1. No category

Sharp weighted estimates for vector-valued singular
integral operators and commutators
Carlos P´erez and Rodrigo Trujillo-Gonz´alez
Tohoku mathematics journal, 55 (2003), 109-129.
Abstract
We prove sharp weighted norm inequalities for vector-valued singular integral
operators and commutators. We first consider the strong (p, p) case with p >
1 and then the weak-type (1, 1) estimate. Our results do not assume any kind
of condition on the weight function and involve iterations of the classical HardyLittlewood maximal function.
1
Introduction and Main Results
The purpose of this paper is to sharpen the results obtained in [5] for vector valued
singular integral operators. Indeed, the method considered in that paper is based on
extrapolation ideas. This method is very general and the results hold for any kind of
operators. Furthermore, it is not possible to derive better results with such a generality.
However, we are going to show that for vector valued singular integral operators we can
improve those results.
To be more precise, we let T be a classical Calder´on-Zygmund operator with kernel
K (see Section 2.2), and let Tq , q > 0, be the vector-valued singular integral operator
associated to T by
!1/q
∞
X
Tq f (x) = |T f (x)|q =
|T fj (x)|q
,
j=1
where, by abuse of notation, we also denote by T the vector valued extension of the scalar
operator T . For any Calder´on-Zygmund singular integral operator T the first author
2000 Mathematics Subject Classification. Primary 42B20; Secondary 42B25.
Key words and phrases: singular integral operators, maximal functions, weights.
Partially supported by DGESIC Grant PB980106 and DGESIC Grant PB98044.
1
proved in [11] that whenever p > 1 and ε > 0,
Z
Z
p
|T f (y)| w(y) dy ≤ C
|f (y)|p ML(log L)p−1+ε (w)(y) dy.
Rn
Rn
See Section 2.3 for the definition and main properties of the maximal operator of ML(log L)α ,
α > 0. On the other hand, it is not hard to see that if we apply the extrapolation method
from [5, Theorem 1.4] to the vector-valued singular operator Tq , we obtain, for p > q > 1
and ε > 0,
Z
Z
p
|Tq f (y)| w(y) dy ≤ C
|f (y)|pq M
(w)(y) dy,
(1.1)
p2
−1+ε
Rn
L(log L)
Rn
q
P
1/q
∞
q
where |f (x)|q =
|f
(x)|
.
j=1 j
We show in this paper that this estimate can be improved by using different techniques.
Our result on (p, p)-type weighted estimates is the following.
Theorem 1.1 Let 1 < p, q < ∞, w(x) be a weight and Tq be a vector-valued singular
integral operator. Suppose that A(t) is a Young function satisfying the condition
Z
(1.2)
c
∞
t
A(t)
p0 −1
dt
<∞
t
for some c > 0. Then there exists a constant C > 0 such that
Z
Z
p
(1.3)
(Tq f (x)) w(x) dx ≤ C
|f (x)|pq MA (w)(x) dx.
Rn
Rn
Observe that (1.2) is independent of q and this is not the case of (1.1).
As a corollary we have the following result.
Corollary 1.2 Let 1 < p, q < ∞, w(x) be a weight and Tq be a vector-valued singular
integral operator.
a) Let ε > 0. Then there exists a constant C > 0 such that
Z
Z
p
(1.4)
(Tq f (x)) w(x) dx ≤ C
|f (x)|pq ML(log L)p−1+ε (w)(x) dx.
Rn
Rn
b) As a consequence, we have that there exists a constant C > 0 such that
Z
Z
p
(1.5)
(Tq f (x)) w(x) dx ≤ C
|f (x)|pq M [p]+1 (w)(x) dx.
Rn
Rn
Estimates (1.5) and (1.4) are sharp because they coincide with the corresponding
scalar results, where the results are already optimal [11].
2
We remark that the first author obtained in [15] an estimate similar to (1.5) (also to
(1.4)) for the vector-valued maximal operator
Mq f (x) =
∞
X
!1/q
(M fj (x))q
.
j=1
The main result from [15] is
Z
Z
p
(Mq f (x)) w(x) dx ≤ C
Rn
|f (x)|pq M [p/q]+1 (w)(x) dx.
Rn
Observe that the operator M [p]+1 is replaced by the pointwise smaller operator M [p/q]+1 .
This result is also sharp, but is different from the corresponding scalar result, namely the
celebrated Fefferman and Stein weighted estimate
Z
Z
p
(M f (x)) w(x)dx ≤ c
|f (x)|p M w(x)dx.
Rn
Rn
As in the scalar situation, the proof of Theorem 1.1 is based on the Calder´on-Zygmund
classical principle which establishes the control of the singular integral operator by the
Hardy-Littlewood maximal operator (see Theorem 1.3 below). Also our approach makes
use of a pointwise estimate between the maximal operators Mδ# and M :
Mδ# (Tq f )(x) ≤ CM (|f |q )(x),
(1.6)
where 0 < δ < 1 (see Lemma 3.1 for details). When δ = 1, estimate (1.6) is false and the
right hand side should be replaced by M (|f |rq )(x)1/r , r > 1. In this case we believe that
this estimate was known, but it is not sharp enough to derive our results.
As a consequence of (1.6), we deduce the following vector-valued version of the classical
estimate of Coifman [2] which is used in the proof of Theorem 1.1.
Theorem 1.3 Let 1 < q < ∞ and 0 < p < ∞. Let w(x) be a weight satisfying the
A∞ condition. Then the following a priori estimate holds: there exists a positive constant
C = C[w]A such that
∞
Z
Z
p
(Tq f (x)) w(x) dx ≤ C
(1.7)
Rn
(M (|f |q )(x))p w(x) dx
Rn
for any smooth function f for which the left hand side is finite. Similarly, we have that
there exists a constant C > 0 such that
(1.8)
kTq f kLp,∞ (w) ≤ C kM (|f |q )kLp,∞ (w)
for any smooth vector function f for which the left hand side is finite.
3
We remark that it is not clear how to prove (1.7) adapting the good-λ inequality
derived in [2].
The weighted weak-type (1, 1) estimate version of (1.3) is the following.
Theorem 1.4 Let 1 < q < ∞ and ε > 0. Then there exists a constant C > 0 such that
for any weight w and λ > 0
Z
C
n
w({y ∈ R : |Tq f (y)| > λ}) ≤
|f (x)|q ML(log L)ε (w)(x) dx,
λ Rn
where f is an arbitrary smooth vector function.
Remark 1.5 As above this result is a vector valued extension of the corresponding scalar
result [11]. When T is replaced by M , the weight on the right hand side is the best possible,
namely M w ([15]). The result can be sharpened by replacing the maximal operator
ML(log L)ε by the maximal operator MA , where A is any Young function satisfying (1.2)
for all p > 1.
We also consider in this paper vector-valued extensions of the by now classical commutator of Coifman-Rochberg-Weiss [h, T ] defined by the formula
Z
[h, T ]f (x) = h(x)T f (x) − T (hf )(x) =
(h(x) − h(y))K(x, y)f (y) dy.
Rn
Here h is a locally integrable function and is usually called the symbol of the operator. T
is any Calder´on-Zygmund operator with kernel K. The main result from [3] establishes
that, whenever the symbol h is a B.M.O. function, the commutator is bounded on Lp (Rn ),
p > 1. Later on this result was extended to the case Lp (w), w ∈ Ap . The first author
has shown in [14] that there is a version of Coifman’s estimate [2] where the role played
by the maximal function M is replaced by M 2 = M ◦ M . This is also the point of view
of [12], where it is shown that commutators with B.M.O. functions are not of weak type
(1, 1) but that they satisfy a L(log L) type estimate. These results show that somehow
commutators with B.M.O. functions carry a higher degree of singularity. We will extend
these estimates to the vector-valued context.
As above, for a sequence f (x) = {fj (x)}∞
j=1 of functions, the vector–valued version of
the commutator [h, T ] is given by the expression
[h, T ]q f (x) = |[h, T ]f (x)|q =
∞
X
!1/q
|[h, T ]fj (x)|q
.
j=1
As in the case of singular integrals, we first need an appropriate version of the
Calder´on-Zygmund principle. The precise estimate is given as follows.
4
Theorem 1.6 Let 1 < q < ∞, 0 < p < ∞, w ∈ A∞ and h ∈ BM O. Then there exists a
constant C = C[w]
such that
A∞
Z
p
p
Z
([h, T ]q f (x)) w(x) dx ≤ C khk
(1.9)
Rn
BM O
Rn
(ML log L (|f |q )(x))p w(x) dx
for any smooth vector function f such that the left hand side is finite.
The proof of this theorem is also based on a pointwise estimate very much in the spirit
of the pointwise estimate (1.6) required for the proof of Theorem 1.3, namely
(1.10)
Mδ# ([h, T ]q f )(x) ≤ CkhkBM O Mε (Tq f )(x) + ML log L (|f |q )(x) ,
where 0 < δ < ε (see Lemma 3.2). This time there is an extra term involving the maximal
operator ML log L , which is pointwise comparable to M 2 . This is optimal and explains why
commutators have a higher degree of singularity.
By arguing as in [14, Theorem 2], we obtain the sharp two-weight estimates where no
assumption is assumed on the weight w.
Theorem 1.7 Let 1 < p, q < ∞, δ > 0 and h ∈ BM O. Then there exists a constant
C > 0 such that for each weight w
Z
Z
p
p
([h, T ]q f (x)) w(x) dx ≤ Ckhk
|f (x)|pq ML(log L)2p−1+δ (w)(x) dx,
(1.11)
Rn
BM O
Rn
where f = {fi }∞
i=1 is any sequence of bounded functions with compact support.
As in the singular integral operator case, (1.11) improves the result obtained for [h, T ]q
as an application of the general extrapolation theorem from [5] derived from the scalar
estimate given in [14, Theorem 2].
The weighted weak-type (1, 1) estimate version of (1.11) is the following.
Theorem 1.8 Let 1 < q < ∞, ε > 0 and h ∈ BM O. Then there exists a constant C > 0
such that for any weight w and λ > 0
Z
|f (x)|q
n
w({x ∈ R : |[h, T ]q f (x)| > λ}) ≤ C
Φ(khkBM O
) ML(log L)1+ (w)(x)dx,
λ
Rn
where Φ(t) = t log(e + t). The constant C is independent of the weight w, f and λ > 0.
This result is a vector valued version of the main result proved in [16].
2
Preliminaries
In this section we introduce the basic tools needed for the proof of the main results.
5
2.1
Ap weights and maximal operators
By a weight we mean a positive and locally integrable function. We say that a weight w
belongs to the class Ap , 1 < p < ∞, if there is a constant C such that
1
|Q|
Z
w(y) dy
Q
1
|Q|
Z
1−p0
w(y)
p−1
≤C
dy
Q
for each cube Q and where as usual 1/p + 1/p0 = 1. A weight w belongs to the class A1
if there is a constant C such that
1
|Q|
Z
w(y) dy ≤ C inf w.
Q
Q
We will denote the infimum of the constants C by [w]Ap . Observe that [w]Ap ≥ 1 by
H¨older’s inequality.
Since the Ap classes are increasing with respect to p, the A∞ class of weights is defined
S
in a natural way by A∞ = p>1 Ap . However, the following characterization is more
interesting in applications: there are positive constants c and ρ such that for any cube Q
and any measurable set E contained in Q
ρ
|E|
w(E)
.
≤c
w(Q)
|Q|
We recall now the definitions of classical maximal operators. If, as usual, M denotes
the Hardy–Littlewood maximal operator, we consider for δ > 0
δ
1/δ
Mδ f (x) = M (|f | )(x)
1
M (f )(x) = sup inf
Q3x c |Q|
#
=
1
sup
Q3x |Q|
Z
δ
|f (y)| dy
1/δ
,
Q
Z
1
|f (y) − c| dy ≈ sup
Q3x |Q|
Q
Z
|f (y) − (f )Q | dy,
Q
where as usual (f )Q denotes the average of f on Q, and a variant of this sharp maximal
operator, which will become the main tool in our scheme, Mδ# f (x) = M # (|f |δ )(x)1/δ .
The main inequality between these operators to be used is a version of the classical
one due to Fefferman and Stein (see [6], [9]).
Theorem 2.1 Let 0 < p, δ < ∞ and w ∈ A∞ . There exists a positive constant C such
that
Z
Z
p
p
(2.1)
Mδ f (x) w(x)dx ≤ C[w]
Mδ# f (x)p w(x)dx
A∞
Rn
Rn
for every function f such that the left hand side is finite.
6
2.2
Calder´
on-Zygmund operators
By a kernel K in Rn × Rn we mean a locally integrable function defined away from the
diagonal. We say that K satisfies the standard estimates if there exist positive and finite
constants γ and C such that, for all distinct x, y ∈ Rn and all z with 2|x − z| < |x − y|,
it verifies
i) |K(x, y)| ≤ C|x − y|−n ,
x−z γ
ii) |K(x, y) − K(z, y)| ≤ C x−y
|x − y|−n ,
x−z γ
iii) |K(y, x) − K(y, z)| ≤ C x−y
|x − y|−n .
We define a linear and continuous operator T : C0∞ (Rn ) → D0 (Rn ) associated to the
kernel K by
Z
K(x, y)f (y)dy,
T f (x) =
Rn
where f ∈ C0∞ (Rn ) and x is not in the support of f . T is called a Calder´on-Zygmund
operator if K satisfies the standard estimates and if it extends to a bounded linear operator
on L2 (Rn ). It is well known that under these conditions T can be extended to a bounded
operator on Lp (Rn ), 1 < p < ∞ and is of weak type–(1,1). For more information on this
subject see [1],[4],[6] or [9].
We next define the vector-valued singular operator Tq associated to the operator T by
Tq f (x) = |T f (x)|q =
∞
X
!1/q
q
|T fj (x)|
.
j=1
It is well-known that, for 1 < q < ∞, Tq is of type-(p, p), 1 < p < ∞, and weak type(1,1). Moreover, the Ap condition also implies the corresponding weighted estimate. For
a complete study on these results see [8, Chapter V].
2.3
Orlicz maximal functions
By a Young function A(t) we shall mean a continuous, nonnegative, strictly increasing
and convex function on [0, ∞) with A(0) = 0 and A(t) → ∞ as t → ∞. In this paper any
Young function A will be doubling, namely A(2t) ≤ C A(t) for t > 0.
We define the A-averages of a function f over a cube Q by
Z
1
|f (x)|
kf kA,Q = inf λ > 0 :
A
dx ≤ 1 .
|Q| Q
λ
7
An equivalent norm, which is often useful in calculations, is (see [10, p. 92] or [17, p.
69]):
Z
µ
|f |
(2.2)
kf kA,Q ≤ inf µ +
A
dx ≤ 2kf kA,Q .
µ>0
|Q| Q
µ
If A, B and C are Young functions such that
A−1 (t)B −1 (t) ≤ C −1 (t),
then
kf gkC,R ≤ 2kf kA,R kgkB,R .
The examples to be considered in our study will be A−1 (t) = log(1+t), B −1 (t) = t/ log(e+
t) and C −1 (t) = t. Then A(t) ≈ et and B(t) ≈ t log(e+t) which gives the H¨older inequality
Z
1
(2.3)
|f g|dx ≤ Ckf kexpL,Q kgkL log L,Q .
|Q| Q
For these examples we recall that, if h ∈ BM O and (h)Q denotes its average on the
cube Q, then
(2.4)
kh − (h)Q kexpL,Q ≤ CkhkBM O
by the classical John-Nirenberg inequality.
Associate to this average, for any Young function A(t) we can define a maximal operator MA given by
MA f (x) = sup kf kA,Q ,
Q3x
where the supremum is taken over all the cubes containing x.
The following result from [13] will be very useful.
Theorem 2.2 Let 1 < p < ∞. Suppose that A is a Young function. Then the following
are equivalent:
i) There exists a positive constant c such that
p−1
Z ∞
t
dt
(2.5)
< ∞.
A(t)
t
c
ii) There exists a constant C such that
Z
Z
w(x)
M w(x)
p
(2.6)
M f (x)
dx ≤ C
f (x)p
dx
p−1
[MA (u)(x)]
u(x)p−1
Rn
Rn
for all non-negative, locally integrable functions f and all weights w and u.
8
3
Pointwise estimates
In this section we prove the basic pointwise estimates for the vector-valued singular integral operator and commutator.
Lemma 3.1 Let 1 < q < ∞ and 0 < δ < 1. Then there exists a constant C > 0 such
that
(3.7)
Mδ# (Tq f )(x) ≤ CM (|f |q )(x)
n
for any smooth vector function f = {fj }∞
j=1 and for every x ∈ R .
Proof. Let f = {fj } be any smooth vector function. Fix x ∈ Rn and let B be a ball
centered at x of radius r. Decompose f = f 1 + f 2 , where f 1 = f χ2B = {fj χ2B }. As
usual, 2B denotes the ball concentric with B and radius two times the radius of B. Set
!1/q
∞
X
c = |(T f 2 )B |q =
|(T fj2 )B |q
.
j=1
Since for any 0 < r < ∞ it follows
(3.8)
|αr − β r | ≤ Cr |α − β|r
for any α, β ∈ C with Cr = max{1, 2r−1 }, we can estimate
1/δ
1/δ
Z
Z
δ
1
1
δ
δ
2
|Tq f (y)| − c dy
|T f (y)|q − |(T f )B |q dy
≤
|B| B
|B| B
1/δ
Z
δ
1
2
≤
T f (y) − (T f )B q dy
|B| B
"
1/δ
Z
1 δ
1
T f (y) dy
≤ C
q
|B| B
1/δ #
Z
2
1
T f (y) − (T f 2 )B δ dy
+
q
|B| B
= I + II.
For I we recall that Tq is weak type-(1,1). Then by Kolmogorov’s inequality ([18, p. 104]),
1
kTq f 1 kL1,∞
|B|
Z
C
≤
|f (y)|q dy
|2B| 2B
I ≤ C
(3.9)
≤ CM (|f |q )(x).
9
To estimate II we will use Jensen’s inequality, the definition of T , the basic estimates
of the kernel K and Minkowski’s inequality to obtain the following:
C
II ≤
|B|
Z
C
=
|B|
Z
C
=
|B|
Z
C
=
|B|
Z
B
2
T f (y) − (T f 2 )B dy
q
∞
X
2
T fj (y) − (T fj2 )B q
B
q !1/q
Z
∞ X
1
2
2
(T
f
(y)
−
T
f
(z))dz
dy
j
j
|B|
B
B
j=1
Z Z
∞ X
1
|B|
B
B
j=1
Z Z Z
C 1
≤
|B| |B|
Z Z Z
B
B
B
B
Rn \2B
Rn \2B
∞ Z
X
k=1
2k r≤|w−x|<2k+1 r
Rn \2B
∞
X
q !1/q
(K(y, w) − K(z, w))fj (w) dwdz dy
!1/q
q
q
|K(y, w) − K(z, w)| |fj (w)|
dwdzdy
j=1
!1/q
∞ X
y − z γq
1
q
dwdzdy
y − w |y − w|nq |fj (w)|
j=1
γq
∞ X
2r
1
|fj (w)|q
k
k
nq
2 r
(2 r)
j=1
Z
∞
X
1
1
≤ C
2kγ (2k+1 r)n 2k+1 B
k=1
(3.10)
dy
j=1
C 1
≤
|B| |B|
≤ C
!1/q
∞
X
!1/q
dw
!1/q
|fj (w)|q
dw
j=1
≤ CM (|f |q )(x)
Finally, (3.7) follows from (3.9) and (3.10) and the proof of the lemma is concluded.
2
As mentioned in the introduction we need a similar estimate for the commutator.
Lemma 3.2 Let h ∈ BM O and let 0 < δ < ε. Then there exists a constant C > 0 such
that
(3.11)
Mδ# ([h, T ]q f )(x) ≤ CkhkBM O Mε (Tq f )(x) + ML log L (|f |q )(x)
n
for any smooth vector function f = {fj }∞
j=1 and for every x ∈ R .
10
Proof. Observe that for any constant λ
[h, T ]f (x) = (h(x) − λ)T f (x) − T ((h − λ)f )(x).
As above we fix x ∈ Rn and let B be a ball centered at x of radius r > 0. We split
f = f 1 + f 2 , where f 1 = f χ2B = {fj χ2B }. Let λ be a constant and c = {cj }∞
j=1 a
sequence of constants to be fixed along the proof.
By (3.8) we have
1
|B|
Z
|[h, T ]q f (y)|δ − |c|δq dy
1/δ
≤ Cδ
B
≤ Cδ
1
|B|
Z
1
|B|
Z
1/δ
δ
||[h, T ]f (y)|q − |c|q | dy
B
|[h, T ]f (y) −
c|δq
1/δ
dy
B
1/δ
Z
1
δ
= Cδ
|(h(y) − λ)T f (y) − T ((h − λ)f )(y) − c|q dy
|B| B
"
1/δ
Z
1
δ
|(h(y) − λ)T f (y)|q dy
≤ Cδ
|B| B
+
+
1
|B|
Z
1
|B|
Z
T ((h − λ)f 1 )(y)δ dy
1/δ
q
B
T ((h − λ)f 2 )(y) − cδ dy
1/δ #
q
B
= I + II + III.
To deal with I, we first fix λ = (h)2B , the average of h on 2B. Then, for any
1 < p < ε/δ, we have
I = Cδ
≤ Cδ
1
|B|
Z
1
|2B|
δ
B
|h(y) − (h)2B |
Z
|T f (y)|δq dy
p0 δ
|h(y) − (h)2B | dy
2B
≤ CkhkBM O Mδp (Tq f )(x)
(3.12)
≤ CkhkBM O Mε (Tq f )(x).
11
1/δ
1/p0 δ 1
|B|
Z
pδ
(Tq f (y)) dy
B
1/pδ
For II we make use of Kolmogorov’s inequality again. Then
Z
1
II ≤ C
|h(y) − (h)2B ||f 1 (y)|q dy
|B| B
Z
1
≤ C
|h(y) − (h)2B ||f (y)|q dy
|2B| 2B
≤ Ckh − (h)2B kexpL,2B k|f |q kL log L,2B
(3.13)
≤ CkhkBM O ML log L (|f |q )(x),
where we have used (2.3) and (2.4).
Finally, for III we first fix the value of c by taking c = {(T ((h − (h)2B )fj2 ))B }∞
j=1 , the
2
average of each T ((h − (h)2B )fj ) on B. Then, by Jensen and Minkowski’s inequalities,
respectively, and the basic estimates of the kernel K, we have
12
1
III ≤ Cδ
|B|
Z
1
|B|
Z
1
= Cδ
|B|
Z
= Cδ
=
C
|B|
B
∞
X
T (h − (h)2B )fj2 )(y) − (T (h − (h)2B )fj2 )B q
B
B
B
j=1
B
j=1
Z Z Z
B
B
Rn \2B
∞ Z
X
k=1
dy
q !1/q
Z
∞ X
1
2
2
(z)
dz
dy
)(y)
−
T
(h
−
(h)
)f
T
(h
−
(h)
)f
2B
2B
j
j
|B|
Z Z
∞ X
1
|B|
Z
!1/q
j=1
B
C 1
≤
|B| |B|
= C
T ((h − λ)f 2 )(y) − c dy
q
2k r≤|w−x|<2k+1 r
Rn \2B
q !1/q
(K(y, w) − K(z, w))(h(w) − (h)2B )fj (w) dwdz dy
!1/q
∞ X
y − z γq
1
q
dwdzdy
y − w |y − w|nq |(h(w) − (h)2B )fj (w)|
j=1
2r
2k r
γ
1
|h(w) − (h)2B ||f (w)|q dw
(2k r)n
Z
∞
X
1
1
≤ C
|h(w) − (h)2B ||f (w)|q dw
2kγ (2k+1 r)n 2k+1 B
k=1
Z
∞
X
1
1
≤ C
|h(w) − (h)2k+1 B ||f (w)|q dw
2kγ (2k+1 r)n 2k+1 B
k=1
Z
∞
X
1
1
|(h)2k+1 B − (h)2B | k+1 n
|f (w)|q dw
+C
2kγ
(2 r) 2k+1 B
k=1
∞
X
1
≤ C
kh − (h)2k+1 B kexpL,2k+1 B k|f |q kL log L,2k+1 B
kγ
2
k=1
!
∞
X
k
+CkhkBM O M (|f |q )(x)
2kγ
k=1
≤ CkhkBM O ML log L (|f |q )(x),
(3.14)
where in the last inequality we have used that |(h)2k+1 B − (h)2B | ≤ 2kkhkBM O .
From (3.12), (3.13) and (3.14) we get (3.11) and the proof is finished.
13
2
4
Proof of the theorems
4.1
Proof of Theorem 1.3
In order to prove
Z
p
Z
(Tq f (x)) w(x) dx ≤ C
(4.15)
Rn
(M (|f |q )(x))p w(x) dx,
Rn
we make some reductions. First, we assume that the right hand side of (4.15) is finite,
since otherwise there is nothing to prove. Next we restrict to a finite number of elements
fm = (f1 , f2 , · · · , fm , 0, · · · ) and prove (4.15) with a constant independent of m. Then we
let m go to ∞. To apply Theorem 2.1, take it for granted that
Z
(4.16)
(Mδ (Tq (fm ))(x))p w(x) dx < ∞.
Rn
Then, since w ∈ A∞ , we can combine Theorem 2.1 together with Lemma 3.1 with 0 <
δ < 1 to get
Z
Z
p
(Tq fm (x)) w(x) dx ≤
(Mδ (Tq fm )(x))p w(x) dx
n
Rn
RZ
p
#
Mδ (Tq fm )(x) w(x) dx
≤ C
n
ZR
≤ C
(M (|fm |q )(x))p w(x) dx.
Rn
It only remains to show (4.16). Indeed, since w ∈ A∞ , there exists r > 1 such that w ∈ Ar
and we can choose δ small enough so that p/δ > r. Then, by Muckenhoupt’s theorem, all
is reduced to checking that kTq fm kLp (w) < ∞. Now, by the classical Coifman [2] estimate
we have
!p/q
Z
m
m Z
X
X
q
|T fj (x)|
w(x) dx ≤ Cm
|T fj (x)|p w(x) dx
Rn
j=1
≤ Cm
n
j=1 R
Z
m
X
j=1
Z
≤ Cm
(M (fj )(x))p w(x) dx
Rn
(M (|f |q )(x))p w(x) dx.
Rn
2
The proof of the theorem is complete.
4.2
Proof of Theorem 1.1
We want to show that the vector valued extension of T is a bounded operator from
Lplq (MA w(x)) into Lplq (w) (the definition of Lplq (µ) is standard, see [8, Chapter V]). A
14
simple duality argument shows that this is equivalent to see that the adjoint operator T ∗
0
0
0
0
is bounded from Lplq0 (w1−p ) into Lplq0 ((MA w(x))1−p ).
So, the estimate to be established is
Z
Z
p0
0
0
∗
1−p0
(4.17)
Tq0 f (x) (MA w(x))
dx ≤ C
|f (x)|pq0 w(x)1−p dx.
Rn
Rn
As above we may restrict to a finite number of elements fm = (f1 , f2 , · · · , fm , 0, · · · ) and
0
show the estimate with a constant independent of m. First we note that (MA w(x))1−p ∈
A∞ (see [11, p. 300]). Thus, since T ∗ is also a Calder´on-Zygmund operator, we can apply
Theorem 1.3 combined with Theorem 2.2 to deduce
Z
Z
p0
0
0
∗
1−p0
Tq0 f (x) (MA w(x))
dx ≤ C
(M (|f |q0 )(x))p (MA w(x))1−p dx
n
Rn
ZR
0
0
≤ C
|f (x)|pq0 w(x)1−p dx,
Rn
whenever
Z
Rn
p0
0
Tq∗0 f (x) (MA w(x))1−p dx < ∞.
To show this we use an argument similar to the proof of Theorem 1.3, where now we make
use of the scalar version of (1.3) derived in [11], since we are assuming that
Z
∞
c
t
A(t)
p0 −1
dt
< ∞.
t
2
4.3
Proof of Theorem 1.4
Fix λ > 0 and let {Qj } be the standard family of nonoverlapping dyadic cubes satisfying
Z
1
(4.18)
λ<
|f (x)|q dx ≤ 2n λ,
|Qj | Qj
maximal with respect to left hand side inequality. Denote by zj and rj the center and
S
side-length of each Qj , respectively. As usual, if we denote Ω = j Qj , then |f (x)|q ≤ λ
a.e. x ∈ Rn \ Ω.
Now we proceed to construct an slightly different version of the classical Calder´onZygmund decomposition. Split f as f = g + b, where g = {gi }∞
i=1 is given by
(
fi (x) for x ∈ Rn \ Ω,
gi (x) =
(fi )Qj for x ∈ Qj ,
15
being, as usual, (fi )Qj the average of fi on the cube Qj , and
∞


X
∞
bij (x)
b(x) = {bi (x)}i=1 =


Qj
i=1
e = ∪j 2Qj . We then have
with bij (x) = (fi (x) − (fi )Qj )χQj (x). Let Ω
e : |Tq g(y)| > λ/2})
w({y ∈ Rn : |Tq f (y)| > λ}) ≤ w({y ∈ Rn \ Ω
e
+w(Ω)
e : |Tq b(y)| > λ/2}).
+w({y ∈ Rn \ Ω
(4.19)
For the first term we invoke Theorem 1.1. Let ε > 0. By choosing 1 < p < 1 + ε, we
have that Aε (t) = t logε (1 + t) satisfies (1.2). Thus,
Z
C
n
e
w({y ∈ R \ Ω : |Tq g(y)| > λ/2}) ≤
(Tq g(y))p w(y) dy
λp Rn \Ωe
Z
C
≤
|g(y)|pq ML(log L)ε (χRn \Ωe w)(y) dy
p
λ Rn
Z
C
≤
|f (y)|pq ML(log L)ε (w)(y) dy
λp Rn \Ω
Z
C
+ p
|g(y)|pq ML(log L)ε (χRn \Ωe w)(y) dy
λ Ω
= I + II
The estimate of I is immediate; since |f (x)|q ≤ λ a.e. x ∈ Rn \ Ω,
Z
C
I≤
|f (y)|q ML(log L)ε (w)(y) dy.
λ Rn
For II, taking into account that for any j
ML(log L)ε (χRn \2Qj w)(y) ≈ ML(log L)ε (χRn \2Qj w)(z)
for all y, z ∈ Qj (see [11, p. 303]), we have by Minkowski’s inequality
16
Z
C X
II =
|g(y)|pq ML(logL)ε (χRn \Ωe w)(y) dy
λp Q Qj
j
!p/q
Z
∞
X
q
C X
(fi )Q =
ML(log L)ε (χRn \Ωe w)(y) dy
j
λp Q Qj i=1
j
q !p/q
Z
∞
C X X 1
|Qj | inf ML(log L)ε (χRn \2Qj w)(y)
≤
f
(z)
dz
i
p
y∈Qj
|Qj | Qj
λ Q
i=1
j
!p
Z
C X
1
≤
|f (z)|q dz |Qj | inf ML(log L)ε (w)(y)
y∈Qj
λp Q
|Qj | Qj
j
!
Z
1
CX
≤
|f (z)|q dz |Qj | inf ML(log L)ε (w)(y)
y∈Qj
λ Q
|Qj | Qj
j
Z
CX
|f (z)|q ML(log L)ε (w)(z)dz
≤
λ Q Qj
j
Z
C
≤
(4.20)
|f (z)|q ML(log L)ε (w)(z)dz.
λ Rn
where the fifth inequality follows by (4.18).
For the second term of (4.19) we proceed as follows. Again by (4.18)
e ≤ C
w(Ω)
X w(2Qj )
Qj
(4.21)
|2Qj |
|2Qj |
Z
C X w(2Qj )
|f (y)|q dy
≤
λ Q |2Qj | Qj
j
Z
CX
≤
|f (y)|q M w(y)dy
λ Q Qj
j
Z
C
≤
|f (y)|q ML(log L)ε w(y)dy
λ Rn
since M w(y) ≤ ML(log L)ε w(y).
Finally, for the third term of (4.19) we recall that each bij has zero average on Qj .
Hence, if zj denotes the center of Qj , we have
e : |Tq b(y)| > λ/2}) ≤
w({y ∈ Rn \ Ω
17
C
≤
λ
Z
C
=
λ
Z
=
C
λ
=
C
λ
≤
C
λ
Tq b(y)w(y) dy
e
Rn \Ω
"
e
Rn \Ω
#1/q
∞
X
|T bi (y)|q
w(y) dy
i=1
q 1/q

Z
∞ X Z
X
 w(y) dy

K(y,
z)b
(z)
dz
ij
n
e
R \Ω
Qj
i=1 Qj
q 1/q

Z
Z
∞
X X
 w(y) dy

(K(y,
z)
−
K(y,
z
))b
(z)
dz
j
ij
e
Rn \Ω
Q
j
i=1 Qj

"∞
#1/q 
Z
XZ
X

|K(y, z) − K(y, zj )|q |bij (z)|q
dz  w(y) dy
e
Rn \Ω
Z
CX
≤
λ Q Qj
Qj
Qj

Z

i=1
"
Rn \2Qj
j
∞
X
#1/q
|K(y, z) − K(y, zj )|q |bij (z)|q

w(y) dy  dz
i=1
!" ∞
#1/q
X
z − zj γ
1
w(y) dy
|bij (z)|q
dz
n
|x − y|
Rn \2Qj z − y
i=1
j
!" ∞
#1/q
γ
Z
∞ Z
X
X
2rj
CX
1
≤
w(y) dy
|bij (z)|q
dz
λ Q Qj k=1 2k rj ≤|y−zj |<2k+1 rj 2k rj
(2k rj )n
i=1
j
#1/q
Z "X
∞
CX
(4.22)≤
|bij (z)|q
M (χRn \2Qj w)(z)dz
λ Q Qj i=1
j
#
"Z
Z
CX
|g(z)|q M (χRn \2Qj w)(z)dz
|f (z)|q M (χRn \2Qj w)(z)dz +
≤
λ Q
Qj
Qj
Z
CX
≤
λ Q Qj
Z
j
= III + IV.
Trivially,
C
III ≤
λ
Z
Rn
|f (z)|q ML(log L)ε w(z)dz.
18
On the other hand,
IV
(4.23)
q #1/q Z
"∞ Z
C X X 1
≤
M (χRn \2Qj w)(y) dy
fi (z)dz |Qj | Qj
λ Q
Qj
i=1
j
Z
CX 1
≤
|f (z)|q dz|Qj | inf M (χRn \2Qj w)(y)
y∈Qj
λ Q |Qj | Qj
j
Z
CX 1
|f (z)|q dz |Qj | inf M (w)(y)
≤
y∈Qj
λ Q |Qj | Qj
j
Z
CX
≤
|f (z)|q M (w)(z)dz
λ Q Qj
j
Z
C
|f (z)|q M (w)(z)dz
≤
λ Rn
Z
C
≤
|f (z)|q ML(log L)ε (w)(z)dz,
λ Rn
2
concluding the proof of the theorem.
4.4
Proof of Theorem 1.6
As in the proof of Theorem 1.3, we fix f = fm = (f1 , f2 , · · · , fm , 0, · · · ) with a finite
amount of smooth functions components with compact support. We also assume that the
right hand side of (1.9) is finite, since otherwise there is nothing to prove.
We will prove the estimates with constant independent of the number of elements of
f . By (2.1), since w ∈ A∞ , it follows from Lemma 3.2 and Lemma 3.1, for 0 < δ < ε < 1,
that
k[h, T ]q f kLp (w) ≤ kMδ ([h, T ]q f )kLp (w)
≤ C[w]A∞ kMδ# ([h, T ]q f )kLp (w)
≤ C[w]A∞ khkBM O kMε (Tq f )kLp (w) + kML log L (|f |q )kLp (w)
2
#
≤ C[w]A khkBM O kMε (Tq f )kLp (w) + kML log L (|f |q )kLp (w)
∞
≤ C[w]2A khkBM O kM (|f |q )kLp (w) + kML log L (|f |q )kLp (w)
∞
≤ C[w]2A khkBM O kML log L (|f |q )kLp (w) ,
∞
whenever we are able to prove that kMδ ([h, T ]q f )kLp (w) and kMε (Tq f )kLp (w) are both finite
as Theorem 2.1 requires.
19
Since w ∈ A∞ , there exists r > 1 such that w ∈ Ar and we can choose δ and ε small
enough so that p/δ, p/ε > r. Then, by Muckenhoupt’s theorem, all is reduced to check
that k[h, T ]q f kLp (w) < ∞ and kTq f kLp (w) < ∞. But this is a consequence of the scalar
situation: k[h, T ]gkLp (w) ≤ kML log L (g)kLp (w) < ∞ [14] and kT gkLp (w) ≤ kM gkLp (w) < ∞
[2], when g is a smooth function with compact support, since the amount of elements on
f is finite. Recall that the right hand side of (1.9) is finite. The theorem is proved.
2
4.5
Proof of Theorem 1.8
A simple homogeneity argument shows that we may assume that khkBM O = 1, and with
this assumption it suffices to show that
Z
|f (x)|q
n
) ML(log L)1+ (w)(x)dx,
w({x ∈ R : |[h, T ]q f (x)| > λ}) ≤ C
Φ(
λ
Rn
where Φ(t) = t log(e + t).
We proceed as in the proof of Theorem 1.4 using essentially the same notation. Let
{Qj } be the family of non-overlapping dyadic cubes which are maximal with respect to
the condition
Z
1
(4.24)
λ<
|f (x)|q dx ≤ 2n λ.
|Qj | Qj
S
For each j we let zj and rj be the center and side-length of Qj . If we denote Ω = j Qj ,
then |f (x)|q ≤ λ a.e. x ∈ Rn \ Ω.
Split f as f = g + b, where g = {gi }∞
i=1 is given by
(
fi (x) for x ∈ Rn \ Ω,
gi (x) =
(fi )Qj for x ∈ Qj .
(fi )Qj being, as usual, the average of fi on the cube Qj , and

∞
X

∞
b(x) = {bi (x)}i=1 =
bij (x)


Qj
i=1
e = ∪j 2Qj . We then have
with bij (x) = (fi (x) − (fi )Qj )χQj (x). Let Ω
e : |[h, T ]q g(y)| > λ/2}) + w(Ω)
e
w({y ∈ Rn : |[h, T ]q f (y)| > λ}) ≤ w({y ∈ Rn \ Ω
e : |[h, T ]q b(y)| > λ/2}).
(4.25)
+w({y ∈ Rn \ Ω
20
Let ε > 0. We use Theorem 1.7, with p and δ such that 1 < p < 1 + ε/2 and
δ = − 2(p − 1) > 0. Then
Z
C
n
e : |[h, T ]q g(y)| > λ/2}) ≤
w({y ∈ R \ Ω
([h, T ]q g(y))p w(y) dy
λp Rn \Ωe
Z
C
|g(y)|pq ML(log L)1+ε (χRn \Ωe w)(y) dy
≤
p
λ Rn
Z
C
≤
|f (y)|pq ML(log L)1+ε (w)(y) dy
λp Rn \Ω
Z
C
+ p
|g(y)|pq ML(log L)1+ε (χRn \Ωe w)(y) dy
λ Ω
= I + II.
The estimate of I is immediate; since |f (x)|q ≤ λ a.e. x ∈ Rn \ Ω,
Z
Z
C
|f (y)|q
ML(log L)1+ (w)(y)dy.
I≤
|f (y)|q ML(log L)1+ε (w)(y) dy ≤ C
Φ
λ Rn
λ
Rn
For II we proceed as in the proof of (4.20) obtaining
Z
C
II ≤
|f (y)|q ML(log L)1+ε (w)(y)dy
λ Rn
Z
|f (y)|q
Φ
≤ C
ML(log L)1+ (w)(y)dy.
λ
Rn
For the second term of (4.25) we proceed as in the proof of (4.21). Then
Z
C
e
|f (y)|q ML(log L)1+ε (w)(y)dy
w(Ω) ≤
λ Rn
Z
|f (y)|q
Φ
≤ C
ML(log L)1+ (w)(y)dy.
λ
Rn
Finally, taking into account the following decomposition
q !1/q
X X
[h, T ]q b(x) =
(h(x) − (h)Qj )T (bij )(x) − T ((h − (h)Qj )bij )(x)
i
j
!1/q
≤
X
j
|h(x) − (h)Qj |
X
|T (bij )(x)|q
i
q #1/q
X X
+
T ( (h − (h)Qj )bij )(x)
"
i
j
= A(x) + B(x),
the third term of (4.25) is estimated by
λ
λ
n
n
e
e
+w
x ∈ R \ Ω : B(x) >
= IV + V.
w
x ∈ R \ Ω : A(x) >
4
4
21
Using the standard estimates of the kernel K and the cancellation of bij over Qj we have
IV
≤
!1/q
XCZ
λ
j
|h(x) − (h)Qj |
X
e
Rn \Ω
C
≤
λ
XZ
C
≤
λ
XZ
C
≤
λ
XZ
i
|K(x, y) − K(x, zj )|
Qj
!1/q
X
|bij (y)|q
X
|bij (y)|q
w(x)dy dx
i
!
Z
|K(x, y) − K(x, zj )||h(x) − (h)Qj |w(x) dx
Rn \2Qj
i
!1/q
X
Qj
j
!1/q
|h(x) − (h)Qj |
Qj
j
w(x)dx
Z
Rn \2Qj
j
|T (bij )(x)|q
|bij (y)|q
i
∞ Z
X
!
y − zj γ
1
x − y |x − y|n |h(x) − (h)Qj |w(x) dx dy
k r ≤|x−z |<2k+1 r
2
j
j
j
k=1

!1/q 
Z
X
X
C

|bij (y)|q
dy 
≤
λ j
Qj
i
Z
∞
X 2−kγ
|h(x) − (h)Qj |χRn \2Qj (x)w(x) dx.
k+1 r )n
(2
k+1 r
j
|x−z
|<2
j
j
k=1
To control the sum on k we use again standard estimates together with the generalized
H¨older inequality and John-Nirenberg’s theorem. Indeed, if y ∈ Qj , we have
∞
X
k=1
2−kγ
(2k+1 rj )n
Z
|x−zj |<2k+1 rj
≤ C
+
≤ C
+
∞
X
k=1
∞
X
2−kγ
(2k+1 rj )n
|(h)2k+1 Qj
k=1
∞
X
k=1
∞
X
|h(x)−(h)Qj |χRn \2Qj (x)w(x) dx
Z
2k+1 Qj
|h(x) − (h)2k+1 Qj |χRn \2Qj (x)w(x) dx
2−kγ
− (h)Qj | k+1 n
(2 rj )
Z
2k+1 Qj
χRn \2Qj (x)w(x) dx
2−kγ kh − (h)2k+1 Qj k exp L,2k+1 Qj kχRn \2Qj wkL log L,2k+1 Qj
2−kγ (k + 1)M (χRn \2Qj w)(y)
k=1
≤ C
ML(log L) (χRn \2Qj w)(y)
∞
X
−kγ
2
k=1
≤ C ML(log L)1+ε (χRn \2Qj w)(y).
22
+ M (χRn \2Qj w)(y)
∞
X
k=1
!
−kγ
2
k
dy
Thus we have
Z
CX
IV ≤
λ j Qj
!1/q
X
|bij (y)|q
ML(log L)1+ε (χRn \2Qj w)(y)dy,
i
and we can continue the estimate of IV in the same way as in the proof of (4.22) with M
replaced by ML(log L)1+ε . We conclude that
IV
C
≤
λ
Z
|f (y)|q ML(log L)1+ε (w)(y)dy
Z
|f (x)|q
Φ
ML(log L)1+ (w)(x)dx.
≤ C
λ
Rn
Rn
To estimate V we will use Theorem 1.4 for singular integrals:
λ
n
e : B(x) >
V = w
x∈R \Ω
4
q #1/q
Z "X X
C
≤
ML(log L) (χRn \Ωe w)(x)dx
(h(x) − (h)Qj )bij (x)
λ Rn i j
!1/q
Z
X
CX
≤
|h(x) − (h)Qj |
|bij (x)|q
ML(log L) (χRn \2Qj w)(x)dx
λ j Qj
i
X
C
≤
inf ML(log L) (χRn \2Qj w)(x)
λ j Qj
!
Z
Z
|h(x) − (h)Qj ||g(x)|q dx
|h(x) − (h)Qj ||f (x)|q dx +
Qj
Qj
= V1 + V2 .
To estimate V2 we combine the argument to prove (4.23), replacing M by ML log L ,
together with the definition of BM O:
Z
|f (x)|q ML(log L) (w)(x)dx.
V2 ≤ C
Rn
For V1 we have by the generalized H¨older inequality (2.3)
Z
CX
V1 =
inf M
(χ
w)(x)
|h(x) − (h)Qj ||f (x)|q dx
λ j Qj L(log L)ε Rn \2Qj
Qj
CX
≤
inf M
(χ
w)(x)|Qj |k|f |q kL log L,Qj .
λ j Qj L(log L)ε Rn \2Qj
23
Now, combining formula (2.2) with (4.24) and recalling that Φ(t) = t log(e + t), we have
Z
1
1
µ
|f (x)|q
|Qj |kfq kL log L,Qj ≤
|Qj | inf {µ +
dx}
Φ
µ>0
λ
λ
|Qj | Qj
µ
Z
|f (x)|q
Φ
≤ |Qj | +
dx
λ
Qj
Z
Z
1
|f (x)|q
≤
dx
|f (x)|q dx +
Φ
λ Qj
λ
Qj
Z
|f (x)|q
≤ 2
Φ
dx.
λ
Qj
Then
Z
V1 ≤ C
Φ
Qj
Z
≤ C
Φ
Rn
|f (x)|q
λ
|f (x)|q
λ
ML(log L) (χRn \2Qj w)(x)dx
ML(log L)1+ε (w)(x)dx.
2
The proof of the theorem is finished.
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24
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´rez
Carlos Pe
´lisis Matema
´tico
Departmento de Ana
´ticas
Facultad de Matema
Universidad de Sevilla
41080 Sevilla
25
Spain
E-mail address: [email protected]
´lez
Rodrigo Trujillo-Gonza
´lisis Matema
´tico
Departmento de Ana
Universidad de La Laguna
38271 La Laguna - S/C de Tenerife
Spain
E-mail address: [email protected]
26