See Section 2.3 for the definition and main properties of the maximal .... that, whenever the symbol h is a B.M.O. function, the commutator is bounded on Lp(Rn),.
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 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| 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) − c δ 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| 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 | = 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 |
