J. Korean Math. Soc. 44 (2007), No. 4, pp. 809–822
ENDPOINT ESTIMATES FOR MAXIMAL COMMUTATORS IN NON-HOMOGENEOUS SPACES
Guoen Hu, Yan Meng, and Dachun Yang
Reprinted from the
Journal of the Korean Mathematical Society Vol. 44, No. 4, July 2007
c
°2007 The Korean Mathematical Society
ENDPOINT ESTIMATES FOR MAXIMAL COMMUTATORS IN NON-HOMOGENEOUS SPACES
Guoen Hu, Yan Meng, and Dachun Yang
Abstract. Certain weak type endpoint estimates are established for maximal commutators generated by Calder´ on-Zygmund operators and Osc
expLr(µ) functions for r ≥ 1 under the condition that the under- lying measure only satisfies some growth condition, where the kernels of Calder´ on-Zgymund operators only satisfy the standard size condition and some H¨ ormander type regularity condition, and Osc
exp Lr(µ) are the spaces of Orlicz type satisfying that Osc
exp Lr(µ) = RBMO(µ) if r = 1 and Osc
exp Lr(µ) ⊂ RBMO(µ) if r > 1.
1. Introduction
It is well known that the doubling condition of the underlying measure is a key assumption in the analysis on spaces of homogeneous type. We recall that µ is said to satisfy the doubling condition if there is a constant C > 0 such that µ (B(x, 2r)) ≤ Cµ (B(x, r)) for all x ∈ R d and r > 0, where B(x, r) = {y ∈ R d : |y − x| < r}. However, during the last several years, many classical results have been proved still valid if the underlying measure µ is a positive Radon measure on R d , which only satisfies the following growth condition that there exist constants C 0 > 0 and n ∈ (0, d] such that for all x ∈ R d and r > 0,
(1.1) µ (B(x, r)) ≤ C 0 r n ;
see [5, 6, 7, 10, 11, 12]. The Euclidean space R d equipped with a Radon measure that only satisfies (1.1) is called a non-homogeneous space since µ may not be doubling. The motivation for developing the analysis on non-homogeneous spaces and some examples of non-doubling measures can be found in [14]. We only point out that the analysis on non-homogeneous spaces played an essential role in solving the long-standing Painlev´e’s problem by Tolsa in [13].
Received January 11, 2006.
2000 Mathematics Subject Classification. Primary 47B47; Secondary 42B20, 43A99.
Key words and phrases. Calder´ on-Zygmund operator, Osc
expLr(µ), maximal commuta- tor, endpoint estimate.
The third author was supported by Program for New Century Excellent Talents in Uni- versity (NCET-04-0142) of China.
c
°2007 The Korean Mathematical Society
809
The purpose of this paper is to establish some weak type endpoint esti- mates for the maximal commutators associated to Calder´on-Zygmund opera- tors, whose kernels satisfy the standard size condition and some weaker regu- larity condition, with Osc expLr(µ) functions, where r ≥ 1. Before stating our results, we first recall some necessary notation and definitions.
Throughout this paper, by a cube Q ⊂ R d , we mean a closed cube whose sides are parallel to the axes and centered at some point of supp(µ), and we denote its side length by l(Q) and its center by x Q . Let α and β be positive constants such that α > 1 and β > α n . For a cube Q, we say that Q is (α, β)- doubling if µ(αQ) ≤ βµ(Q), where αQ denotes the cube concentric with Q and having side length αl(Q). In what follows, for definiteness, if α and β are not specified, by a doubling cube we mean a (2, 2 d+1 )-doubling cube. Especially, for any given cube Q, we denote by e Q the smallest doubling cube in the family {2 k Q} k≥0 . For two cubes Q 1 ⊂ Q 2 , set
K Q1, Q
2 = 1 +
N
Q1, Q2X
k=1
µ(2 k Q 1 )
£ l(2 k Q 1 ) ¤ n ,
where N Q1, Q
2 is the first positive integer k such that l(2 k Q 1 ) ≥ l(Q 2 ); see [9]
for some basic properties of K Q1, Q
2.
Definition. For r ≥ 1, a locally integrable function f is said to belong to the space Osc exp Lr(µ) if there is a constant C 1 > 0 such that
(i) for any Q,
° °
°f − m Q e (f )
° °
° exp Lr, Q
= inf
½
λ > 0 : 1 µ(2Q)
Z
Q
exp ³ |f − m Q e (f )|
λ
´ r dµ ≤ 2
¾
≤ C 1 ,
(ii) for any two doubling cubes Q 1 ⊂ Q 2 , |m Q1(f ) − m Q2(f )| ≤ C 1 K Q1, Q
2, where m Q e (f ) is the mean value of f on e Q, namely, m Q e (f ) = µ( e 1 Q) R
(f )| ≤ C 1 K Q1, Q
2, where m Q e (f ) is the mean value of f on e Q, namely, m Q e (f ) = µ( e 1 Q) R
Q e f (x) dµ(x).
The minimal constant C 1 satisfying (i) and (ii) is defined to be the Osc exp Lr(µ) norm of f and denoted by kf k Oscexp Lr(µ) .
(µ) .
The space Osc exp Lr(µ) is an analogy of the classical Osc exp Lr(R d ) space which was introduced by P´erez and Trujillo-Gonz´alez in [8]. Obviously, for any r 2 > r 1 > 1, Osc exp Lr2(µ) ⊂ Osc exp Lr1(µ) ⊂ RBMO(µ). Moreover, from the John-Nirenberg inequality established by Tolsa in [9], it follows that Osc expL1(µ) is just the space RBMO(µ) of Tolsa in [9].
(R d ) space which was introduced by P´erez and Trujillo-Gonz´alez in [8]. Obviously, for any r 2 > r 1 > 1, Osc exp Lr2(µ) ⊂ Osc exp Lr1(µ) ⊂ RBMO(µ). Moreover, from the John-Nirenberg inequality established by Tolsa in [9], it follows that Osc expL1(µ) is just the space RBMO(µ) of Tolsa in [9].
(µ) ⊂ RBMO(µ). Moreover, from the John-Nirenberg inequality established by Tolsa in [9], it follows that Osc expL1(µ) is just the space RBMO(µ) of Tolsa in [9].
Let K ∈ L 1 loc (R d × R d \ {(x, y) : x = y}) and there exists a constant C > 0 such that for all x, y ∈ R d with x 6= y,
(1.2) |K(x, y)| ≤ C|x − y| −n ,
and for all y, y 0 ∈ R d , (1.3)
Z
|x−y|≥2|y−y
0|
{|K(x, y) − K(x, y 0 )|
+|K(y, x) − K(y 0 , x)|} dµ(x) ≤ C.
For any ² > 0, define the truncated operators T ² by (1.4) T ² (f )(x) =
Z
|x−y|>²
K(x, y)f (y) dµ(y), and the maximal Calder´on-Zygmund operator T ∗ by
(1.5) T ∗ (f )(x) = sup
²>0 |T ² (f )(x)| .
It is well known that if the operators T ² are bounded on L 2 (µ) uniformly for
² > 0, then there is an operator T which is the weak limit as ² → 0 of some subsequence of the uniformly bounded operators T ² . The operator T is also bounded on L 2 (µ) and satisfies that for f ∈ L 2 (µ) with suppf 6= R d , and almost all x ∈ R d \ supp f ,
T (f )(x) = Z
R
dK(x, y)f (y) dµ(y).
For m ∈ N and b 1 , b 2 , . . . , b m ∈ RBMO(µ), define the multilinear commutator T ~b by
T ~b f (x) = [b m , [b m−1 , . . . , [b 1 , T ] · · ·]] (f )(x), where ~b = (b 1 , b 2 , . . . , b m ) and [b 1 , T ] is defined by
(1.6) [b 1 , T ](f )(x) = b 1 (x)T (f )(x) − T (b 1 f )(x).
For the case of m = 1, we denote T ~b simply by T b . When the kernel K satisfies the size condition (1.2) and the standard regularity condition: there exist two constants α ∈ (0, 1] and C > 0 such that for all x, y, y 0 ∈ R d with
|x − y| ≥ 2|y − y 0 | and x 6= y,
(1.7) |K(x, y) − K(x, y 0 )| + |K(y, x) − K(y 0 , x)| ≤ C |y − y 0 | α
|x − y| n+α , Tolsa [9] proved that if the operators T ² are bounded on L 2 (µ) uniformly for ² >
0, then T b is bounded on L p (µ) for any p ∈ (1, ∞). In [1], we generalized this result of Tolsa and proved that if K satisfies (1.2) and (1.7) and the operators T ² are bounded on L 2 (µ) uniformly for ² > 0, then for any m ∈ N, T ~b is also bounded on L p (µ) with p ∈ (1, ∞), and satisfies a weak type endpoint estimate, namely, there exists a constant C > 0 such that for all λ > 0 and all bounded functions f with compact support,
µ ³n
x ∈ R d : |T ~b (f )(x)| > λ o´
≤ Cϕ 1/r
³ Y m
i=1
kb i k Oscexp Lri(µ)
´ Z
R
dϕ 1/r
³ |f(x)|
λ
´
dµ(x),
where C > 0 is a constant, 1/r = P m
i=1 1/r i and ϕ σ (t) = t log σ (2 + t) for σ, t > 0.
We now define the maximal commutator associated with the operator T ~b . For m ∈ N and b 1 , b 2 , . . . , b m ∈ RBMO(µ), the maximal commutator T ~b ∗ is defined by
(1.8) T ~b ∗ (f )(x) = sup
²>0
¯ ¯
¯T ²,~b (f )(x)
¯ ¯
¯
= sup
²>0
¯ ¯
¯ ¯
¯ Z
|x−y|>²
Y m i=1
[b i (x) − b i (y)]K(x, y)f (y)dµ(y)
¯ ¯
¯ ¯
¯ . Repeating the proof of Lemma 4.1 in [4], we can prove that if K satisfies (1.2) and the following H¨ormander-type condition
(1.9) sup
y, y
0∈R
d, r≥|y−y
0|
X ∞ l=1
l m Z
2
lr<|x−y|≤2
l+1r
n
|K(x, y) − K(x, y 0 )|
+|K(y, x) − K(y 0 , x)| o
dµ(x) < ∞,
and if the operators T ² are bounded on L 2 (µ) uniformly on ² > 0, then for b i ∈ RBMO(µ) with i = 1, 2, . . . , m, the operator T ~b ∗ is bounded on L p (µ) for any p ∈ (1, ∞). It was also proved in [3] that if K satisfies (1.2) and (1.7), and if the operators T ² are bounded on L 2 (µ) uniformly for ² > 0, then for m = 1, the maximal commutator T ~b ∗ satisfies the weak type endpoint estimate, namely, there exists a constant C > 0 such that for all λ > 0 and all bounded functions f with compact support,
µ
³n
x ∈ R d : |T ~b ∗ (f )(x)| > λ o´
≤ Cϕ 1/r
³
kb 1 k Oscexp Lr(µ)
´ Z
R
dϕ 1/r
³ |f(x)|
λ
´ dµ(x).
In this paper, we will further prove that if K satisfies (1.2) and (1.9), then for any m ∈ N, the maximal commutator T ~b ∗ enjoys the same endpoint estimate.
Our result can be stated as follows.
Theorem 1.1. Let m ∈ N, r i ≥ 1 and b i ∈ Osc exp Lri(µ) for i = 1, 2, . . . , m, and T b ∗ be the same as in (1.8) with the kernel K satisfying (1.2) and (1.9). If the operators T ² are bounded on L 2 (µ) uniformly for ² > 0, then there exists a constant C > 0 such that for all λ > 0 and all bounded functions f with compact support,
(1.10)
µ
³n
x ∈ R d : |T ~b ∗ (f )(x)| > λ o´
≤ Cϕ 1/r
³ Y m
i=1
kb i k Oscexp Lri(µ)
´ Z
R
dϕ 1/r
³ |f(x)|
λ
´ dµ(x).
Throughout this paper, for any index p ∈ [1, ∞], we denote by p 0 its conju-
gate index, namely, 1/p + 1/p 0 = 1. For f ∼ g, we mean that the ratio f /g is
bounded and bounded away from zero by constants independent of the relevant variables in f and g. Similar is f . g. Constants with subscripts, such as C 0 , are positive constants independent of the main parameters involved but whose values may not be the same at each occurrence.
2. Proof of Theorem 1.1
We begin with a generalization of the H¨older inequality. For r > 0, a cube Q and an appropriate function f , define
kf k L(logL)r, Q = inf n
λ > 0 : 1 µ(2Q)
Z
Q
|f (x)|
λ log r µ
2 + |f (x)|
λ
¶
dµ(x) ≤ 1 o
, and
kf k expLr, Q = inf n
λ > 0 : 1 µ(2Q)
Z
Q
exp
µ |f (x)|
λ
¶ r
dµ(x) ≤ 2 o . Then for any cube Q,
(2.1) 1
µ(2Q) Z
Q
¯ ¯
¯ Y m i=1
b i (x)f (x)
¯ ¯
¯ dµ(x) ≤ C Y m i=1
kb i k exp Lri, Q kf k L(log L)
1/r, Q , where r i ≥ 1 and 1/r = P m
i=1 1/r i ; see Lemma 3.2 in [8] and the related references there.
Lemma 2.1. Let m ∈ N, r i ≥ 1 and b i ∈ Osc exp Lri(µ) for i = 1, 2, . . . , m, and M ~b be defined by
(2.2) M ~b (f )(x) = sup
r>0
1 r n
Z
|y−x|≤r
Y m i=1
|b i (x) − b i (y)||f (y)| dµ(y).
Then there exists a constant C > 0 such that for all λ > 0 and all bounded functions f with compact support,
µ ¡©
x ∈ R d : M ~b (f )(x) > λ ª¢
≤ Cϕ 1/r
³ Y m
i=1
kb i k Oscexp Lri(µ)
´ Z
R
dϕ 1/r
µ |f (x)|
λ
¶ dµ(x), where r and ϕ 1/r are the same as in Theorem 1.1.
For the special case m = 1, Lemma 2.1 was proved in [3]. For m ≥ 2, Lemma 2.1 can be proved in a similar way. We omit the details for brevity.
Proof of Theorem 1.1. By the homogeneity, we may assume that for i = 1, . . . , m, kb i k Oscexp Lri(µ) = 1. We carry out the argument by induction on m.
Step I. m = 1. In this case, we denote T ~b ∗ simply by T b ∗ . For each fixed bounded function f with compact support and each λ > 0 (with λ >
2 d+1 kf k L1(µ) /kµk if kµk < ∞; note that if kµk < ∞ and λ ≤ 2 d+1 kf k L
1(µ) /kµk,
then the inequality (1.10) is trivial), applying the Calder´on-Zygmund decom- position to f at level λ (see [9]), we can obtain a sequence of cubes {Q j } j∈N
with bounded overlaps (that is, P
j χ Q
j(x) . 1) such that (a) 2d+1λ < µ(2Q 1
j
)
R
Q
j|f (x)| dµ(x);
(b) µ(2ηQ 1
j
)
R
ηQ
j|f (x)| dµ(x) ≤ 2d+1λ for any η > 2;
(c) |f (x)| ≤ λ µ-a. e. on R d \ ∪ j Q j ;
(d) for each fixed j, let R j be the smallest (6, 6 n+1 )-doubling cube of the form 6 k Q j , k ≥ 1. Set w j = χ Qj/ P
k χ Q
k. Then there is a function θ j
with supp θ j ⊂ R j and satisfying Z
R
dθ j (x) dµ(x) = Z
Q
jf (x)w j (x) dµ(x), kθ j k L∞(µ) µ(R j ) . Z
Q
j|f (x)| dµ(x), and P
j |θ j (x)| . λ.
Set g(x) = f (x)χ Rd\∪
jQ
j(x) + P
j θ j (x) and h(x) = f (x) − g(x) = X
j
{f (x)w j (x) − θ j (x)} = X
j
h j (x).
Obviously,
T b ∗ (f )(x) ≤ T b ∗ (g)(x) + T b ∗ (h)(x).
Note that kgk L∞(µ) . λ, and kgk L
1(µ) . kf k L
1(µ) . The L 2 (µ)-boundedness of T b ∗ tells us that
(2.3) µ ¡©
x ∈ R d : T b ∗ (g)(x) > λ ª¢
. λ −2 Z
R
d|g(x)| 2 dµ(x) . λ −1
Z
R
d|f (x)| dµ(x).
Taking into account the fact that (2.4) µ (∪ j 2Q j ) . λ −1
Z
R
d|f (x)| dµ(x),
we see that the proof of Theorem 1.1 can be reduced to proving that (2.5) µ ¡©
x ∈ R d \ ∪ j 2Q j : T b ∗ (h)(x) > λ ª¢
. Z
R
dϕ 1/r
³ |f(x)|
λ
´ dµ(x).
To this end, by the vanishing moment conditions of h j for j ∈ N, we can write T b ∗ (h)(x)χ Rd\ ∪ j 2Q j (x)
= sup
²>0
¯ ¯
¯ ¯
¯ ¯ X
j
Z
R
d½
K ² (x, y) [b(x) − b(y)]
−K ² (x, x Qj) h
b(x) − m Q ej(b) i ¾
h j (y) dµ(y)
¯ ¯
¯ ¯ χ Rd\ ∪ j 2Q j (x)
≤ X
j
¯ ¯
¯b(x) − m Q fj(b)
¯ ¯
¯
× sup
²>0
¯ ¯
¯ ¯ Z
R
d£ K ² (x, y) − K ² (x, x Qj) ¤
h j (y) dµ(y)
¯ ¯
¯ ¯ χ Rd\ ∪ j 2Q j (x) + T ∗ ³ X
j
h
b − m Q fj(b) i
h j
´ (x)
= E(x) + F(x),
where T ∗ is defined by (1.5), and K ² for ² > 0 is defined by K ² (x, y) = K(x, y)χ {|x−y|>²} (x, y).
Theorem 1.1 in [2] tells us that if T ² are bounded on L 2 (µ) uniformly on ² > 0, and K satisfies (1.2) and (1.3), then T ∗ is bounded from L 1 (µ) to weak L 1 (µ).
Thus, µ ¡©
x ∈ R d : |F(x)| > λ ª¢
. λ −1 X
j
Z
R
d¯ ¯
¯b(x) − m Q fj(b)
¯ ¯
¯ |f (x)w j (x)| dµ(x)
+ λ −1 X
j
Z
R
d¯ ¯
¯b(x) − m Q fj(b)
¯ ¯
¯ |θ j (x)| dµ(x)
= G + H.
Note that R j is also (2, 2 n+1 )-doubling and R j = f R j . A trivial computation gives us that K Q fj, R
j . 1. Thus,
H . λ −1 X
j
kθ j k L∞(µ)
(Z
R
j¯ ¯b(x) − m Rj(b) ¯
¯ dµ(x)
+µ(R j )
¯ ¯
¯m Rj(b) − m Q f
j
(b)
¯ ¯
¯ o
. λ −1 X
j
kθ j k L∞(µ) µ(R j )
. λ −1 Z
R
d|f (x)| dµ(x).
On the other hand, by the generalization of H¨older inequality (2.1), we obtain G . λ −1 X
j
µ(2Q j )kf k L(log L)1/r, Q
jkb − m Q f
j
(b)k expLr, Q
j
. λ −1 X
j
µ(2Q j ) inf
t>0
(
t + t µ(2Q j )
Z
Q
j|f (x)|
t log 1/r
³
2 + |f (x)|
t
´ dµ(x)
)
. Z
R
d|f (x)|
λ log 1/r ³
2 + |f (x)|
λ
´
dµ(x).
It remains to estimate E(x). Note that for y ∈ R j and x ∈ R d \ 2R j ,
|x − x Qj| ∼ |x − y|. A straightforward computation indicates sup
²>0
Z
R
d¯ ¯K ² (x, y) − K ² (x, x Qj) ¯
¯ |h j (y)| dµ(y)
≤ Z
R
d¯ ¯K(x, y) − K(x, x Qj) ¯
¯ |h j (y)| dµ(y)
+ sup
²>0
Z
R
dn
|K(x, y)|χ {|x−y|>²}∩{|x−xQj|≤²} (x, y) +|K(x, x Q
j)|χ {|x−y|≤²}∩{|x−xQj|>²} (x, y)
|>²} (x, y)
o
|h j (y)| dµ(y) .
Z
R
d¯ ¯K(x, y) − K(x, x Qj) ¯
¯ |h j (y)| dµ(y) + M (h j )(x), where M is the Hardy-Littlewood maximal operator defined by
M h(x) = sup
Q3x
1 [l(Q)] n
Z
Q
|h(y)| dµ(y).
Therefore, E(x) . X
j
¯ ¯
¯b(x) − m Q fj(b)
¯ ¯
¯ Z
R
d¯ ¯K(x, y) − K(x, x Qj) ¯
¯ |h j (y)| dµ(y)χ Rd\2R
j(x)
+ M ~b ³ X
j
|h j | ´
(x) + M ³ X
j
¯ ¯b − m Q f
j
(b) ¯
¯|h j | ´ (x)
+ X
j
¯ ¯
¯b(x) − m Q fj(b)
¯ ¯
¯ T ∗ (h j )(x)χ 2Rj\2Q
j(x)
+ X
j
¯ ¯
¯b(x) − m Q fj(b)
¯ ¯
¯ sup
²>0
Z
R
d¯ ¯K ² (x, x Qj)h j (y) ¯
¯ dµ(y)χ 2Rj\2Q
j(x)
= E 1 (x) + E 2 (x) + E 3 (x) + E 4 (x) + E 5 (x).
An application of Lemma 2.1 gives us that µ ¡©
x ∈ R d : E 2 (x) > λ ª¢
. Z
R
dϕ 1/r ³ P
j |f ω j (x)|
λ
´ dµ(x)
+ Z
R
dϕ 1/r
³ P j |θ j (x)|
λ
´ dµ(x)
= I + J.
Obviously,
I . Z
R
dϕ 1/r ³ |f(x)|
λ
´ dµ(x).
Recall that P
j |θ j (x)| . λ. It then follows that J . λ −1 X
j
kθ j k L∞(µ) µ(R j ) . λ −1 Z
R
d|f (x)| dµ(x),
and so
µ ¡©
x ∈ R d : E 2 (x) > λ ª¢
. Z
R
dϕ 1/r ³ |f(x)|
λ
´ dµ(x).
Since the Hardy-Littlewood maximal operator M is bounded from L 1 (µ) to weak L 1 (µ), similar to the estimate for F(x), it follows that
µ ¡©
x ∈ R d : E 3 (x) > λ ª¢
. Z
R
dϕ 1/r
³ |f(x)|
λ
´ dµ(x).
To estimate E 4 (x), observing that 1 ≤ k ≤ N 2Qj, 2R
j, K Q f
j
, ^ 2
k+1Q
j. K Qj, R
j . 1, we can easily obtain that
µ ¡©
x ∈ R d : E 4 (x) > λ ª¢
. 1 λ
X
j
N
2Qj , 2RjX
k=1
Z
2
k+1Q
j\2
kQ
j× Z
R
d|b(x) − m Q f
j
(b)|
|x − y| n {|f (y)w j (y)| + |θ j (y)|} dµ(y)dµ(x) . 1
λ X
j
N
2Qj , 2RjX
k=1
µ(2 k+2 Q j ) l(2 k Q j ) n K Q f
j
, ^ 2
k+1Q
j× (Z
Q
j|f (y)| dµ(y) + kθ j k L∞(µ) µ(R j ) )
. 1 λ
X
j
Z
Q
j|f (y)| dµ(y), and similarly,
µ ¡©
x ∈ R d : E 5 (x) > λ ª¢
. 1 λ
X
j
Z
Q
j|f (y)| dµ(y).
For E 1 (x), another application of the generalized H¨older inequality (2.1) yields Z
R
d\2R
j¯ ¯
¯b(x) − m Q fj(b)
¯ ¯
¯ ¯
¯K(x, y) − K(x, x Qj) ¯
¯ dµ(x)
≤ X ∞ k=1
¯ ¯
¯m 2 ^k+1R
j
(b) − m Q fj(b)
¯ ¯
¯ Z
2
k+1R
j\2
kR
j¯ ¯K(x, y) − K(x, x Qj) ¯
¯ dµ(x)
+ X ∞ k=1
Z
2
k+1R
j\2
kR
j¯ ¯
¯b(x) − m 2k+1^ R
j
(b)
¯ ¯
¯ ¯
¯K(x, y) − K(x, x Qj) ¯
¯ dµ(x)
. X ∞ k=1
K Q f
j
, ^ 2
k+1R
jZ
2
k+1R
j\2
kR
j¯ ¯K(x, y) − K(x, x Qj) ¯
¯ dµ(x)
+ X ∞ k=1
µ ¡ 2 k+2 R j
¢ ° °
°b − m 2 ^k+1R
j
(b)
° °
° exp Lr(µ), 2
k+1R
j
× °
° ©
K(·, y) − K(·, x Qj) ª
χ 2k+1R
j\2
kR
j(·) °
° L(log L)1/r(µ), 2
k+1R
j. Let
λ k = £ µ ¡
2 k+2 R j
¢¤ −1 ³ k
Z
2
k+1R
j\2
kR
j¯ ¯K(x, y) − K(x, x Qj) ¯
¯ dµ(x) + 2 −k ´ . By (1.2), we then have that for y ∈ R j ,
1 µ(2 k+2 R j )
Z
2
k+1R
j\2
kR
j|K(x, y) − K(x, x Qj)|
λ k
× log 1/r
³
2 + |K(x, y) − K(x, x Qj)|
λ k
´ dµ(x)
. 1
µ(2 k+2 R j ) Z
2
k+1R
j\2
kR
j|K(x, y) − K(x, x Qj)|
λ k
× log 1/r
³
2 + 1
λ k |x − y| n + 1 λ k |x − x Qj| n
´ dµ(x)
. k
µ(2 k+2 R j ) Z
2
k+1R
j\2
kR
j|K(x, y) − K(x, x Qj)|
λ k
dµ(x) . 1.
Thus,
° ° ©
K(·, y) − K(·, x Qj) ª
χ 2k+1R
j\2
kR
j(·) °
° L(log L)1/r(µ), 2
k+1R
j . λ k . This via (1.9) tells us that
Z
R
d\2R
j¯ ¯
¯b(x) − m Q fj(b)
¯ ¯
¯ ¯
¯K(x, y) − K(x, x Qj) ¯
¯ dµ(x)
. X ∞ k=1
à k
Z
2
k+1R
j\2
kR
j¯ ¯K(x, y) − K(x, x Qj) ¯
¯ dµ(x) + 2 −k
!
. 1.
Therefore, µ ¡©
x ∈ R d : E 1 (x) > λ ª¢
. 1 λ
X
j
Z
R
jZ
R
d\2R
j¯ ¯
¯b(x) − m Q fj(b)
¯ ¯
¯
× ¯
¯K(x, y) − K(x, x Qj) ¯
¯ |h j (y)| dµ(x) dµ(y) . 1
λ X
j
Z
R
j|h j (y)| dµ(y)
. 1 λ
Z
R
d|f (y)| dµ(y),
which along with the estimates for E 2 (x), E 3 (x), E 4 (x) and E 5 (x) gives the desired estimate for E(x). Combining the estimates for E(x) and F(x) yields the estimate (2.5) and then completes the proof of m = 1.
Step II. m ≥ 2. In this case, we need more notation. For 0 ≤ i ≤ m, we de- note by C i m the family of all finite subsets σ = {σ(1), . . . , σ(i)} of {1, 2, . . . , m}
with i different elements. For σ ∈ C i m , the complementary sequence σ 0 is given by σ 0 = {1, 2, . . . , m}\σ. If σ = ∅, we set σ 0 = {1, . . . , m}. For any i−tuple r = (r 1 , r 2 , . . . , r i ), we write 1/r σ = 1/r σ(1) + · · · + 1/r σ(i) and 1/r σ0 = 1/r − 1/r σ , where 1/r = 1/r 1 + · · · + 1/r m . Let ~b = (b 1 , b 2 . . . , b m ) be a finite family of locally integrable functions. For σ ∈ C i m , we set ~b σ = (b σ(1) , . . . , b σ(i) ) and the product b σ = b σ(1) · · · b σ(i) . For σ = ∅, we define b σ = 1. With this notation, we write
k~b σ k Oscexp Lrσ(µ) = kb σ(1) k Osc
exp Lrσ(1)
(µ) · · · kb σ(i) k Osc
exp Lrσ(i)
(µ) . For i ∈ {1, . . . , m} and σ ∈ C i m , we set
[b(y) − b(z)] σ = £
b σ(1) (y) − b σ(1) (z) ¤
· · · £
b σ(i) (y) − b σ(i) (z) ¤ and
h
m Q e (b) − b(y) i
σ = h
m Q e (b σ(1) ) − b σ(1) (y) i
· · · h
m Q e (b σ(i) ) − b σ(i) (y) i ,
where Q is any cube in R d and y, z ∈ R d . For any σ ∈ C i m , define
T ~b ∗
σ
(f )(x) =
¯ ¯
¯ sup
²>0
Z
R
dK ² (x, y) Y i j=1
£ b σ(j) (x) − b σ(j) (y) ¤
f (y) dµ(y)
¯ ¯
¯.
When σ = {1, . . . , m}, we denote T ~b ∗
σ
simply by T ~b ∗ .
Now let m ≥ 2 be an integer. We assume that (1.10) holds for any 1 ≤ i ≤ m − 1 and any subset σ ∈ C i m . For any fixed f and λ > 2 d+1 kf k L1(µ) /kµk, let Q j , R j , θ j , w j , g, h and h j be the same as in Step I. By an argument similar to the estimates for (2.3) and (2.4), it suffices to verify that
(2.6) µ ³n
x ∈ R d \ ∪ j 2Q j : |T ~b ∗ h(x)| > λ o´
. Z
R
dϕ 1/r ³ |f(x)|
λ
´ dµ(x).
With the aid of the formula that for y, z ∈ R d , Y m
i=1
h
m Q e (b i ) − b i (z) i
= X m i=0
X
σ∈C
im[b(y) − b(z)] σ0
h
m Q e (b) − b(y) i
σ
and the vanishing moment conditions satisfied by h j for j ∈ N, it is easy to see that
T ~b ∗ h(x) = sup
²>0
¯ ¯
¯ ¯
¯ ¯ X
j
Z
R
d(
K ² (x, y) Y m i=1
[b i (x) − b i (y)]
−K ² (x, x Qj) Y m i=1
h
b i (x) − m Q ej(b i ) i )
h j (y) dµ(y)
¯ ¯
¯ ¯
¯
≤ sup
²>0
X
j
Y m i=1
¯ ¯
¯b i (x) − m Q e
j
(b i )
¯ ¯
¯
× Z
R
d¯ ¯K ² (x, y) − K ² (x, x Qj) ¯
¯ |h j (y)| dµ(y)
+ T ∗ ³ X
j
Y m i=1
h
b i − m Q ej(b i ) i h j
´ (x)
+
m−1 X
i=1
X
σ∈C
imT ~b ∗
σ0
³ X
j
h
b − m Q f
j
(b) i
σ h j
´ (x)
= T ~b ∗, I (h)(x) + T ~b ∗, II (h)(x) +
m−1 X
i=1
X
σ∈C
miT ~b ∗, III
σ0
(h)(x).
Similar to the estimate for E(x) in Step I, we have µ
³n
x ∈ R d \ ∪ j 2Q j : T ~b ∗, I (h)(x) > λ o´
. Z
R
dϕ 1/r
µ |f (x)|
λ
¶ dµ(x).
On the other hand, the same argument as that for F(x) in Step I leads to that µ
³n
x ∈ R d \ ∪ j 2Q j : T ~b ∗, II (h)(x) > λ o´
. Z
R
dϕ 1/r
µ |f (x)|
λ
¶ dµ(x).
For each fixed i with 1 ≤ i ≤ m − 1, our induction hypothesis now states that µ
³n
x ∈ R d : T ~b ∗, III
σ0
(h)(x) > λ o´
. X
j
Z
R
dϕ 1/rσ0
³ ¯ ¯
¯ h
b(x) − m Q fj(b) i
σ
¯ ¯
¯ |w j (x)f (x)|
λ
´ dµ(x)
+ Z
R
dϕ 1/rσ0³ X
j
¯ ¯
¯ h
b(x) − m Q f
j
(b) i
σ
¯ ¯
¯ |θ j (x)|
λ
´ dµ(x)
= M σ + N σ . Applying the inequality
ϕ 1/rσ0(t 0 t 1 · · · t i ) . ϕ 1/r (t 0 ) + exp t r 1σ(1)+ · · · + exp t r iσ(i), t 0 , t 1 , . . . , t i > 0
+ · · · + exp t r iσ(i), t 0 , t 1 , . . . , t i > 0
(see Lemma 2.2 in [8]), and the fact (a) in the Calder´on-Zygmund decomposi- tion, we then deduce that
M σ . X
j
Z
R
dϕ 1/r ³
k~b σ k Oscexp Lrσ(µ) |χ Q
j(x)f (x)|
λ
´ dµ(x)
+ X
j
X i l=1
Z
R
dexp ³ |b σ(l) (x) − m Q fj(b σ(l) )|
kb σ(l) k Osc
exp Lrσ(l)
(µ) χ Q
j(x)
´ rσ(l)
dµ(x)
. Z
R
d|f (x)|
λ log 1/r ³
2 + |f (x)|
λ
´
dµ(x) + X
j
µ(2Q j )
. Z
R
d|f (x)|
λ log 1/r
³
2 + |f (x)|
λ
´ dµ(x).
To estimate N σ , let r j = λ −1 |θ j |, and Λ ⊂ N be a finite index set. The convexity of ϕ 1/rσ0 says that
ϕ 1/rσ0
³ X
j∈Λ
¯ ¯
¯ h
b(x) − m Q fj(b) i
σ
¯ ¯
¯ |θ j (x)|
λ
´
≤ X
j∈Λ
³ P r j l∈Λ r l
´
ϕ 1/rσ0³ ¯ ¯
¯ h
b(x) − m Q f
j
(b) i
σ
¯ ¯
¯ χ Rj(x) X
l∈Λ
r l
´
. 1
P
l∈Λ r l
ϕ 1/rσ0³ X
l∈Λ
r l
´ X
j∈Λ
r j ϕ 1/rσ0³¯
¯ £
b(x) − m Q f
j
(b) ¤
σ
¯ ¯χ Rj(x) ´
. log 1/rσ0
³ 2 + X
l∈Λ
r l
´ X
j∈Λ
r j ϕ 1/rσ0
³¯ ¯ £
b(x) − m Q fj(b) ¤
σ
¯ ¯χ Rj(x)
´
. X
j∈Λ
r j ϕ 1/rσ0³¯
¯ £
b(x) − m Q f
j
(b) ¤
σ
¯ ¯χ Rj(x) ´ . This in turn leads to that
N σ . λ −1 X
j
kθ j k L∞(µ) Z
R
jϕ 1/rσ0
³¯ ¯
¯ h
b(x) − m Q f
j
(b) i
σ
¯ ¯
¯
´ dµ(x)
. λ −1 X
j
kθ j k L∞(µ) µ(R j )
. λ −1 Z
R
d|f (x)| dµ(x).
Therefore, for 1 ≤ i ≤ m − 1 and σ ∈ C i m , we have µ ³n
x ∈ R d : T ~b ∗, III
σ0
