A matrix weighted $T1$ theorem for matrix kernelled Calderon Zygmund operators - I

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by  Joshua Isralowitz

A MATRIX WEIGHTED T1 THEOREM FOR MATRIX
KERNELLED CALDERO´N-ZYGMUND OPERATORS - I
4
1 JOSHUA ISRALOWITZ, HYUN KYOUNG KWON, AND SANDRA POTT
0
2
Abstract. In this series of two papers, we will prove a natural
n
matrix weighted T1 theorem for matrix kernelled CZOs. In the
a
J current paper, we will prove matrix weighted norm inequalities

5 for matrix symbolled paraproducts via a general matrix weighted
2
Carlesonembedding theorem. Along the way, we will also provide

a stopping time proof ofthe identification of Lp(W) as a weighted
]
A Triebel-Lizorkin space when W is a matrix A weight.
p
C
.
h
t
a
m
[ 1. Introduction

1 WeightednorminequalitiesforCaldero´n-Zygmundoperators(orCZOs
v
0 for short) acting on ordinary Lp(Rd) is a classical topic that goes back
7
to the 1970’s with the seminal works [7,9]. On the other hand, it is
5
6 well known that proving matrix weighted norm inequalities for CZOs
.
1 is a very difficult task, and because of this, matrix weighted norm in-
0
equalities for certain CZOs have only recently been investigated (see
4
1 [21,22] for specific details of these difficulties). In particular, if n and d
v: are natural numbers and if W : Rd → M (C) is positive definite a. e.
n
Xi (where as usual M (C) is the algebra of n×n matrices with complex
n
r scalar entries), then define Lp(W) for 1 < p < ∞ to be the space of
a
measurable functions f~: Rd → Cn with norm
kf~kp = |Wp1(x)f~(x)|pdx.
Lp(W)
Rd
Z
It was proved by F. Nazarov and S. Treil, M. Goldberg, and A.
Volberg, respectively in [8,14,22] that certain CZOs are bounded on
2010 Mathematics Subject Classification. 42B20.
Key words and phrases. paraproducts, weighted norm inequalities, matrix
weights.
1
2 JOSHUAISRALOWITZ,HYUNKYOUNGKWON,AND SANDRAPOTT
Lp(W) when 1 < p < ∞ if W is a matrix A weight, which means that
p
p
sup 1 1 kW1p(x)W−p1(t)kp′dt p′ dx < ∞ (1.1)
|I| |I|
I⊂Rd ZI (cid:18) ZI (cid:19)
I isacube
where p′ is the conjugate exponent of p (note that an operator T acting
on scalar functions can be canonically extended to Cn valued functions
via the action of T on its coordinate functions.)
Now note that CZOs with matrix valued kernels acting on Cn val-
ued functions appear very naturally in various branches of mathemat-
ics (and as a particular example see [10] for extensive applications of
matrix kernelled CZOs to geometric function theory.) Despite this
and the fact that the theory of matrix weights has numerous applica-
tions to Toeplitz operators, multivariate prediction theory, and even to
the study of finitely generated shift invariant subspaces of unweighted
Lp(Rd) (see [14,16,22]), virtually nothing is known regarding matrix
weighted norm inequalities for matrix kernelled CZOs or related oper-
ators.
The purpose of this series of two papers is therefore to investigate
the boundedness of matrix kernelled CZOs on Lp(W) when W is a
matrixA weight. Wewill needtointroducesomemorenotationbefore
p
we state our main result. It is well known (see [8] for example) that
for a matrix weight W, a cube I, and any 1 < p < ∞, there exists
positive definite matrices VI and VI′ such that |I|−p1kχIWp1~ekLp ≈ |VI~e|
and |I|−p1′kχIW−p1~ekLp′ ≈ |VI′~e| for any ~e ∈ Cn, where k · kLp is the
canonicalLp(Rd;Cn)normandthenotationA ≈ B asusualmeansthat
two quantities A and B are bounded above and below by a constant
multiple of each other. Note that it is easy to see that kV V′k ≥ 1
I I
for any cube I. We will say that W is a matrix A weight if the
p
product V V′ has uniformly bounded matrix norm with respect to all
I I
cubes I ⊂ Rd (note that this condition is easily seen to be equivalent
1
to (1.1).) Also note that when p = 2 we have VI = (mIW)2 and
VI′ = (mI(W−1))12 where mIW is the average of W on I, so that the
matrix A condition takes on a particularly simple form that is very
2
similar to the scalar A condition.
2
Now let T : L2(Rd;Cn) → L2(Rd;Cn) be a densely defined operator
where the dense domain contains at least the indicator function of all
cubes. If 1 < p < ∞ and W is a matrix A weight, then we will call T
p
MATRIX WEIGHTED T1 THEOREM PART I 3
a “W-weighted CZO” with associated matrix kernel K : Rd×Rd\∆ →
M (C) (where as usual ∆ ⊂ Rd ×Rd is the diagonal) if the following
n
three conditions are true: first,
~ ~
Tf(x) = K(x,y)f(y)dy, x 6∈ supp (f)
Rd
Z
~
for all f in the dense domain of T with compact support. Second,
for each cube I ⊂ Rd, assume that the matrix function V K(x,y)V−1
I I
satisfies the “standard kernel estimates”
C
|V K(x,y)V−1| ≤ ,
I I |x−y|d
|x−x′|δ
|V (K(x,y)−K(x′,y))V−1|+|V (K(y,x)−K(y,x′))V−1| ≤ C
I I I I |x−y|d+δ
for all x,x′,y ∈ Rd with |x− y| > 2|x− x′| where δ,C > 0 are inde-
pendent of I. Third, assume that T satisfies the “weak boundedness
property”
1
sup khT(1 ),1 i k < ∞
|I| I I L2
I⊂Rd
I isacube
where 1 is the indicator function of the cube I and k·k is any matrix
I
norm on M (C). Moreover, if 1 < p < ∞ and W is a matrix A
n p
p
weight, then let BMO be the space of locally integrable functions
W
B : Rd → M (C) where
n
sup |I1| kWp1(x)(B(x)−mIB)VI−1kpdx < ∞ : if 2 ≤ p < ∞
 I⊂Rd ZI
I isacube .
 sup |I1| kW−p1(x)(B∗(x)−mIB∗)(VI′)−1kp′dx < ∞ : if 1 < p ≤ 2
I⊂Rd ZI
I isacube


Our main goal in these two papers will be to prove the following

theorem
Theorem 1.1. Let 1 < p < ∞. If W is a matrix A weight and T
p
is a W -weighted CZO, then T is bounded on Lp(W) if and only if
T1 ∈ BMOp and T∗1 ∈ BMOp′ .
W W1−p′
In this paper, however, we will focus our attention towards proving
matrix weighted norm inequalities for dyadic paraproducts, which will
be used to prove Theorem 1.1 in part II. In particular, let D be a
4 JOSHUAISRALOWITZ,HYUNKYOUNGKWON,AND SANDRAPOTT
dyadic system of cubes in Rd and let {hi} be a system
I I∈D, i∈{1,...,2d−1}
of Haar functions adapted to D. Given a locally integrable function
B : Rd → M (C), define the dyadic paraproduct π with respect to a
n B
dyadic grid D by
~ ~
π f = B (m f)h (1.2)
B I I I
I∈D
X
where B is the matrix of Haar coefficients of the entries of B with
I
respect to I. In this paper will prove the following theorem
Theorem 1.2. Let 1 < p < ∞. If W is a matrix A weight then
p
π is bounded on Lp(W) if and only if B ∈ BMOp (where here the
B W
supremum defining BMOp is taken over all I ∈ D instead of all cubes
W
I.)
Let us comment that restricting oneself to W-weighted CZOs is in
fact quite natural. In particular, note that Theorem 1.1 is false for
general matrix A weights and matrix kernelled CZOs, and in the last
p
section we will construct a very simple example, for each 1 < p < ∞,
of a matrix A weight W and a matrix kernelled CZO T with T1 =
p
T∗1 = 0 but where T is not bounded on Lp(W).
Moreover, let A = {A } ⊂ M (C) be a sequence of matrices. We
I I∈D n
will then prove (see Section 4) that given a matrix A weight W, the
p
Haar multiplier
~ ~
f 7→ A f h
I I I
I∈D
X
is bounded on Lp(W) if and only if sup kV A V−1k < ∞. On the
I∈D I I I
other hand, in the last section we will exhibit a very simple example
of a sequence A and a matrix A weight W, for each 1 < p < ∞,
p
where sup kV A V−1k = ∞. Similarly in the last section we will
I⊂D I I I
construct a matrix function B ∈ BMO (the ordinary John-Nirenberg
BMO space) and a matrix A weight W for each 1 < p < ∞ where
p
B 6∈ BMOp .
W
The proof of Theorem 1.2 will require the following matrix weighted
Carleson embedding theorem, which is obviously of independent inter-
est itself.
Theorem 1.3. Let 1 < p < ∞. If W is a matrix A weight and A :=
p
{A } is a sequence of matrices, then the following are equivalent:
I I∈D
MATRIX WEIGHTED T1 THEOREM PART I 5
(a) The operator Π defined by
A
ΠAf~:= VIAImI(W−p1f~)hI
I∈D
X
is bounded on Lp(Rd;Cn)
(b)
1
sup kV A V−1k2 < ∞
|J| I I I
J∈D
I∈D(J)
X
(c) There exists C > 0 independent of J ∈ D such that
1
A∗V2A < CV2
|J| I I I J
I∈D(J)
X
if 2 ≤ p < ∞, and
1
A (V′)2A∗ < C(V′)2
|J| I I I J
I∈D(J)
X
if 1 < p ≤ 2.
Furthermore, the operator norm in (a) and the supremums in (b) and
(c) are equivalent in the sense that they are independent of the sequence
A. Finally, a matrix function B ∈ BMOp if and only if the sequence of
W
Haar coefficients of B satisfies any of the above equivalent conditions.
Despite the perhaps strange appearance of Theorem 1.3, first note
that if w is a scalar A weight, then clearly a locally integrable scalar
p
function b is in BMOp if and only if b is in BMO (i. e. the classical
W
John-Nirenberg BMO space), which in this case is also equivalent to
1
sup w(x)|b(x)−m b|dx < ∞ (1.3)
I
w(I)
I⊂Rd ZI
I isacube
where w(I) = w(x)dx. In fact, it is well known (and easy to prove)
I
that if w is a scalar A weight, then b satisfies (1.3) if and only if
R ∞
b ∈ BMO (see [13] for details.)
Furthermore, note that when p = 2, the implication (c) ⇒ (a) in
~ 1 ~
Theorem 1.3 gives us that (after replacing f with W2f and replacing
AI with (mIW)−21AI )
|A (m f~)|2 . Ckf~k2
I I L2(W)
I∈D
X
6 JOSHUAISRALOWITZ,HYUNKYOUNGKWON,AND SANDRAPOTT
whenever W is a matrix A weight and {A } is a “W-Carleson
2 I I∈D
sequence” of matrices in the sense that
A∗A < C W(x)dx
I I
I∈D(J) ZJ
X
holds for all J ∈ D.
Interestingly, note that this “weighted Carleson embedding Theo-
rem” in the scalar p = 2 setting appears as Lemma 5.7 in [17] for scalar
A weights and was implicitly used in sharp form by O. Beznosova [2]
∞
(see (2.4) and(2.5) in[2]) to prove sharp weighted norm inequalities for
scalar paraproducts. We will further discuss the connection between
Theorem 1.3 and sharp matrix weighted norm inequalities for scalar
kernelled CZOs in the final section.
The main tool for proving Theorem 1.3 will be an adaption of the
stopping time arguments from [11,19] to the matrix weighted p 6= 2
setting. Moreover, the proof will require the identification from [14,22]
of Lp(W) as a weighted Triebel-Lizorkin space for 1 < p < ∞ when W
is a matrix A weight, which more precisely says that
p
p
|V f~ |2 2
kf~kp ≈ I I χ (x) dx (1.4)
Lp(W) |I| I
Rd !
Z I∈D
X
~ ~
where f is the Haar coefficient of f. Note that we will use our stopping
I
timetogiveanewandmoreclassicalstoppingtimeproofof(1.4),which
could be thought of as the third contribution of the current paper (and
which provides a simpler approach when compared to the ones in [14,
22].) It is hoped that our proofs of Theorem 1.3 and (1.4) will convince
the reader of the overall usefulness of our stopping approach to matrix
weightednorminequalities andwillgenerateinterest inextending other
stopping time arguments to the matrix weighted setting (such as the
ones pioneered by M. Lacey, S. Petermichl, and M. C. Reguera in [12],
which will be discussed further in the last section.)
It is also hoped that the results in this series of two papers will con-
vince the reader of the following philosophy: what is true in the scalar
A /scalar CZOsetting shouldlargelybetrueinthematrixsetting after
p
one takes noncommutativity into account.
We will end this introduction by outlining the contents of each sec-
tion. In the next section we will extend the stopping time arguments in
[11,19]tothep 6= 2matrixweightedsettingandshowthatthisstopping
MATRIX WEIGHTED T1 THEOREM PART I 7
time is a decaying stopping time in the sense of [11], which will then be
used to prove (1.4). In the third section, we will prove Theorems 1.2
and 1.3 by utilizing (1.4) in conjunction with our stopping time argu-
ments. FinallyinSectionfour, wewillconstructtheexamplesdiscussed
earlier in this introduction. Moreover, we will present some interesting
open problems, including other related matrix weighted BMO spaces
p
and their possible equivalences to BMO .
W
2. Weighted Haar multipliers and stopping times
We will now describe the Haar multipliers and the stopping time
that will be needed throughout this paper. Define the constant Haar
multiplier M by
W,p
~ ~
M f := V f h .
W,p I I I
I∈D
X
Note that trivially πB is bounded on Lp(W) if and only if W1pπBW−p1
is bounded on Lp(Rd;Cn), and note that
Wp1πBW−p1 = Wp1(MW,p)−1 MW,pπBW−p1 .
(cid:16) (cid:17)
The main goal of this section will be to extend the stopping time
arguments in [11,19] to the matrix weighted setting and then use these
1
arguments to prove that Wp(MW,p)−1 is bounded and invertible on
Lp(Rd;Cn) if W is a matrix A weight, so that one only needs to deal
p
with MW,pπBW−p1 in order to prove Theorem 1.2. Furthermore, note
that dyadic Littlewood-Paley theory immediately says that the bound-
1
edness and invertibility of Wp(MW,p)−1 is equivalent to (1.4).
Now assume that W is a matrix A weight. For any cube I ∈ D, let
p
J(I) be the collection of maximal J ∈ D(I) such that
kV V−1kp > λ or kV−1V kp′ > λ (2.1)
J I 1 J I 2
forsomeλ ,λ > 1tobespecifiedlater. Also,letF(I)bethecollection
1 2
of dyadic subcubes of I not contained in any cube J ∈ J(I), so that
clearly J ∈ F(J) for any J ∈ D(I).
Let J0(I) := {I} and inductively define Jj(I) and Fj(I) for
j ≥ 1 by Jj(I) := J(J) and Fj(I) := F(J).
J∈Jj−1(I) J∈Jj−1(I)
Clearly the cubes in Jj(I) for j > 0 are pairwise disjoint. Fur-
S S
thermore, since J ∈ F(J) for any J ∈ D(I), we have that D(I) =
∞ Fj(I). We will slightly abuse notation and write J(I) for the
j=0
S S
8 JOSHUAISRALOWITZ,HYUNKYOUNGKWON,AND SANDRAPOTT
set J and write | J(I)| for | J|. We will now show
J∈J(I) J∈J(I)
that J is a decaying stopping time in the sense of [11].
S S S
Lemma 2.1. Let 1 < p < ∞ and let W be a matrix A weight. For
p
λ ,λ > 1 large enough, we have that | Jj(I)| ≤ 2−j|I| for every
1 2
I ∈ D.
S
Proof. By iteration, it is enough to prove the lemma for j = 1. For
I ∈ D, let G(I) denote the collection of maximal J ∈ D(I) such that
the first inequality (but not necessarily the second inequality) in (2.1)
holds. Then by maximality and elementary linear algebra, we have
that
1 C |I|
J = |J| . kWp1(y)V−1kpdy ≤ 1
(cid:12) (cid:12) λ I λ
(cid:12)J∈G(I) (cid:12) J∈G(I) 1 J∈G(I)ZJ 1
(cid:12) [ (cid:12) X X
(cid:12) (cid:12)
for some C > 0 only depending on n and d.
(cid:12) 1 (cid:12)
(cid:12) (cid:12)
On the other hand, let For I ∈ D, let G(I) denote the collection of
maximal J ∈ D(I) such that the second inequality (but not necessarily
the first inequality) in (2.1) holds. Theneby the matrix A condition
p
we have
′
p
C C′kWkp
J ≤ 2 kW−p1(y)VIkp′dy ≤ 2 Ap|I|
(cid:12) (cid:12) λ λ
(cid:12)(cid:12)J∈[Ge(I) (cid:12)(cid:12) 2 JX∈Ge(I)ZJ 2
(cid:12) (cid:12)
for som(cid:12)e C′ on(cid:12)ly depending on n and d. The proof is now completed
(cid:12) 2 (cid:12) ′
p
by setting λ = 4C and λ = 4C′kWkp . (cid:3)
1 1 2 2 Ap
While we will not have a need to discuss matrix A weights in
p,∞
detail in this paper, note that in fact Lemma 3.1 in [22] immediately
gives us that Lemma 2.1 holds for matrix A weights (with possibly
p,∞
larger λ of course.)
2
The next main result will be an “Lp Cotlar-Stein lemma” (Lemma
2.2) that is a vector version of Lemma 8 in [11]. We will need a few
preliminary definitions before we state this result. Fix J ∈ D with
0
side-length 1 and with 0 ∈ J and let Jj := Jj(J ) and Fj :=
0 0
Fj(J ). Now for each j ∈ N let ∆ be defined by
0 j
~ ~
∆ f := f h ,
j I I
IX∈Fj
~ ~
and write f := ∆ f.
j j
MATRIX WEIGHTED T1 THEOREM PART I 9
Lemma 2.2. Let the Fj’s be as above and write T := T∆ for any
j j
linear operator T acting on Cn valued functions defined on Rd. If T =
∞ T , and if there exists C > 0 and 0 < c < 1 such that
j=1 j
P |Tjf~|p2|Tkf~|p2 dx . c|j−k|kf~jkLp2pkf~kkLp2p
Rd
Z
for every j,k ∈ N, then T is bounded on Lp(Rd;Cn).
Proof. It follows directly from Lemma 7 in [11] and elementary linear
algebra that
∞
kf~kp . kf~kp
j Lp Lp
j=1
X
whenever f ∈ Lp(Rd;Cn). The proof of Lemma 2.2 is now identical to
the proof of Lemma 8 in [11]. (cid:3)
Theorem 2.3. Let 1 < p < ∞. If W is a matrix A weight, then
p
Wp1M−1 is bounded on Lp(Rd;Cn).
W,p
Proof. Obviously it is enough to prove that the operator T defined by
Tf~:= Wp1VI−1f~IhI
I∈XD(J0)
is bounded on Lp(Rd;Cn). Note that we also clearly have T = ∞ T .
j=1 j
For each I ∈ D, let
P
M f~:= V−1f~ h
I J J J
J∈F(I)
X
so that
~ 1 ~
Tjf = WpMIf.
I∈XJj−1
Since V M is a constant Haar multiplier and since kV V−1kp ≤ kWk
I I I J Ap
if J ∈ F(I), we immediately have that
kV M f~kp . kWk kf~kp .
I I Lp Ap Lp
Now we will show that each T is bounded. To that end, we have
j
that
|T f|pdx = |T f|pdx+ |T f|pdx
j j j
ZRd ZSJj−1\SJj ZSJj
:= (A)+(B).
10 JOSHUAISRALOWITZ,HYUNKYOUNGKWON,AND SANDRAPOTT
1
Since kWp(x)V−1kp . 1 on J\ J(J), we can estimate (A) first as
J
follows:
S
(A) = |T f~|pdx
j
J∈XJj−1ZJ\SJ(J)
= |Wp1(x)MJf~(x)|pdx
J∈XJj−1ZJ\SJ(J)
≤ kW1p(x)VJ−1kp|VJMJf~(x)|pdx
J∈XJj−1ZJ\SJ(J)
. |V M f~|pdx
J J
J∈XJj−1ZJ
. kWk kf~kp
Ap j Lp
. kWk kf~kp .
Ap Lp
~
As for (B), note that M f is constant on I ∈ J(J), and so we will
J
~
refer to this constant by M f(I). We then estimate (B) as follows:
J
(B) = |T f~|pdx
j
ZSJj
≤ |Wp1(x)MJf~|pdx
J∈XJj−1I∈XJ(J)ZI
1
≤ |I||VJMJf~(I)| |I| kWp1(x)VJ−1kpdx
J∈XJj−1I∈XJ(J) (cid:18) ZI (cid:19)
~
. |I||V M f(I)|
J J
J∈XJj−1I∈XJ(J)
= |V M f~(x)|pdx
J J
J∈XJj−1I∈XJ(J)ZI
. kWk kf~kp
Ap j Lp
. kWk kf~kp . (2.2)
Ap Lp
To finish the proof, we claim that there exists 0 < c < 1 such that
|T f~|pdx . ck−jkf~kp
j j Lp
ZSJk−1