + and KI+ fI − − KI− fI + :
Z
Z
KI+ fI − + KI− fI + = KI+
f + KI−
f
I−
I+
Z
Z
Z
Z
1
1
+
+
−
−
KI + KI
KI − KI
f −
f
f+
f−
=
2
2
I+
I−
I−
I+
p
1
1
(2.29)
=
KI+ + KI− |I| mI f −
KI+ − KI−
|I| hf ; hI i
2
2
and, similarly, replacing KI− by −KI− ,
(2.30)
KI+ fI − − KI− fI + =
p
1
1
|I| hf ; hI i .
KI+ − KI− |I| mI f −
KI+ + KI−
2
2
We plug (2.29) and (2.30) in (2.28) and obtain a formal representation of the perfect dyadic
operator T :
p
1X
KI+ + KI− |I| mI f χI − KI+ − KI−
T f (x) =
|I| hf ; hI i χI
4
I∈D
(2.31)
+ KI+ − KI− |I|3/2 mI f hI − KI+ + KI− |I| hf ; hI i hI .
We can represent T ∗ in a similar way:
X
T ∗ f (x) =
KI− fI − χI + (x) + KI+ fI + χI − (x)
I∈D
X1
p
1
KI− fI − + KI+ fI + χI (x) +
KI− fI − − KI+ fI +
|I| hI (x)
2
2
I∈D
p
1X
=
|I| hf ; hI i χI (x)
KI+ + KI− |I| mI f χI (x) + KI+ − KI−
4
I∈D
− KI+ − KI− |I|3/2 mI f hI (x) − KI+ + KI− |I| hf ; hI i hI (x).
=
(2.32)
3. Decomposition of T , additional assumptions.
In this section we prove Theorem 1 modulo the unweighted boundedness of the operator
T1 , which is done in a more general weighted case in the next section.
We assume, in addition to T being perfect dyadic, that T (1) and T ∗ (1) are both in the
dyadic BM Od , with the BM Od norm of at most Q. We will also assume that T satisfies
testing conditions hT hI , hI i 6 Q for all dyadic intervals I.
PERFECT DYADIC OPERATORS: WEIGHTED T (1) THEOREM AND TWO WEIGHT ESTIMATES.
7
3.1. The BM Od condition. A function b(x), or a sequence {bI }I∈D = {hb; hI i}I∈D is in the
dyadic BM Od whenever the sequence b2I is a Carleson sequence, i.e. there is a finite constant
C such that for all dyadic intervals J ∈ D we have
1 X 2
bI 6 C.
|J|
I∈D(J)
The best such constant C is called the dyadic BM Od norm of b.
Let T (1), T ∗ (1) ∈ BM Od . We want to show that
1 X
(KI+ − KI− )2 |I|3 6 16Q.
|J|
I∈D(J)
First let us plug f (x) = 1(x) in T and T ∗ :
1X
KI+ + KI− |I| χI (x) + KI+ − KI− |I|3/2 hI (x)
T (1) =
4
I∈D
and
1X
KI+ + KI− |I| χI (x) − KI+ − KI− |I|3/2 hI (x).
4
I∈D
1 P
2
A function f (x) belongs to the dyadic class BM Od whenever supJ∈D |J|
I∈D(J) hf ; hI i ≤
T ∗ (1) =
Q, the smallest such constant is the BM Od -norm of the function f (x). So, let us find
hT (1); hJ i and hT ∗ (1); hJ i:
1X
hT (1); hJ i =
KI+ + KI− |I| hχI (x); hJ (x)i + KI+ − KI− |I|3/2 hhI (x); hJ (x)i
4
I∈D
=
(3.33)
1
4
+
X
I∈D(J + )
|I|
|I|
1 X
−
KI+ + KI− |I| p
KI+ + KI− |I| p
|J| 4 I∈D(J − )
|J|
1
KJ+ − KJ− |J|3/2 .
4
Similarly,
hT ∗ (1); hJ i =
1X
KI+ + KI− |I| hχI (x); hJ (x)i − KI+ − KI− |I|3/2 hhI (x); hJ (x)i
4
I∈D
=
(3.34)
1
4
−
X
I∈D(J + )
|I|
|I|
1 X
KI+ + KI− |I| p
−
KI+ + KI− |I| p
4
|J|
|J|
I∈D(J − )
1
KJ+ − KJ− |J|3/2 .
4
Let
αJ :=
X
I∈D(J + )
X
|I|2
|I|2
KI+ + KI− p
−
KI+ + KI− p
|J| I∈D(J − )
|J|
and
βJ :=
KJ+ − KJ− |J|3/2 ,
8
O. BEZNOSOVA
then
1
1
(αJ + βJ ) ,
hT ∗ (1); hJ i = (αJ − βJ ) .
4
4
d
If T (1) ∈ BM O with kT (1)kBM Od ≤ Q, then for every dyadic interval J ∈ D
X
1 X
1
(3.35)
hT (1); hI i2 =
(αI + βI )2 ≤ Q.
|J|
16 |J|
hT (1); hJ i =
I∈D(J)
I∈D(J)
Similarly, T ∗ (1) ∈ BM Od with kT ∗ (1)kBM Od ≤ Q implies that
X
1
1 X
hT ∗ (1); hI i2 =
(αI − βI )2 ≤ Q.
(3.36)
∀J ∈ D
|J|
16 |J|
I∈D(J)
I∈D(J)
Let us see how conditions (3.35) and (3.36) imply that
2
1 X 2
1 X
∀J ∈ D
βI =
KI+ − KI− |I|3 ≤ 16Q.
|J|
|J|
I∈D
I∈D(J)
The sum
1
|J|
2
I∈D(J) βI
P
1 X 2
βI =
|J|
I∈D(J)
=
can be written as:
αI
βI
αI 2
1 X
βI
−
+
+
|J|
2
2
2
2
I∈D(J)
1 X
αI 2
2 X
αI
βI
αI
1 X
αI 2
βI
βI
βI
−
+
−
+
+
+
|J|
2
2
|J|
2
2
2
2
|J|
2
2
I∈D(J)
I∈D(J)
I∈D(J)
X
1
≤ 8Q +
(βI − αI ) (βI + αI ) .
2 |J|
I∈D(J)
By Cauchy-Schwarz
1 

1
|J|
X
I∈D(J)
1
(βI − αI ) (βI + αI ) ≤ 
|J|
2
X
2
(βI − αI )
I∈D(J)
 1
|J|
1
2
X
(βI + αI )
2
≤ 16Q.
I∈D(J)
Hence, for every dyadic interval J ∈ D
2
1 X 2
1 X
(3.37)
βI =
KI+ − KI− |I|3 ≤ 16Q.
|J|
|J|
I∈D
I∈D(J)
3.2. Testing conditions. We assume that T satisfies testing conditions
(3.38)
∀J ∈ D
hT hJ ; hJ i ≤ Q.
Let us write hT hJ ; hJ i as follows:
p
1 X
|hT hJ ; hJ i| =
KI+ + KI− hhJ ; χI i hχI ; hJ i − KI+ − KI−
|I| hhJ ; hI i hχI ; hJ i
4
I∈D
p
+
−
+
−
+ KI − KI
|I| hhJ ; χI i hhI ; hJ i − KI + KI |I| hhJ ; hI i hhI ; hJ i
PERFECT DYADIC OPERATORS: WEIGHTED T (1) THEOREM AND TWO WEIGHT ESTIMATES.
9
!2
!2
X
1 X
1
|I|
|I|
p
KI+ + KI−
KI+ + KI− − p
= +
4
4
|J|
|J|
I∈D(J + )
I∈D(J − )
1
− KJ+ + KJ− |J|
4
X
1
1
2
+
+
−
−
KJ + KJ |J| ≤ Q.
KI + KI |I| −
= 2
4 |J| I∈D(J)
Note that by standard size condition on the kernel K, we know that KJ+ + KJ− |J| ≤ 2Q, so
1 X
2
+
−
KI + KI |I| ≤ 8Q.
(3.39)
|J|
I∈D(J)
So, a perfect dyadic Calderón-Zygmund singular integral operator T satisfying the T (1) conditions kT (1)kBM O2 , kT ∗ (1)kBM Od 6 Q and the dyadic testing conditions (hT hJ ; hJ i ≤ Q ∀J ∈ D),
can be written as:
X
T f (x) =
KI+ fI − χI + (x) + KI− fI + χI − (x)
I∈D
=
(3.40)
p
1X
KI+ + KI− |I| mI f χI (x) − KI+ − KI−
|I| hf ; hI i χI (x)
4
I∈D
+ KI+ − KI− |I|3/2 mI f hI (x) − KI+ + KI− |I| hf ; hI i hI (x).
Moreover, coefficients KI+ and KI− satisfy the following conditions:
size conditions
+
K + K − |I| ≤ 2Q,
(3.41)
∀J ∈ D
J
J
T (1) conditions
(3.42)
∀J ∈ D
1 X
|J|
KI+ − KI−
2
|I|3 ≤ 16Q
I∈D(J)
and testing conditions
(3.43)
∀J ∈ D
1 X
2
+
−
K
+
K
|I|
I
I
|J|
≤ 8Q.
I∈D(J)
By (3.40) operator T can be written as
(3.44)
T =
1
(T1 − T2 + T3 − T4 ) ,
4
where
(3.45)
T1 f (x) =
X
I∈D
KI+ + KI− |I| mI f χI (x),
10
O. BEZNOSOVA
(3.46)
X
T2 f (x) =
KI+ − KI−
p
|I| hf ; hI i χI (x),
I∈D
(3.47)
T3 f (x) =
X
KI+ − KI− |I|3/2 mI f hI (x),
I∈D
(3.48)
T4 f (x) =
X
KI+ + KI− |I| hf ; hI i hI (x),
I∈D
since all four operators are known well-defined linear operators bounded on L2 . Operators
T3 and T2 are dyadic paraproduct and its adjoint, that P
are both well defined and bounded in
L2 since the sequence bI = (KI+ − KI− )|I|3/2 satisfies I∈D(J) b2I 6 C|J| for any J ∈ D by
(3.42). T4 is a martingale transform with symbol bounded uniformly by (3.41). In the next
section we show that T1 is bounded on L2 and on all weighted spaces for weights in Ad2 .
This completes the proof of Theorem 1.
4. Weighted T (1) theorem for perfect dyadic operators
Let w be a weight, i.e. almost everywhere positive locally integrable function on the real
line. Let a weight w be such that w−1 is a weight as well. Assume that w is in the dyadic
Muckenhoupt class Ad2 , i.e.
[w]Ad := sup mI w mI (w−1 ) < ∞.
2
I∈D
In order to show that the weighted L2 (w) norm of the operator T depends on the A2 constant of the weight w at most linearly, it is enough to show the linear bound for each of
the operators Ti , i = 1, 2, 3, 4.
First let us show that T1 is bounded on L2 (w) and its norm depends on the A2 -constant
of the weight w at most linearly. We will do it by duality, we will show that ∀f ∈ L2 (w) and
∀g ∈ L2 (w−1 )
X
hT1 f ; gi =
KI+ + KI− |I|2 mI f mI g ≤ C[w]Ad kf kL2 (w) kgkL2 (w−1 )
2
I∈D
or, alternatively, ∀f, g ∈ L2
X
(4.49)
KI+ + KI− |I|2 mI f w−1/2 mI gw1/2 ≤ C[w]Ad kf kL2 kgkL2 .
2
I∈D
Without loss of generality we may assume that coefficients (KI+ + KI− ) are all non-negative.
We are going to use the following version of the bilinear embedding theorem from [28].
Theorem 5 (Nazarov, Treil, Volberg). Let v and w be weights. Let {aI } be a sequence of
nonnegative numbers, s.t. for all dyadic intervals J ∈ D the following three inequalities hold
with some constant Q > 0:
1 X
aI mI w mI v ≤ Q,
(4.50)
|J|
I∈D(J)
(4.51)
1 X
aI mI w ≤ Q mI w,
|J|
I∈D(J)
PERFECT DYADIC OPERATORS: WEIGHTED T (1) THEOREM AND TWO WEIGHT ESTIMATES.
11
1 X
aI mI v ≤ Q mI v,
|J|
(4.52)
I∈D(J)
then for any two nonnegative functions f, g ∈ L2
X
(4.53)
aI mI f v 1/2 mI gw1/2 ≤ C Q kf kL2 kgkL2 .
I∈D
Applying this theorem with v = w−1 and aI = KI+ + KI− |I|2 , we can see that the desired
bound (4.49) will hold if we can prove the following three inequalities for every dyadic interval
J ∈ D:
1 X
KI+ + KI− |I|2 mI w mI (w−1 ) ≤ C[w]Ad ,
(4.54)
2
|J|
I∈D(J)
(4.55)
1 X
|J|
KI+ + KI− |I|2 mI w ≤ C[w]Ad mJ w,
2
I∈D(J)
(4.56)
1 X
|J|
KI+ + KI− |I|2 mI (w−1 ) ≤ C[w]Ad mJ (w−1 ).
2
I∈D(J)
It turns out that all these inequalities follow from the testing conditions (3.43). It is
easy
to see that inequality
(4.54) follows from (3.43) and the definition of A2 -constant
−1
mI w mI (w ) ≤ [w]Ad .
2
In order to see that inequalities (4.55) and (4.56) are true, we will need the following
lemma, which can be found, for example, in [4]:
Lemma 6. Let v be a weight, such that v −1 is a weight as well, and let {λI } be a Carleson
sequence of nonnegative numbers, that is, there exists a constant Q > 0 s.t.
1 X
∀J ∈ D
λI ≤ Q,
|J|
I∈D(J)
then
∀J ∈ D
1 X
λI
≤ 4Q mJ v
|J|
mI (v −1 )
I∈D(J)
and, therefore, if v ∈ A2 then for any J ∈ D we have
1 X
mI v λI ≤ 4Q kvkAd mJ v.
2
|J|
I∈D(J)
Applying this lemma to λI = KI+ + KI− |I|2 and
−1v= w, we obtain that (4.55) follows
−1
from (3.43). If we take v = w and observe that w Ad = kwkAd , we can see that (4.56)
2
2
follows from (3.43) as well.
So,
kT1 kL2 (w)→L2 (w) ≤ C kwkAd .
2
12
O. BEZNOSOVA
Let us analyze operators T3 and T2 now.
X
T3 f (x) =
KI+ − KI− |I|3/2 mI f hI (x),
I∈D
T2 f (x) =
X
KI+ − KI−
p
|I| hf ; hI i χI (x),
I∈D
We first note that T3 is a paraproduct operator and T2 is its adjoint:
X
T3 f (x) = πb f (x) =
mI f bI hI (x),
I∈D
T2 f (x) = πb∗ f (x) =
X
bI hf ; hI i
I∈D
KI+
KI−
1
χI (x)
|I|
3/2
with the sequence bI =
−
|I| .
In order for dyadic paraproduct
and
its adjoint to be bounded in L2 we need b to be in
2
d
the BM O , i.e. we need bI I∈D to be a Carleson sequence, which is
2
1 X
KI+ − KI− |I|3 ≤ Q
∀J ∈ D
|J|
I∈D
and coincides with T (1) conditions (3.42).
It has been shown in [4] that the norm of the dyadic paraproduct (and, by symmetry
of A2 -constant, of its adjoint as well) depends on the A2 -constant of the weight w at most
linearly, i.e.
kT3 kL2 (w)→L2 (w) + kT2 kL2 (w)→L2 (w) ≤ C Q1/2 [w]Ad .
2
And, finally, the linear bound on the operator T4 ,
X
T4 f (x) =
KI+ + KI− |I| hf ; hI i hI (x),
I∈D
is implied by the uniform boundedness of the symbol (i.e. size conditions (3.41)) and Wittwer’s weighted linear bound on the martingale transform (see [38]).
Since all parts of T obey linear bounds on their L2 (w) norms with respect to the A2 constant of the weight w, bound on the norm of T is linear as well,
kT kL2 (w)→L2 (w) ≤ C kwkA2 ,
where constant C depends on the constant in the size condition (2.24), dyadic BM Od -norms
of T (1) and T ∗ (1) (at most linearly) and the constant in testing conditions (3.38) (at most
linearly as well). Therefore, we have the following theorem.
Theorem 7. Let T be a perfect dyadic operator on R such that
kT (1)kBM Od 6 Q and kT ∗ (1)kBM Od 6 Q
and
. Then T is bounded on
L2 (w)
∀I ∈ D hT hI , hI i 6 Q
and
kT f kL2 (w) 6 CQkf kL2 (w) ,
with some constant C independent of the operator T .
PERFECT DYADIC OPERATORS: WEIGHTED T (1) THEOREM AND TWO WEIGHT ESTIMATES.
13
Theorem 7 in the case w(x) = 1 is a T (1) theorem, so we can view it as a weighted version
of the T (1) theorem. It is also interesting that the L2 (w) norm of the operator T after proper
normalization (we assume that the decay constant of the kernel is 1) only depends the BM Od
norms of T (1) and T ∗ (1) and the constant in testing conditions. It also depends on these
constants at most linearly.
5. Two weight boundedness of perfect dyadic operators.
We start with the decomposition (3.44),
1
T f (x) = (T1 f (x) + T2 f (x) + T3 f (x) + T4 f (x)).
4
First we consider
X
T1 f =
(KI+ + KI− )|I|mI f χI (x).
I∈D
Similarly to the one weight case, by duality using Theorem 5, we know it is bounded from
L2 (v) in L2 (u) (with the norm bounded by CQ1 ) whenever for every dyadic interval J ∈ D
the following three conditions hold simultaneously
1 X
(KI+ + KI− )|I|2 mI (v −1 )mI u 6 Q1 ,
(5.57)
|J|
I∈D(J)
1 X
(KI+ + KI− )|I|2 mI u 6 Q1 mJ u,
|J|
(5.58)
I∈D(J)
(5.59)
1 X
(KI+ + KI− )|I|2 mI (v −1 ) 6 Q1 mJ (v −1 ).
|J|
I∈D(J)
Second, consider the paraproduct T3 and its adjoint T2 . In [5] we have sufficient conditions
for the two weight boundedness for the dyadic paraproduct:
Theorem 8. Let πb be the dyadic paraproduct operator associated to the sequence b =
{bI }I∈D :
X
πb f :=
mI f bI hI .
I∈D
Let (v, u) be a pair of measurable functions on R such that u and v −1 , the reciprocal of v, are
weights on R and such that
(i) (v, u) ∈ Ad2 , that is [v, u]Ad := supI∈D mI (v −1 ) mI u < ∞.
2
(ii) Assume that there is a constant Cv,u > 0 such that
X
|∆I u|2 |I| mI (v −1 ) ≤ Cv,u u(J)
for all J ∈ D,
I∈D(J)
where ∆I u := mI+ u − mI− u, and I± are the right and left children of I.
Assume that b ∈ Carlv,u , i.e. there is a constant Bv,u > 0 such that
X |bI |2
≤ Bv,u v −1 (J)
for all J ∈ D.
mI u
I∈D(J)
14
O. BEZNOSOVA
Then πb , the dyadic paraproduct associated to b, is bounded from L2 (v) in L2 (u). Moreover,
there exist C > 0 such that
q
q
p
[v, u]Ad + Cv,u kf kL2 (v) .
kπb f kL2 (u) ≤ C [v, u]Ad Bv,u
2
2
Therefore, in order to be able to bound the paraproduct T3 that has symbol bI = (KI+ −
we need the following three conditions to hold:
KI− )|I|3/2 ,
[v, u]Ad := sup mI (v −1 ) mI u < ∞,
(5.60)
(5.61)
2
X
I∈D
|∆I u|2 |I| mI (v −1 ) ≤ Cv,u u(J)
for all J ∈ D,
I∈D(J)
(5.62)
X (K + − K − )2 |I|3
I
I
≤ Bv,u v −1 (J)
mI u
for all J ∈ D.
I∈D(J)
The operator T2 is adjoint to T3 , T2 = T3∗ , so T2 is bounded from L2 (v) in L2 (u) whenever
the paraproduct T3 is bounded from L2 (u−1 ) in L2 (v −1 ). Therefore, in order for T2 to be
bounded from L2 (v) in L2 (u) we need the following three conditions to hold:
[u−1 , v −1 ]Ad := sup mI (u) mI (v −1 ) < ∞,
(5.63)
(5.64)
2
X
I∈D
|∆I (v −1 )|2 |I| mI u ≤ Cu−1 ,v−1 v −1 (J)
for all J ∈ D,
I∈D(J)
(5.65)
X (K + − K − )2 |I|3
I
I
≤ Bu−1 ,v−1 u(J)
mI (v −1 )
for all J ∈ D.
I∈D(J)
Finally, consider T4 , the martingale transform with symbol σ = (KI+ + KI− )|I|, which is
uniformly bounded by (3.41),
X
T4 f (x) =
KI+ + KI− |I| hf ; hI i hI (x).
I∈D
Necessary and sufficient conditions were obtained in [28]:
Theorem 9. Let σ = {σI }I∈D be a sequence of signs ±. The family of operators Tσ of
martingale transforms with symbol σ is uniformly bounded from L2 (v) to L2 (u) if and only if
the following assertions hold simultaneously:
∀J ∈ D mJ u mJ v −1 6 C < ∞,
(5.66)
(5.67)
∀J ∈ D
1 X
(∆I u)2 mI (v −1 )|I| 6 CmI u,
|J|
I∈D(J)
(5.68)
∀J ∈ D
1 X
(∆I (v −1 ))2 mI u|I| 6 CmI (v −1 ),
|J|
I∈D(J)
PERFECT DYADIC OPERATORS: WEIGHTED T (1) THEOREM AND TWO WEIGHT ESTIMATES.
15
P
∆I (v−1 )∆I u and an operator T0 , defined by T0 f := I∈D mI (v−1 )mI u mI f χI (x), is bounded from L2 (v)
to L2 (u), or, equivalently
2
X ∆I u∆I (v −1 ) χI (x) udx 6 CmJ (v −1 )

mI u

(5.69)
1
|J|
Z
1
∀J ∈ D
|J|
Z
∀J ∈ D
J
I∈D(J)
and
2
X ∆I u∆I (v −1 ) 
 −1
mI (v −1 ) χI (x) v dx 6 CmJ u.

(5.70)
J
I∈D(J)
Now we need to put all conditions we obtained for the operators T1,2,3,4 , conditions (5.57),
(5.58), (5.59), (5.60), (5.61), (5.62), (5.63), (5.64), (5.65), (5.66)(5.67)(5.68)(5.69)(5.70), together. Note that (5.60), (5.63) and (5.66) are all the same joint dyadic Ad2 condition, while
condition (5.57) follows from the joint Ad2 and (3.43) under no additional assumptions. Note
also that conditions (5.61) and (5.67) are the same, conditions (5.64) and (5.68) also coincide.
Therefore, we obtain the following theorem.
Theorem 10. Let (v, u) be a pair of measurable functions, such that u and v −1 , the reciprocal
of v, are weights on R such that
(5.71)
(5.72)
(v, u) ∈ Ad2 , i.e. [v, u]Ad := sup mI (v −1 )mI u < ∞,
2
∀J ∈ D
I∈D
1 X
(∆I u)2 mI (v −1 )|I| 6 CmI u,
|J|
I∈D(J)
(5.73)
1 X
(∆I (v −1 ))2 mI u|I| 6 CmI (v −1 ),
|J|
∀J ∈ D
I∈D(J)
and operator T0 is bounded from L2 (v) in L2 (u), i.e.

2
Z
−1
X
∆I u∆I (v ) 1
χI (x) u(x)dx 6 CmJ (v −1 )

(5.74)
∀J ∈ D
|J| J
mI u
I∈D(J)
and
2
X ∆I u∆I (v −1 )  −1

mI (v −1 ) χI (x) v (x)dx 6 CmJ u.

(5.75)
1
∀J ∈ D
|J|
Z
J
I∈D(J)
Let T be a perfect dyadic operator with kT (1)kBM Od , kT ∗ (1)kBM Od 6 Q and for every I ∈ D
we have hT hI ; hI i 6 Q.
Then T is bounded from L2 (v) to L2 (u) whenever for any dyadic interval J ∈ D
(5.76)
1 X
(KI+ + KI− )|I|2 mI u 6 CmJ u,
|J|
I∈D(J)
16
(5.77)
O. BEZNOSOVA
1 X
(KI+ + KI− )|I|2 mI (v −1 ) 6 CmJ (v −1 ),
|J|
I∈D(J)
1 X (KI+ − KI− )2 |I|3
≤ CmJ (v −1 )
|J|
mI u
(5.78)
I∈D(J)
and
1 X (KI+ − KI− )2 |I|3
≤ CmJ u.
|J|
mI (v −1 )
(5.79)
I∈D(J)
6. Sufficient conditions for the two weight boundedness of perfect dyadic
operators under additional assumptions that weighted are in Ad∞
In this section we will assume that v −1 and u are Ad∞ weights and show that conditions of
Theorem 10 can be reduced in this case. A weight w belongs to the class Ad∞ whenever
[w]Ad∞ := sup mI we−mI (ln w) < ∞.
I∈D
From [Be1] we have the following lemma
Lemma 11. Let w be a weight, such that log w ∈ L1loc , let {λI }I∈D be a Carleson sequence:
1 X
∃Q > 0 s.t. ∀J ∈ D
λI 6 Q.
|J|
I∈D(J)
Then for every dyadic interval J ∈ D
1 X mI (log w)
e
λI 6 4QmJ w
|J|
I∈D(J)
and therefore, if w ∈
Ad∞
then for every dyadic interval J ∈ D we have:
1 X
mI w λI 6 4Q[w]Ad∞ mI w.
|J|
I∈D(J)
In particular, u, v −1 ∈ Ad∞ implies that u and v −1 are in RH1d (see [6]), where RH1d is
defined as follows
w
w
d
w ∈ RH1 ⇐⇒ [w]RH d := sup mI
log
< ∞.
1
mI w
mI w
I∈D
We also know from [6] that [w]RH d 6 log 16[w]Ad∞ and we have the following theorem that
1
first appeared in [7] without the sharp constant and in [6] it was stated in stronger form and
the constant was traced. Here we state the strong form from [6].
Theorem 12. Let w be almost everywhere positive locally integrable function on the real line
and J be any interval. Then up to a numerical constant
∆I w 2
1 X
mJ (w log w) − mJ w log mJ w ≈
mI w|I|.
|J|
mI w
I∈D(J)
PERFECT DYADIC OPERATORS: WEIGHTED T (1) THEOREM AND TWO WEIGHT ESTIMATES.
17
In particular case when w is a weight in RH1d , we have that for every dyadic interval J ∈ D
1 X
∆I w 2
(6.80)
mI w|I| 6 C[w]RH d mJ w.
1
|J|
mI w
I∈D(J)
Firstly, note that conditions (5.72) and (5.73) follow from the fact that u and v −1 are in
Ad∞ , Theorem 12 and the joint Ad2 condition (6.81).
Secondly, conditions (5.76) and (5.77) by Lemma 11 follow from the fact that u, v −1 ∈
d
A∞ and 3.43. Therefore, under additional assumptions that u, v −1 ∈ Ad∞ , we can simplify
Theorem 10 as follows.
Theorem 13. Let (v, u) be a pair of measurable functions, such that u and v −1 , the reciprocal
of v, are Ad∞ weights on R such that
(v, u) ∈ Ad2 , i.e. [v, u]Ad := sup mI (v −1 )mI u < ∞,
(6.81)
2
I∈D
and operator T0 is bounded from L2 (v) in L2 (u), i.e.

2
Z
X ∆I u∆I (v −1 ) 1
χI (x) u(x)dx 6 CmJ (v −1 )

(6.82)
∀J ∈ D
|J|
mI u
J
I∈D(J)
and
2
X ∆I u∆I (v −1 ) 
 −1
mI (v −1 ) χI (x) v (x)dx 6 CmJ u.

(6.83)
∀J ∈ D
1
|J|
Z
J
I∈D(J)
Let T be a perfect dyadic operator with kT (1)kBM Od , kT ∗ (1)kBM Od 6 Q and for every I ∈ D
we have hT hI ; hI i 6 Q.
Then T is bounded from L2 (v) to L2 (u) whenever for any dyadic interval J ∈ D
(6.84)
1 X (KI+ − KI− )2 |I|3
≤ CmJ (v −1 )
|J|
mI u
I∈D(J)
and
(6.85)
1 X (KI+ − KI− )2 |I|3
≤ CmJ u.
|J|
mI (v −1 )
I∈D(J)
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Oleksandra Beznosova, Department of Mathematics, University of Alabama, 345 Gordon
Palmer Hall, Tuscaloosa, AL 35487
E-mail address: [email protected]