An Interpolation Theorem for Sublinear Operators on Non-homogeneous Metric Measure Spaces

0.14 MB·English

by  Haibo Lin

An Interpolation Theorem for Sublinear Operators on
2 Non-homogeneous Metric Measure Spaces
1
0
2 Haibo Lin and Dongyong Yang∗

n
a
J

0
3
Abstract. Let (X,d,µ) be a metric measure space and satisfy the so-called upper dou-

bling condition and the geometrically doubling condition. In this paper, the authors
]
P establish an interpolation result that a sublinear operator which is bounded from the
A Hardy space H1(µ) to L1,∞(µ) and from L∞(µ) to the BMO-type space RBMO(µ) is
. alsoboundedonLp(µ)forallp∈(1, ∞). Thisextensionisnotcompletelystraightforward
h
t and improves the existing result.
a
m
1 Introduction
[

1
v Spaces of homogeneous type were introduced by Coifman and Weiss [3] as a general
7 framework in which many results from real and harmonic analysis on Euclidean spaces
9
have their natural extensions; see, for example, [4, 5, 6]. Recall that a metric space (X, d)
0
6 equipped with a nonnegative Borel measure µ is called a space of homogeneous type if
. (X, d, µ) satisfies the following measure doubling condition that there exists a positive
1
0 constant C , depending on µ, such that for any ball B(x, r) ≡{y ∈ X : d(x, y) < r} with
µ
2
x ∈ X and r ∈ (0, ∞),
1
: 0 < µ(B(x, 2r)) ≤ Cµµ(B(x, r)). (1.1)
v
i
X Themeasuredoublingcondition(1.1)playsakeyroleintheclassicaltheoryofCalder´on-
r Zygmund operators. However, recently, many classical results concerning the theory of
a
Calder´on-Zygmund operators and function spaces have been proved still valid if the dou-
bling condition is replaced by a less demanding condition such as the polynomial growth
condition; see, for example [14, 16, 17, 15, 18] and the references therein. In particular,
let µ be a non-negative Radon measure on Rn which only satisfies the polynomial growth
condition that there exist positive constants C and κ ∈(0,n] such that for all x ∈ Rn and
r ∈ (0, ∞),
µ({y ∈ Rn : |x−y| < r})≤ Crκ. (1.2)
Such a measure does not need to satisfy the doubling condition (1.1). We mention that
theanalysis withnon-doublingmeasuresplayed astrikingroleinsolvingthelong-standing
open Painlev´e’s problem by Tolsa in [18].
2010 Mathematics Subject Classification. Primary 42B35; Secondary 42B25, 47B38.
Key words and phrases. upperdoubling, geometrically doubling, RBMO(µ), sublinear, interpolation.
∗Corresponding author.
1
Because measures satisfying (1.2) are only different, not more general than measures
satisfying (1.1), the Calder´on-Zygmund theory with non-doubling measures is not in all
respects a generalization of the corresponding theory of spaces of homogeneous type. To
include the spaces of homogeneous type and Euclidean spaces with a non-negative Radon
measure satisfying a polynomial growth condition, Hyto¨nen [8] introduced a new class
of metric measure spaces which satisfy the so-called upper doubling condition and the
geometrically doubling condition (see, respectively, Definitions 2.1 and 2.2 below), and
a notion of the regularized BMO space, namely, RBMO(µ) (see Definition 2.4 below).
Since then, more and more papers focus on this new class of spaces; see, for example
[11, 12, 10, 1, 9, 7, 13].
Let(X, d, µ)beametricmeasurespacesatisfyingtheupperdoublingconditionandthe
geometrically doubling condition. In [10], the atomic Hardy space H1(µ) (see Definition
2.5 below) was studied and the duality between H1(µ) and RBMO(µ) of Hyto¨nen was
established. Some of results in [10] were also independently obtained by Anh and Duong
[1] via different approaches. Moreover, Anh and Duong [1, Theorem 6.4] established an
interpolation result that a linear operator which is bounded from H1(µ) to L1(µ) and
from L∞(µ) to RBMO(µ) is also bounded on Lp(µ) for all p ∈ (1, ∞). The purpose of
this paper is to generalize and improve the interpolation result for linear operators in [1]
to sublinear operators in the current setting (X, d, µ), which is stated as follows.
Theorem 1.1. Let T be a sublinear operator that is bounded from L∞(µ) to RBMO(µ)
and from H1(µ) to L1,∞(µ). Then T extends boundedly to Lp(µ) for every p ∈(1, ∞).
In Section 2, we collect preliminaries we need. In Section 3, for r ∈ (0,1), we first
♯
show that the maximal function M (f), which is a variant of the sharp maximal function
r
M♯(f)in[1],isboundedfromRBMO(µ)toL∞(µ),thenweestablishaweaktypeestimate
between the doubling maximal function N(f) and M♯(f), and we also establish a weak
typeestimate for N (f)with r ∈ (0,1), avariant of N(f). Usingthese results we establish
r
Theorem 1.1. We remark that the method for the proof of Theorem 1.1 is different from
that of [1, Theorem 6.4]. Precisely, in the proof of [1, Theorem 6.4], the fact that the
composite operator M♯◦T of the sharp maximal function M♯ and a linear operator T is
a sublinear operator was used. However, as far as we know, when T is sublinear, whether
the composite operator M♯ ◦T is a sublinear operator is unclear and so the proof of [1,
Theorem 6.4] is not available.
Throughoutthis paper, we denote by C a positive constant which is independentof the
main parameters involved but may vary from line to line. The subscripts of a constant
indicate the parameters it depends on. The notation f . g means that there exists a
constant C > 0 such that f ≤ Cg. Also, for a µ-measurable set E, χ denotes its
E
characteristic function.
2 Preliminaries
In this section, we will recall some necessary notions and notation and the Calder´on-
Zygmund decomposition which was established in [1]. We begin with the definition of
upper doubling space in [8].
2
Definition 2.1. A metric measure space (X, d, µ) is called upper doubling if µ is a Borel
measure on X and there exists a dominating function λ : X × (0, ∞) → (0, ∞) and a
positive constant C such that for each x ∈ X, r → λ(x, r) is non-decreasing, and for all
λ
x ∈ X and r ∈ (0, ∞),
µ(B(x, r))≤ λ(x, r)≤ C λ(x, r/2).
λ
Remark 2.1. (i) Obviously, a space of homogeneous type is a special case of the upper
doubling spaces, where one can take the dominating function λ(x, r)≡ µ(B(x, r)). More-
over, let µ be a non-negative Radon measure on Rn which only satisfies the polynomial
growth condition (1.2). By taking λ(x, r)≡ Crκ, we see that (Rn, |·|, µ) is also an upper
doubling measure space.
(ii)Itwasprovedin[10]thatthereexistsadominatingfunctionλrelatedtoλsatisfying
the property that there exists a positive constant Cλe such that λ ≤ λ, Cλe ≤ Cλ, and for
all x, y ∈ X with d(x, y) ≤ r, e
λ(x, r) ≤ Ceλ(y, r). e (2.1)
λ
Basedonthis,inthispaper,wealways assumethatthedominatingfunctionλalsosatisfies
e e
(2.1).
Throughout the whole paper, we also always assume that the underlying metric space
(X, d) satisfies the following geometrically doubling condition introduced in [8].
Definition 2.2. A metric space (X, d) is called geometrically doubling if there exists
some N ∈ N+ ≡ {1, 2, ···} such that for any ball B(x, r) ⊂ X, there exists a finite ball
0
covering {B(x , r/2)} of B(x, r) such that the cardinality of this covering is at most N .
i i 0
The following coefficients δ(B, S) for all balls B and S were introduced in [8] as ana-
logues of Tolsa’s numbers K in [16]; see also [10].
Q,R
Definition 2.3. For all balls B ⊂ S, let
dµ(x)
δ(B, S) ≡ .
λ(c ,d(x, c ))
Z(2S)\B B B
where and in that follows, for a ball B ≡ B(c , r ) and ρ ∈(0, ∞), ρB ≡ B(c , ρr ).
B B B B
p
In what follows, for each p ∈ (0,∞), L (µ) denotes the set of all functions f such
loc
that |f|p is µ-locally integrable.
Definition 2.4. Let η ∈ (1,∞) and p ∈ (0,∞). A function f ∈ Lp (µ) is said to be in
loc
the space RBMOp(µ) if there exists a non-negative constant C and a complex number f
η B
for any ball B such that for all balls B,
1
|f(y)−f |pdµ(y) ≤ Cp
B
µ(ηB)
ZB
and that for all balls B ⊂ S,
|f −f |≤ C[1+δ(B, S)].
B S
Moreover, the RBMOp(µ) norm of f is defined to be the minimal constant C as above
η
and denoted by kfkRBMOpη(µ).
3
When p = 1, we write RBMO1(µ) simply by RBMO(µ), which was introduced by
η
Hyto¨nen in [8]. Moreover, the spaces RBMOp(µ) and RBMO(µ) coincide with equivalent
η
norms, which is the special case of [7, Corollary 2.1].
Proposition 2.1. Let η ∈ (1,∞) and p ∈ (0,∞). The spaces RBMOp(µ) and RBMO(µ)
η
coincide with equivalent norms.
Remark 2.2. It was proved in [8, Lemma 4.6] that the space RBMO(µ) is independent
of the choice of η. By this and Proposition 2.1, it is obvious that the space RBMOp(µ) is
η
independent of the choice of η.
We now recall the definition of the atomic Hardy space introduced in [10]; see also [1].
Definition 2.5. Let ρ ∈ (1,∞) and p ∈ (1, ∞]. A function b ∈ L1(µ) is called a (p, 1) -
λ
atomic block if
(i) there exists some ball B such that supp(b) ⊂ B;
(ii) b(x)dµ(x) = 0;
X
(iii) fRor j = 1, 2, there exist functions a supported on balls B ⊂ B and λ ∈ C such
j j j
that
b = λ a +λ a ,
1 1 1 2
and
ka k ≤ [µ(ρB )]1/p−1[1+δ(B , B)]−1.
j Lp(µ) j j
Moreover, let
|b| ≡ |λ |+|λ |.
H1,p(µ) 1 2
atb
A function f ∈L1(µ) is said to belong to the atomic Hardy space H1,p(µ) if there exist
atb
(p, 1)λ-atomic blocks {bj}j∈N such thatf = ∞j=1bj and ∞j=1|bj|H1,p(µ) < ∞. Thenorm
atb
1,p
of f in H (µ) is defined by
atb P P
kfk ≡ inf |b | ,
H1,p(µ)  j H1,p(µ)
atb atb
Xj 
where the infimum is taken over all the possible decompositions of f as above.
Remark 2.3. It was proved in [10] that for each p ∈ (1, ∞], the atomic Hardy space
1,p 1,p
H (µ) is independent of the choice of ρ, and that for all p ∈ (1, ∞), the spaces H (µ)
atb atb
1,∞ 1,p
and H (µ) coincide with equivalent norms. Thus, in the following, we denote H (µ)
atb atb
simply by H1(µ).
At the end of this section, we recall the (α, β)-doubling property of some balls and the
Calder´on-Zygmund decomposition established by Anh and Duong [1, Theorem 6.3].
Given α, β ∈ (1, ∞), a ball B ⊂ X is called (α, β)-doubling if µ(αB) ≤ βµ(B). It was
log α
proved in [8] that if a metric measure space (X, d, µ) is upper doubling and β > C 2 ≡
λ
4
αν, then for every ball B ⊂ X, there exists some j ∈ Z ≡ N ∪ {0} such that αjB is
+
(α, β)-doubling. Moreover, let (X, d) begeometrically doubling, β > αn with n ≡ log N
2 0
and µ a Borel measure on X which is finite on boundedsets. Hyto¨nen [8] also showed that
for µ-almost every x ∈ X, there exist arbitrarily small (α, β)-doubling balls centered at
x. Furthermore, the radius of these balls may be chosen to be of the form α−jr for j ∈ N
and any preassigned number r ∈ (0, ∞). Throughout this paper, for any α ∈ (1, ∞) and
ball B, Bα denotes the smallest (α, β )-doubling ball of the form αjB with j ∈ Z , where
α +
β ≡max{αn, αν}+30n +30ν = αmax{n,ν}+30n +30ν. (2.2)
α
e
Lemma 2.1. Let p ∈ [1, ∞), f ∈ Lp(µ) and ℓ ∈ (0,∞) (ℓ > ℓ ≡ γ kfk /µ(X) if
0 0 Lp(µ)
µ(X) < ∞, where γ is any fixed positive constant satisfying that γ > max{C3log26, 63n},
0 0 λ
C is as in (2.2) and n = log N ). Then
λ 2 0
(i) there exists an almost disjoint family {6B } of balls such that {B } is pairwise
j j j j
disjoint,
1 ℓp
|f(x)|pdµ(x) > forallj,
µ(62B ) γ
j ZBj 0
1 ℓp
|f(x)|pdµ(x) ≤ foralljandallη > 1,
µ(62ηB ) γ
j ZηBj 0
and
|f(x)| ≤ ℓ forµ−almosteveryx ∈ X \(∪ 6B );
j j
(ii) for each j, let S be a (3×62, Clog2(3×62)+1)-doubling ball concentric with B sat-
j λ j
isfying that r > 62r , and ω ≡ χ /( χ ). Then there exists a family {ϕ } of
Sj Bj j 6Bj k 6Bk j j
functions such that for each j, supp(ϕ ) ⊂ S , ϕ has a constant sign on S and
j j j j
P
ϕ (x)dµ(x) = f(x)ω (x)dµ(x),
j j
ZX Z6Bj
|ϕ (x)| ≤ γℓ forµ−almosteveryx ∈ X,
j
j
X
where γ is some positive constant depending only on (X, µ), and there exists a positive
constant C, independent of f, ℓ and j, such that
kϕjkL∞(µ)µ(Sj) ≤C |f(x)ωj(x)|dµ(x),
ZX
and if p ∈ (1, ∞),
1/p
|ϕ (x)|pdµ(x) [µ(S )]1/p′ ≤ C |f(x)ω (x)|pdµ(x);
j j ℓp−1 j
(ZSj ) ZX
(iii) if for any j, choosing S in (ii) to be the smallest (3×62, Clog2(3×62)+1)-doubling
j λ
ball of (3×62)B , then h ≡ (fω −ϕ ) ∈ H1,p(µ) and there exists a positive constant
j j j j atb
C, independent of f and ℓ, such that
P
C
p
khk ≤ kfk .
Ha1t,bp(µ) ℓp−1 Lp(µ)
5
3 Proof Theorem 1.1
To prove Theorem 1.1, we also need some maximal functions in [8, 1] as follows. Let
f ∈ L1 (µ) and x ∈ X. The doubling Hardy-Littlewood maximal function N(f)(x) and
loc
the sharp maximal function M♯(f)(x) are respectively defined by setting,
1
N(f)(x) ≡ sup |f(y)|dµ(y),
µ(B)
B(6,β6B)−∋dxoubling ZB
and
1
M♯(f)(x) ≡ sup |f(y)−me (f)|dµ(y)
µ(5B) B6
B∋x ZB
|m (f)−m (f)|
B S
+ sup ,
1+δ(B, S)
x∈B⊂S
B,S(6,β6)−doubling
where for any f ∈ L1 (µ) and ball B, m (f) means its average over B, namely, m (f) ≡
loc B B
1
f(x)dµ(x). It was showed in [9, Lemma 2.3] that for any p ∈ [1, ∞), Nf is
µ(B)
boundZeBd from Lp(µ) to Lp,∞(µ).
Lemma 3.1. Let f ∈ RBMO(µ), r ∈ (0,1) and M♯(f) ≡ [M♯(|f|r)]1/r. Then we have
r
M♯f ∈ L∞(µ), and moreover,
r
kMr♯fkL∞(µ) . kfkRBMO(µ).
Proof. From Remark 2.2, we deduce that for any ball B,
1
fBe6 −mBe6(f) ≤ µ(B6)ZBe6 f(x)−fBe6 dµ(x) . kfkRBMO(µ).
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
On the other hand, by Proposition 2.1 and Remark 2.2, we see that
e
kfkRBMO(µ) ∼ kfkRBMOr(µ).
5
From these facts, it follows that
1
|f(x)|r −me (|f|r) dµ(x)
µ(5B) B6
ZB
1 (cid:12) (cid:12)
≤ (cid:12) |f(x)|r − me(cid:12)(f) r + me (f) r −me (|f|r) dµ(x)
µ(5B) B6 B6 B6
ZB
1 (cid:2)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)(cid:12) (cid:12) (cid:12)(cid:3)
. µ(5B) |f(cid:12)(x)−fB|r(cid:12) dµ(x)+(cid:12) (cid:12)fB(cid:12)−(cid:12) fBe6 r +(cid:12) fBe6 −mBe6(f(cid:12)) r
ZB
1 (cid:12) (cid:12) (cid:12) (cid:12)
+ f(x)−fe r dµ(x)(cid:12) (cid:12) (cid:12) (cid:12)
µ(B6) ZBe6 B6
(cid:12) r (cid:12)
. 1+δ B,B(cid:12)6 kfkr (cid:12) . kfkr ,
e RBMO(µ) RBMO(µ)
h (cid:16) (cid:17)i
e
6
where the last inequality follows from the fact that δ(B,B6) . 1, which holds by [10,
Lemma 2.1].
On the other hand, for any (6,β )-doubling balls B ⊂ S,e
6
|m (|f|r)−m (|f|r)| ≤ |m (|f|r)−|f |r|+||f |r −|f |r|+||f |r −m (|f|r)|
B S B B B S S S
≤ m (|f −f |r)+|f −f |r +m (|f −f |r)
B B B S S S
. [1+δ(B,S)]rkfkr .
RBMO(µ)
Combining these two inequalities finishes the proof of Lemma 3.1.
Lemma 3.2. Let p ∈ [1, ∞) and f ∈ L1 (µ) such that f(x)dµ(x) = 0 if µ(X) < ∞.
loc X
If for any R > 0,
R
sup ℓpµ({x ∈ X : N(f)(x) > ℓ})< ∞,
0<ℓ<R
we then have
supℓpµ({x ∈ X : N(f)(x) > ℓ}) . supℓpµ x ∈ X : M♯(f)(x) > ℓ .
ℓ>0 ℓ>0
(cid:16)n o(cid:17)
Proof. Recall the λ-good inequality in [1] that for some fixed constant ν ∈ (0,1) and all
ǫ ∈ (0,∞), there exists some δ > 0 such that for any ℓ > 0,
µ x ∈ X : N(f)(x) > (1+ǫ)ℓ, M♯(f)(x) ≤ δℓ ≤ νµ({x ∈X : N(f)(x) > ℓ}).
(cid:16)n o(cid:17)
From this, it then follows that for R large enough and any ǫ > 0,
sup ℓpµ({x ∈ X : N(f)(x) >ℓ})
0<ℓ<R
≤ sup [(1+ǫ)ℓ]pµ({x ∈ X : N(f)(x) > (1+ǫ)ℓ})
0<ℓ<R
≤ ν(1+ǫ)p sup ℓpµ({x ∈ X : N(f)(x) > ℓ})
0<ℓ<R
+(1+ǫ)psupℓpµ x ∈ X : M♯(f)(x) > δℓ .
ℓ>0
(cid:16)n o(cid:17)
Choosing ǫ small enough such that ν(1+ǫ)p < 1, our assumption then implies that
sup ℓpµ({x ∈ X : N(f)(x) >ℓ}) . supℓpµ x ∈ X : M♯(f)(x) >ℓ .
0<ℓ<R ℓ>0
(cid:16)n o(cid:17)
LettingR → ∞thenleadstotheconclusion, whichcompletes theproofofLemma3.2.
Lemma 3.3. Let r ∈ (0, 1) and N (f) ≡ [N(|f|r)]1/r. Then for any p ∈ [1, ∞), there
r
exists a positive constant C, depending on r, such that for suitable function f and any
ℓ > 0,
µ({x ∈ X : N (f)(x) > ℓ}) ≤ Cℓ−psupτpµ({x ∈ X : |f(x)| > τ}).
r
τ≥ℓ
7
Proof. For each fixed ℓ > 0 and function f, decompose f as
f(x)= f(x)χ (x)+f(x)χ ≡ f (x)+f (x).
{x∈X:|f(x)|≤ℓ} {x∈X:|f(x)|>ℓ} 1 2
By the boundedness of N form Lp(µ) to Lp,∞(µ), we obtain that
µ({x ∈X : N (f)(x)> 21/rℓ}) ≤ µ({x ∈ X : N(|f |r)(x) > ℓr})
r 2
. ℓ−rp |f (x)|rpdµ(x)
2
ZX
. µ({x ∈ X : |f(x)| > ℓ})
∞
+ℓ−rp τrp−1µ({x ∈ X : |f(x)|> τ})dτ
Zℓ
. µ({x ∈ X : |f(x)| > ℓ})
+ℓ−psupτpµ({x ∈ X : |f(x)| > τ}),
τ>ℓ
which implies our desired result.
Proof of Theorem 1.1. BytheMarcinkiewiczinterpolationtheorem,weonlyneedtoprove
that for all f ∈ Lp(µ) with p ∈ (1, ∞) and ℓ > 0,
µ({x ∈ X : |Tf(x)| > ℓ}) .ℓ−pkfkp . (3.1)
Lp(µ)
We further consider the following two cases.
Case (i)µ(X)= ∞. LetL∞(µ)bethespace of bounded functions with bounded supports
b
and
L∞ (µ)≡ f ∈ L∞(µ) : f(x)dµ(x) = 0 .
b,0 b
(cid:26) ZX (cid:27)
Then in this case, L∞ (µ) is dense in Lp(µ) for all p ∈ (1,∞). Let r ∈ (0, 1) and
b,0
N (g) ≡ [N(|g|r)]1/r for any g ∈ Lr (µ). Notice that |Tf|≤ N (Tf) µ-almost everywhere
r loc r
on X. Then by a standard density argument, to prove (3.1), it suffices to prove that for
all f ∈ L∞ (µ) and p ∈ (1, ∞),
b,0
supℓpµ({x ∈ X : N (Tf)(x) > ℓ}) . kfkp . (3.2)
r Lp(µ)
ℓ>0
For each fixed ℓ > 0, applying Lemma 2.1, we obtain that f = g + h, where h is as
Lemma 2.1 and g ≡ f −h, such that
kgkL∞(µ) . ℓ, h ∈ H1(µ) (3.3)
and
khk . ℓ1−pkfkp . (3.4)
H1(µ) Lp(µ)
For each r ∈ (0, 1), define M♯(f) ≡ {M♯(|f|r)}1/r. Then, (3.3) together with the bound-
r
edness of T from L∞(µ) to RBMO(µ) and Lemma 3.1 shows that the function M♯(Tg) is
r
bounded by a multiple of ℓ. Hence, if c is a sufficiently large constant, we have
0
x ∈ X : M♯(Tg)(x) > c ℓ = ∅. (3.5)
r 0
n o
8
On the other hand, since both f and h belong to H1(µ), we see that g ∈ H1(µ) and
kgk ≤ kfk +khk . kfk +ℓ1−pkfkp .
H1(µ) H1(µ) H1(µ) H1(µ) Lp(µ)
By this together with the boundedness of T from H1(µ) to L1,∞(µ) and Lemma 3.3, we
have that for any p ∈ (1, ∞) and R > 0,
sup ℓpµ({x ∈ X : N (Tg)(x) > ℓ}) . sup ℓp−1supτµ({x ∈ X : |Tg(x)| > τ}) < ∞.
r
0<ℓ<R 0<ℓ<R τ≥ℓ
From this, (3.5), Lemma 3.2 and the fact that N ◦T is quasi-linear, we deduce that there
r
exists a positive constant C such that
supℓpµ x ∈ X : N (Tef)(x)> Cc ℓ
r 0
ℓ>0
(cid:16)n o(cid:17)
≤ supℓpµ({x ∈X : Nr(Tg)(x)e> c0ℓ})+supℓpµ({x ∈X : Nr(Th)(x) > c0ℓ})
ℓ>0 ℓ>0
. supℓpµ x ∈X : M♯(Tg)(x) > c ℓ +supℓpµ({x ∈ X : N (Th)(x) > c ℓ})
r 0 r 0
ℓ>0 ℓ>0
(cid:16)n o(cid:17)
. supℓpµ({x ∈X : N (Th)(x) > ℓ}). (3.6)
r
ℓ>0
From the boundednessof N from L1(µ)to L1,∞(µ) andthe boundednessof T fromH1(µ)
to L1,∞(µ), it follows that
µ({x ∈X : N (Th)(x) > ℓ})
r
ℓr
≤ µ x ∈ X : N |Th|rχ >
1
{x∈X:|(Th)(x)|>ℓ/2r} 2
(cid:18)(cid:26) (cid:27)(cid:19)
(cid:16) (cid:17)
r
. ℓ−r (Th)(x)χ (x) dµ(x)
1
{x∈X:|(Th)(x)|>ℓ/2r}
ZX (cid:12) (cid:12)
(cid:12) ℓ(cid:12)/2r1
. ℓ−rµ (cid:12)x ∈ X : |(Th)(x)| > ℓ/21r (cid:12) sr−1ds
(cid:16)n∞ o(cid:17)Z0
+ℓ−r sr−1µ({x ∈ X : |(Th)(x)| > s})ds
1
Zℓ/2r
1
1
. µ x ∈ X : |(Th)(x)| > ℓ/2r + sup sµ({x ∈ X : |(Th)(x)| > s})
ℓ
1
(cid:16)n o(cid:17) s≥ℓ/2r
khk
. H1(µ) .ℓ−pkfkp ,
ℓ Lp(µ)
which together with (3.6) yields (3.2).
Case (ii) µ(X) < ∞. In this case, assume that f ∈ L∞(µ). Notice that if ℓ ∈ (0,ℓ ],
b 0
where ℓ is as in Lemma 2.1, then (3.1) holds trivially. Thus, we only have to consider the
0
♯
case when ℓ > ℓ . Let r ∈ (0, 1), N (f) be as in Lemma 3.3 and M as in Case (i). For
0 r r
each fixed ℓ > ℓ , applying Lemma 2.1, we obtain that f = g+h with g and h satisfying
0
9
(3.3) and (3.4), which together with the boundedness of T from L∞(µ) to RBMO(µ) and
♯
Lemma 3.1 yields (3.5) for M (Tg). We now claim that
r
1
F ≡ |Tg(x)|rdµ(x) . ℓr, (3.7)
µ(X)
ZX
where the constant depends on µ(X) and r. In fact, since µ(X) < ∞, we regard X as a
ball. Then g ≡ g − 1 g(x)dµ(x) ∈ H1(µ). On the other hand, |T1|r ∈ RBMO(µ)
0 µ(X) X
because of the fact that T1 ∈ RBMO(µ) and Lemma 3.1. This together with µ(X) < ∞
R
implies that
|T1(x)|rdµ(x) < ∞.
ZX
Then by the boundedness of T from H1(µ) to L1,∞(µ) and (3.3), we have
r
1
|Tg(x)|rdµ(x) ≤ |Tg (x)|r + T g(y)dµ(y) (x) dµ(x)
0
µ(X)
ZX ZX (cid:26) (cid:12) (cid:20) ZX (cid:21) (cid:12) (cid:27)
. r kg0kH1(µ)/µ(X)(cid:12)(cid:12)(cid:12)tr−1µ({x ∈ X : |Tg0(x)| > t(cid:12)(cid:12)(cid:12)}) dt
Z0
∞
+r tr−1µ({x ∈ X : |Tg (x)| > t}) dt+ℓr
0
Zkg0kH1(µ)/µ(X)
. µ(X) kg0kH1(µ)/µ(X)tr−1dt+kg k ∞ tr−2dt+ℓr
0 H1(µ)
Z0 Zkg0kH1(µ)/µ(X)
. [µ(X)]1−rkg kr +ℓr . ℓr,
0 H1(µ)
which implies (3.7).
Observe that (|Tg|r −F)dµ(x) = 0 and for any R > 0,
X
sRup ℓpµ({x ∈ X : N(|Tg|r −F)(x) >ℓ})≤ Rpµ(X) < ∞.
0<ℓ<R
♯
From this together with Lemma 3.2, M (F) = 0, (3.7) and an argument similar to that
r
used in Case (i), we conclude that there exists a positive constant c such that
supℓpµ({x ∈ X : N (Tf)(x) > cc ℓ}) ≤ supℓpµ({x ∈X : N(|Tg|r −F)(x) > (c ℓ)r})
r 0 0
e
ℓ>ℓ0 ℓ>ℓ0
+supℓpµ({x ∈ X : N (Th)(x) > c ℓ})
e r 0
ℓ>ℓ0
. supℓpµ x ∈ X : M♯(Tg)(x) > c ℓ
r 0
ℓ>0
(cid:16)n o(cid:17)
+supℓpµ({x ∈ X : N (Th)(x) > c ℓ})
r 0
ℓ>0
. supℓpµ({x ∈ X : N (Th)(x) > ℓ}) . kfkp ,
r Lp(µ)
ℓ>0
whereinthefirstinequalitywechoosec largeenoughsuchthatF≤ (c ℓ)r. Thiscompletes
0 0
the proof of Theorem 1.1.
10