Sharp weighted estimates for multi-linear Calder\'{o} n-Zygmund ...

SHARP WEIGHTED ESTIMATES FOR MULTI-LINEAR ´ CALDERON-ZYGMUND OPERATORS ON NON-HOMOGENEOUS SPACES ABHISHEK GHOSH, PARASAR MOHANTY, AND SAURABH SHRIVASTAVA

arXiv:1609.08995v1 [math.CA] 28 Sep 2016

Abstract. We establish sparse domination of multi-linear Calder´ on-Zygmund operators on non-homogeneous spaces. As a consequence we obtain sharp weighted bounds for multi-linear Calder´ on-Zygmund operators with respect to multi-linear AP~ weights.

1. Introduction and Preliminaries The theory of multi-linear singular integral operators has been developed extensively in past two decades. In 1970’s Coifman and Meyer [1, 16] were one of the first to adopt a multi-linear point of view to study the commutators and paraproduct operators. In [4] Grafakos and Torres provided a systematic account of multi-linear Calderón-Zygmund operators. Recently, in [13] Lerner et al. developed a suitable notion of multi-linear weights and introduced an appropriate multi-linear maximal function. This maximal function plays the role of the classical Hardy-Littlewood maximal and helps in obtaining weighted estimates for the multi-linear Calderón-Zygmund operators. There have been many developments about weighted estimates for multi-linear operators in recent times. The most recent approach is the method of sparse domination of operators under consideration. This approach yields sharp weighted estimates for maximal operators and singular integral operators. In this paper, we extend the method of sparse domination to multi-linear Calderón-Zygmund operators defined on non-homogeneous spaces. Recently in [15, 17] sparse domination technique have been introduced for CalderónZygmund operators in non-homogeneous setting using David-Mattila cells. In order to describe the results is details, we need to first recall the notion of upper doubling and geometrical doubling metric measure spaces. Definition 1.1. [6, 15] We say that a metric measure space (X, d, µ) is upper doubling if there exist a dominating function λ : X × (0, ∞) → (0, ∞) and a constant Cλ > 0 such that for every x ∈ X the function r → λ(x, r) is non-decreasing and µ(B(x, r)) ≤ λ(x, r) ≤ Cλ λ(x, r/2) for all x ∈ X, r > 0. Here B(x, r) denotes the ball of radius r with center at x ∈ X. Remark 1.1. In [8] authors showed that with the help of the dominating function λ, one ˜ satisfying an additional property can define a new domination function λ ˜ r) ≤ Cλ λ(y, ˜ r) for all x, y ∈ X with |x − y| ≤ r (1) λ(x, Therefore, without loss of any generality we may always assume that the dominating function λ satisfies the property (1). Date: September 29, 2016. 1991 Mathematics Subject Classification. Primary: 42B20; Secondary: 42B25. Key words and phrases. Multi-linear operators, Calder´ on-Zygmund operators, sparse operators, Ap weights. 1

2

ABHISHEK GHOSH, PARASAR MOHANTY, AND SAURABH SHRIVASTAVA

Definition 1.2. We say that a metric space (X, d) is geometrically doubling and has doubling dimension n if for given 0 < r ≤ R and any ball B ⊂ X of radius R, the r−separated subsets in B have the cardinality at most C( Rr )n . We follow the notion of Ap weights presented in [19] and define multi-linear AP~ (µ) weights as follows. Let 1 6 p1 , . . . , pm < ∞ be such that p1 = p11 + · · · + p1m . Let P~ denote the m−tuple P~ = (p1 , . . . , pm ). Given an m−tuple of weights w ~ = (w1 , w2 , . . . , wm ), set vw~ =

m Y

p/pj

wj

.

j=1

Definition 1.3. Let ρ ∈ [1, ∞). An m−tuple of weights w ~ = (w1 , w2 , . . . , wm ) is said to belong to the multilinear AρP~ (µ)−class if it satisfies the multilinear AρP~ (µ)−condition 1/p Y 1/p′j Z Z m ′ 1 1 1−pj (2) sup ≤ K < ∞. vw~ dµ wj dµ µ(ρB) B µ(ρB) B B j=1 where the supremum is being taken over all balls.

Here we follow the standard interpretation of the average (µ ess inf B wj )−1 when pj = 1.

1 µ(ρB)

′

w 1−pj dµ B

R

1′ p

j

as

1.1. Multi-linear Calderon-Zygmund operators. Definition 1.4. An m−linear operator T which is bounded (weak-type) from L1 (X) × · · · × L1 (X) into L1/m,∞ (X) and is represented for bounded functions fj , 1 ≤ j ≤ m, with bounded supports by Z m Y T (f1 , f2 , . . . , fm )(x) = K(x, y1 , y2, . . . , ym ) fj (yj )dµ(yj ) Xm

j=1

∩m j=1 supp(fj ),

for all x ∈ / where the kernel K is locally integrable function defined away from the diagonal x = y1 = y2 = · · · = ym in X m+1 and satisfies • Size condition: 1 (3) |K(x, y1 , y2, . . . , ym )| . min i=1,2,...,m λ(x, d(x, yj ))m • Regularity conditions: (4) ! ′ 1 d(x, x ) |K(x, y1 , y2 , . . . , ym ) − K(x′ , y1 , y2 , . . . , ym )| . min ω Pm j=1,2,...,m λ(x, d(x, yi ))m j=1 d(x, yi ) whenever d(x, x′ ) ≤

1 max d(x, yi ). 2 j=1,2,...,m

Also a similar regularity condition for each j, (5) 1 ω |K(x, . . . , yj , . . . , ym) − K(x, . . . , yj , . . . , ym )| . min j=1,2,...,m λ(x, d(x, yi ))m whenever d(yj , yj′ ) ≤

1 max d(x, yi ). 2 j=1,2,...,m

where ω is a modulus of continuity satisfying the Dini condition.

d(yj , yj′ ) Pm j=1 d(x, yi )

!

´ MULTI-LINEAR CALDERON-ZYGMUND OPERATORS ON NON-HOMOGENEOUS SPACES

3

For r > 0, we define the truncated operator Tr as, Z m Y Tr (f1 , f2 , . . . , fm )(x) = K(x, y1 , y2 , . . . , ym ) fj (yj )dµ(yj . Pm

j=1

d(x,yj )2 >r 2

j=1

The truncated maximal operator is given by T ∗ (f1 , f2 , . . . , fm )(x) = sup |Tr (f1 , f2 , . . . , fm )(x)| r>0

1.2. David-Mattila cells. The notion of dyadic lattice in Rn with a non-doubling measure µ was introduced by David-Mattila [3]. In [15], authors observed that the same construction of David-Mattila cells works in general in the case of a geometrically doubling metric measure space. The David-Mattila cells are the key to obtain the sparse domination of Calderón-Zygmund operators in [15]. In this paper we exploit their ideas and extend this to multi-linear Calderón-Zygmund operators defined on a geometrically doubling metric measure space (X, d, µ). Let us first recall the notion of David-Mattila cells. Lemma 1.2. [3, 15, 17] Let (X, d, µ) be a geometrically doubling metric measure space with doubling dimension n and locally finite Borel measure µ. If W denotes the support of µ and C0 > 1 and A0 > 5000C0 are two given constants, then for each integer k, there exists a partition of W into Borel sets Dk = {Q}Q∈Dk with the following properties • For each k ∈ Z, the set W is disjoint union W = ∪Q∈Dk Q. Moreover, if k < l, Q ∈ Dl , and R ∈ Dk , then either Q ∩ R = ∅ or Q ⊂ R. • For each k ∈ Z and each cube Q ∈ Dk , there exists a ball B(Q) = B(zQ , r(Q)) with −k zQ ∈ W such that A−k and W ∩ B(Q) ⊂ Q ⊂ W ∩ 28B(Q) = 0 ≤ r(Q) ≤ C0 A0 W ⊂ B(zQ , 28r(Q)). Further, the collection {5B(Q)}Q∈Dk is pairwise disjoint. • Let Dkdb denote the collection of cubes Q in Dk satisfying doubling property µ(100B(Q)) ≤ C0 µ(B(Q)). • For the non-doubling cubes Q ∈ Dk \ Dkdb , we have that rQ = A−k 0 and µ(cB(Q)) ≤ C0−1 µ(100cB(Q)) for all 1 ≤ c ≤ C0 . 1.3. Multi-linear sparse operators. A family of measurable sets S = {Q} in X is said to be η-sparse, 0 < η < 1, if for every Q ∈ S there exists a measurable set EQ ⊂ Q such that • µ(EQ ) ≥ ηµ(Q) and • the sets {EQ }Q∈S are pairwise disjoint. Given a sparse family S in X and a large number α ≥ 200, a multi-linear version of the sparse operator is defined by ! Z m X Y 1 AS (f~)(x) = |fi |dµ χQ . µ(αB(Q)) 30B(Q) i=1 Q∈S 2. Main results The main result of this paper is the following.

4


Theorem 2.1. Let T be a multi-linear Calderón-Zygmund operator defined on an upper doubling, geometrically doubling metric measure space (X, d, µ). Then for all exponents m P 1 and any multiple weight w ~ ∈ AP~ (µ), we have 1 < p1 , . . . , pm < ∞ satisfying p1 = pi i=1

||T (f~)||pLp (vw~ ) . Cω

m Y

||fi ||pLpi (wi ) ,

i=1

where       supQ        Cω = supQ          supQ   

p′0

vw ~ (Q)

m Q

pp′0 ′ σi (αQ) pi

i=1 ′

µ(αB(Q))mp µ(Q)mp(p0 −1) m Q σi (αB(Q)) vw ~ (Q) i=1 1 m p′ µ(αB(Q)) vw ~ (EQ ) ′

p vw ~ (Q) 0

m Q

m Q

if m1 < p ≤ 1 and p0 = mini pi

1 σi (EQ ) pi

if p ≥ maxi p′i

i=1 ′

σi (αB(Q))p0

i=1

µ(αB(Q))m p′0 vw ~ (EQ

p′ 0 ) p′

m Q

σi (EQ

p′ 0 ) pi

if p′0 = max{p′1 , ..., p′m }.

i=1

We would like to remark that if the underlying measure is a doubling measure then in 1 p

max(1,

p′1 p

,...,

p′m ) p

each of the above cases mentioned above, we get that Cω . [w] . Therefore, ~ A ~ (µ) P in view of [14], we obtain sharp constants for the case of doubling measures. We shall follow the approach of Kranich and Volberg [15] and establish the above theorem using the method of sparse domination for multi-linear Calderón-Zygmund operators. To be precise, we shall show that Theorem 2.2. Let T be a multi-linear Calderón-Zygmund operator defined on an upper doubling, geometrically doubling metric measure space (X, d, µ). If α ≥ 200 and X ′ ⊂ X is a bounded set, then for integrable functions fj , 1 ≤ j ≤ m with support contained in X ′ , we can find sparse families Sk , k ≥ 0, of David-Mattila cells such that the sparse domination ∞ X ∗ ~ (6) T (f )(x) ≤ 100−k ASk (f~)(x) k=0

holds pointwise µ−almost everywhere on X ′ with implicit constants independent of X ′ .

Next, we will obtain sharp weighted estimates for the multi-linear sparse operator AS and thereby obtain Theorem 2.1. m P 1 . Then for any multiple Proposition 2.3. Let 1 < p1 , . . . , pm < ∞ be such that p1 = pi i=1

weight w ~ ∈ AP~ (µ) we have

||AS (f~)||pLp (vw~ ) . Cw

m Y

||fi ||pLpi (wi ) ,

i=1

where Cω as in Theorem 2.1.

3. Proofs of results 3.1. Domination by sparse operators. In this section we establish the domination of multi-linear Calderón-Zygmund operators by sparse operators and thereby obtain a proof


5

of Theorem 2.2. For notational convenience, we shall write the proof in bilinear setting only. Given a David-Mattila cell Q0 ∈ D, we shall denote by D(Q0 ) the collection of cells Q contained in Q0 . Here we consider X 2 with the metric d((x1 , x2 ), (y1 , y2 )) = max d(xi , yi ). Let Q ∈ D i=1,2

and x ∈ Q, define the truncated operator by Z F (x, Q) = K(x, y1 , y2 )f1 (y1 )f2 (y2 ), X 2 \30B(Q)2

where B(Q)2 is a ball in (X 2 , d) which is B(Q) × B(Q). For a fixed cell Q0 and x ∈ Q0 , the localized version of bilinear grand maximal truncated operator is defined by MT,Q0 (f1 , f2 )(x) =

sup

sup |F (y, P )|.

x∈P,P ∈D(Q0 ) y∈P

The bilinear grand maximal truncated operator MT,Q0 is the key tool in obtaing the sparse domination. We would require to prove the end-point weak-type estimates for this operator. This is achieved by proving a pointwise relation between the operators MT,Q0 and T ∗ . In the process we need to study another bilinear maximal operator Mλ , which may be thought of as an analogue of the classical Hardy-Littlewood maximal operator in bilinear setting. This is defined by Z Z 1 Mλ (f1 , f2 )(x) = sup |f1 (y1 )|dµ(y1) |f1 (y2 )|dµ(y2). 2 r>0 λ(x, r) B(x,r) B(x,r) 1

It is easy to see that Mλ is bounded from L1 (µ) × L1 (µ) to L 2 ,∞ (µ) Lemma 3.1. For any cell Q0 ∈ D, we have |MT,Q0 (f1 , f2 )(x) − T ∗ (f1 , f2 )(x)| ≤ C(||ω||Dini + CK )Mλ (f1 , f2 )(x) for a.e x ∈ Q0 , with constant independent of Q0 . Proof. Let Q ∈ D be a cell and x, x′ ∈ Q. Denote Jk = (2k+1 30B(Q))2 \ (2k 30B(Q))2 we have, |F (x, Q) − F (x′ , Q)| Z ≤ |K(x, y1 , y2) − K(x′ , y1 , y2)||f1 (y1 )||f2 (y2 )|dµ(y1)dµ(y2) X 2 \30B(Q)2

≤

XZ k≥0 J

|K(x, y1 , y2) − K(x′ , y1 , y2)|f1 (y1 )||f2 (y2 )|dµ(y1)dµ(y2)

k

! d(x, x′ ) 1 ω P min |f1 (y1 )||f2 (y2 )|dµ(y1)dµ(y2) ≤ i=1,2 λ(x, d(x, yi ))2 d(x, y i) j=1,2 k≥0 J k Z X 56r(Q) 1 |f1 (y1 )||f2(y2 )|dµ(y1)dµ(y2) ω .λ k r(Q))2 k r(Q) λ(x, 2 2 k≥0 XZ

(2k+1 30B(Q))2

.λ ||ω||DiniMλ (f1 , f2 )(x). Denote T˜r (f1 , f2 )(x) =

Z

X 2 \Sr (x)

K(x, y1 , y2 )f1 (y1 )f2 (y2 )dµ(y1)dµ(y2),

6


where Sr (x) = {~y ∈ X 2 : max d(x, yi ) ≤ r}. Let x ∈ Q0 and fix a cell P ∈ D(Q0 ) such i=1,2

that x ∈ P . Choose r > 0 to be such that r ∼ diam(P ). Then,

≤ ≤ ≤ ≤

|T˜ (f , f )(x) − F (x, P )| Zr 1 2 |K(x, y1 , y2 )||f1 (y1 )||f2(y2 )|dµ(y1)dµ(y2) 30B(P )2 \Sr (x) Z 1 |f (y )||f2(y2 )|dµ(y1)dµ(y2 ) min 2 1 1 30B(P )2 \Sr (x) i=1,2 λ(x, |x − yi |) Z 1 |f (y )||f2 (y2 )|dµ(y1)dµ(y2) CK 2 1 1 30B(P )2 \Sr (x) λ(x, 30r(P )) Z 1 |f1 (y1 )||f2 (y2 )|dµ(y1)dµ(y2) CK,λ λ(x, 30r(P ))2 30B(P )2

≤ CK,λMλ (f1 , f2 )(x).

Let y ∈ P be any point, then by combining the above estimates, we get (7) |T˜r (f1 , f2 )(x) − F (y, P )| ≤ (CK,λ + ||ω||Dini)Mλ (f1 , f2 )(x) Thus from the Eq.(7) and the above estimate one can easily see that |Tr (f1 , f2 )(x) − F (y, P )| ≤ (C˜K,λ + ||ω||Dini)Mλ (f1 , f2 )(x) sup |F (y, P )| ≤ T ∗ (f1 , f2 )(x) + (CK,λ + ||ω||Dini)Mλ (f1 , f2 )(x) y∈P

Finally, taking the supremum over all cells P containing x, we obtain: |MT,Q0 (f1 , f2 )(x)| − T ∗ (f1 , f2 )(x)| ≤ (CK,λ + ||ω||Dini)Mλ (f1 , f2 )(x). 1

Following the arguments in [5, 18], T ∗ is bounded from L1 (µ) × L1 (µ) to L 2 ,∞ (µ). 1 Thus MT,Q0 is is bounded from L1 (µ) × L1 (µ) to L 2 ,∞ (µ). This completes the proof of Lemma 3.1. The proof of the sparse domination (6) is constructive and follows a recursive argument. We shall prove a recursive formula involving the operator MT,Q0 . Before proceeding further, we set some notation. For α ≥ 200 and a cell Q, we write for bilinear averages Z 2 Y 1 ~ A(f , Q) = |fi |dµ. µ(αB(Q)) 30B(Q) i=1

Also, for a given cell Q, we use the notation

µ(αB(Q))2 Θ(Q) = λ(zQ , r(Q))2 We shall prove the following recursive formula. ˆ we have Lemma 3.2. For any two nested cells Q ⊂ Q, (8)

~ ˆ ˆ MT,Qˆ (f1 130B(Q) ˆ , f2 130B(Q) ˆ )(x) ≤ CΘ(Q)A(f , Q) + MT,Q (f1 130B(Q) , f2 130B(Q) )(x)

ˆ implies 30B(Q) ⊂ 30B(Q). ˆ We construct the David-Mattila cells such that Q ⊂ Q


7

ˆ with x ∈ P , we have Proof. For every cell Q ∈ D, x ∈ Q, and every cell P ∈ D(Q) Z |F (y, P )| = | K(y, z1, z2 )f1 (z1 )f2 (z2 )dµ(z1 )dµ(z2 )| ˆ 2 \30B(P )2 30B(Q) Z = | K(y, z1, z2 )f1 (z1 )f2 (z2 )dµ(z1 )dµ(z2 ) 30B(Q)2 \30B(P )2 Z + K(y, z1, z2 )f1 (z1 )f2 (z2 )dµ(z1 )dµ(z2 )| ˆ 2 \30B(Q)2 30B(Q) Z 1 |f1 (z1 )||f2 (z2 )|dµ(z1)dµ(z2 ). . MT,Q (f1 130B(Q) , f2 130B(Q) )(x) + λ(x, r(Q))2 30B(Q) ˆ 2

Therefore, we get the recursive formula (8).

We know from [3, 15] that the quantities Θ(Q) are bounded by 1 and they decrease exponentially for a nested sequence of non-doubling cells. More precisely, we know that Lemma 3.3. [3, 15] Let l0 be the maximal number with 100l0 < Cα0 and suppose that ⌈log A⌉ l /2 C00 > Cλ 2 , where Cλ is the doubling constant of the dominating function λ. Let Q0 = Qˆ1 ⊃ Q1 = Qˆ2 ⊃ Q2 = Qˆ3 ⊃ . . . be a nested family of cubes such that Q1 , Q2 , . . . are non-doubling. Then, Θ(Qk ) . C0−kl0 Θ(Q0 ) Lemma 3.4. Let Q0 ∈ D db be a doubling cell. Then there exists a subset Ω ⊂ Q0 , a collections of pairwise disjoint cubes Cn (Q0 ) ⊂ D, n = 1, 2, 3, . . . contained in Ω, and a collection of pairwise disjoint doubling cubes F (Q0 ) ⊂ D db contained in Ω such that 1 1. µ(Ω) ≤ µ(Q0 ) 2 2. For every P ∈ F and Q ∈ Cn (Q0 ) either P ⊂ Q or P ∩ Q = ∅, ∞ X X X 3. MT,Q0 (f~)1Q0 ≤ MT,P (f~)1P + CA(f~, Q0 ) + C 100−n A(f~, Q)1Q . n=1

P ∈F (Q0 )

Q∈Cn (Q0 )

Proof. The weak-type (1, 1, 1/2) boundedness of the operator MT,Q0 implies that for large enough M > 0, the set Ω = {x ∈ Q0 : MT,Q (f1 , f2 )(x) > MA(f~, Q0 )} 0

satisfies the desired property (1). Since Ω can be decomposed into a collection of maximal and hence disjoint DavidMattila cells, say C0 (Q0 ), we get X 1 µ(Q) ≤ µ(Q0 )· 2 Q∈C0 (Q0 )

For a cell Q ∈ C0 (Q0 ) and x ∈ Q, the maximality of Q implies that ˜ ≤ MA(f~, Q0 ) sup |F (x, Q)| ˜ y∈Q

˜ such that Q ⊂ Q ˜ ⊂ Q0 . Therefore, we have for any Q MT,Q (f1 , f2 )(x) ≤ MA(f~, Q0 ) + M 0

˜ (f1 , f2 )(x). T,Q

8


The above estimate together with the recursive formula (8) yield ˆ ˆ + MT,Q (f1 130B(Q) , f2 130B(Q) )(x). MT,Q0 (f1 , f2 )(x) ≤ MA(f~, Q0 ) + CΘ(Q)A( f~, Q) Next, we separate the cells with doubling property and set F = {Q ∈ C0 (Q0 ) : Q is doubling} and C1 = C0 (Q0 ) \ F . ˆ ∈ C1 , if Q is doubling we will put it in the basket F Now for a cell Q ∈ D such that Q otherwise it goes to C2 . We will follow this process inductively. Now by Lemma 3.3 we know that the quantities Θ(Q), Q ∈ Cn decay exponentially thus summing over them, we get the desired result. Recall Lemma 3.1 and note that in order to prove sparse domination of the operator T , it is sufficient to do so for the maximal operator Mλ . The next lemma establishes 0 this sparse domination. We require to consider the localized version Md,Q of Mλ for λ doubling cell Q0 , which is defined by taking the supremum over cells contained in D(Q0 ), that is, ∗

0 ~ Md,Q (f )(x) λ

=

sup

2 Y

x∈P,P ∈D(Q0) i=1

1 λ(zP , r(P ))

Z

|fi |dµ. 30B(P )

Lemma 3.5. For every doubling cell Q0 ∈ D db there exist a subset Ω ⊂ Q0 , a collections of pairwise disjoint cubes Cn (Q0 ) ⊂ D, n = 1, . . . contained in Ω, and a collection of pairwise disjoint doubling cubes F (Q0 ) ⊂ D db contained in Ω such that 1 1. µ(Ω) ≤ µ(Q0 ) 2 2. For every P ∈ F and Q ∈ Cn (Q0 ) then either P ⊂ Q or P ∩ Q = ∅, ∞ X X X d,P ~ −n 0 ~ ~ 3. Md,Q ( f )χ ≤ M ( f )χ + CA( f , Q ) + C 100 Q0 P 0 λ λ n=1

P ∈F (Q0 )

A(f~, Q)χQ

Q∈Cn (Q0 )

The proof of this lemma is similar to the previous lemma. Thus we skip the proof. To have recursive application of Lemma 3.4 we will encounter cubes which are doubling as well as non-doubling. At this point we follow the arguments as given in [15] to find a sparse collection. This completes the proof of Theorem 2.2.

3.2. Sharp weighted inequalities for multi-linear oparse operators. We begin with the following observation about multiple weights. For details one can look into [14]. Lemma 3.6. Let 1 < p1 , . . . , pm < ∞ be such that ′ , wi+1, . . . , wm ) AP~ (µ), then w ~ = (w1 , . . . , wi−1 , vw1−p ~ i ′ P~ = (p1 , . . . , pi−1 , p , pi+1 , . . . , pm ).

i

1 p

=

m P

i=1

1 . pi

If w ~ = (w1 , . . . , wm ) ∈ i

p′i p

∈ AP~ i with [w ~ ]AP~ i = [w] ~ A ~ , where P


Proof of Proposition 2.3: We first consider the case p1 = min {p1 , p2 }. Now, i=1,2 Z AS (f~σ )p vw~ dµ X !p Z 2 X Y 1 ≤ |fi |σi vw~ (Q) µ(αB(Q)) 30B(Q) i=1 Q∈S ≤ (sup KQ ) Q

X

Q∈S

µ(B) vw~ (Q)

′ p1 −1

′

σi (αQ)

< p ≤ 1. Let’s assume

2 Z Y

mp(p1 −1) 2 Q

1 m

′ pp 1 ′ p i

|fi |σi

30B(Q)

i=1

!p

i=1

where

vw~ (Q) KQ =

′ p1

2 Q

σi (αQ)

′ pp1 ′ p i

i=1 ′

µ(αB(Q))mp µ(Q)mp(p1 −1)

Now we observe that µ(Q)

′ mp(p1 −1)

. µ(E(Q))

′ mp(p1 −1)

≤ vw~ (Q)

′ p1 −1

2 Y

σi (EQ )

′ p(p1 −1) ′ p i

i=1

and ′ p1 −1

′ p1 −1

vw~ (Q) ≤ vw~ (Q) and σi (EQ ) Now using all the estimates above, we get Z AS (f~σ )p vw~ dµ

′ p(p1 −1) − pp ′ i p i

≤ σi (Q)

′ pp1 ′ −p p i

X

2 XY

p Z p 1 |fi |σi σi (EQ ) pi ≤ (sup KQ ) σi (αB(Q)) 30B(Q) Q Q∈S i=1 pi Z 2 X Y p 1 |fi |σi σi (EQ )} pi ≤ (sup KQ ) { σi (αB(Q)) 30B(Q) Q i=1 Q∈S ! pp 2 i Y XZ Mσi dµ (fi )pi σi dµ ≤ (sup KQ ) Q

≤ (sup KQ ) Q

≤ (sup KQ ) Q

i=1

2 Y

Q∈S

EQ

||Mσi dµ (fi )||pLpi (σi dµ)

i=1

2 Y

||fi ||pLpi (σi dµ)

i=1

Write vw~ (Q) Cω = sup KQ = sup Q

Q

′ p1

2 Q

σi (αQ)

9

′ pp1 ′ p i

i=1 ′

µ(αB(Q))mp µ(Q)mp(p1 −1)

10


If the underlying measure is doubling then µ(αB(Q)) and µ(Q) are comparable. vw~ (αB(Q))

2 Q

′ p1

σi (αB(Q))

′ pp1 ′ p i

p

i=1

Cω = sup KQ ≤

′

µ(αB(Q))mp µ(αB(Q))mp(p1 −1)

Q

′

≤ [w] ~ A1~ (µ) P

Therefore, for doubling measure, we have ′ p1 p

||AS (f~)||Lp (vw~ ) . [w] ~ A ~ (µ) P

m Y

||fi ||Lpi (wi ) ,

i=1

which is sharp for doubling measures. Next, consider the case when p ≥ maxi p′i . It is sufficient to prove m Y

||AS (f~σ )||Lp (vw~ ) . [w] ~ AP~ (µ)

||fi ||Lpi (σi )

i=1

for all fj in a dense class. Since, p > 1, we use the duality to estimate the above. For a ′ non-negative function g ∈ Lp (vw~ ) consider Z

AS (f~σ )gvw~ dµ ≤

X

XZ

Q∈S

=

gvw~ dµ ·

Q

m Y i=1

m Q

1 µ(αB(Q))

X vw~ (Q) i=1 σi (200B(Q)) µ(αB(Q))m

Q∈S

vw~ (Q)

|fi |σi dµ

30B(Q)

1 vw~ (Q)

Z

gvw~ dµ

Q

σi (200B(Q))

i=1

. sup Q

m Q

Z

1

µ(αB(Q))m vw~ (EQ ) p′

m Q

1

σi (EQ ) pi

i=1

"

i=1

" m Y X i=1

1 ( σi (200B(Q)) Q∈S

Z

m Y

1 σi (200B(Q))

1 ( v ~ (Q) Q∈S w X

# p1

i

|fi |σi )pi σi (EQ )

30B(Q)

.

Denote vw~ (Q)

m Q

σi (200B(Q))

i=1

LQ =

1

µ(αB(Q))m vw~ (EQ ) p′

m Q

1

σi (EQ ) pi

i=1

Now using the facts that, µ(Q)

m(p−1)

. µ(EQ )

m(p−1)

. vw~ (EQ )

p−1 p

m Y i=1

σi (EQ )

p−1 ′ p 1

Z

′

Z

gvw~ )p vw~ (EQ ) Q

|fi |σi dµ

30B(Q)

# 1′ p

´ MULTI-LINEAR CALDERON-ZYGMUND OPERATORS ON NON-HOMOGENEOUS SPACES p ′

p ′

′

11

−1

As p ≥ maxi pi and EQ ⊂ Q, we have σi (EQ ) pi ≤ σi (200B(Q)) pi σi (EQ ) for all i = 1, . . . , m. Hence, Z m Y D ~ ′ AS (fσ )gvw~ dµ . (sup LQ )||Mvw~ dµ (g)||Lp (v dµ) ||Mσi dµ (fi )||Lpi (σi ) w ~

Q

R

i=1

. (sup LQ )||g||Lp′ (v

w ~ dµ)

Q

m Y

||fi ||Lpi (σi ) .

i=1

Similar to the previous case if we restrict ourselves to doubling measures one can see that Cω = (sup LQ ) ≤ [w] ~ AP~ . Therefore, by duality we get the result. Q

Finally, we consider the the general case of exponents. Note that the operator AS is a multi-linear self-adjoint operator with respect to each j, 1 ≤ j ≤ m. Hence without loss of generality we may assume that p′1 ≥ max(p, p′2 , ..., p′m ). The self-adjointness of the operator together with the previous estimate and Lemma 3.6 yields ||AS ||Lp1 (w1 )×···×Lpm (wm )→Lp (vw~ ) = ||AS ||

′

′

1−p′ 1)

1−p )×···×Lpm (wm )→Lp1 (w1 Lp′ (vw ~

. [w ~ 1]AP~ 1 p′i p

= [w] ~ A~ . P

This completes the proof of Proposition 2.3. References [1] R. R. Coifman; Y. Meyer, On commutators of singular integrals and bilinear singular integrals. Trans. Amer. Math. Soc. 212 (1975), 315-331. [2] W. Dami´ an; A. K. Lerner; C. Pérez, Sharp weighted bounds for multilinear maximal functions and Calder´ on-Zygmund operators. J. Fourier Anal. Appl. 21 (2015), no. 1, 161-181. 42B20 (42B25) [3] G. David; P. Mattila, Removable sets for Lipschitz harmonic functions in the plane. Rev. Mat. Iberoamericana 16 (2000), no. 1, 137-215. 31A15 (30C85 42B20) [4] L. Grafakos; R. H. Torres, Multilinear Calder´ on-Zygmund theory. Adv. Math. 165 (2002), no. 1, 124-164. 42B25 (35S05 47G10) [5] Grafakos, Loukas; Torres, Rodolfo H. Maximal operator and weighted norm inequalities for multilinear singular integrals. Indiana Univ. Math. J. 51 (2002), no. 5, 1261-1276. 42B25 [6] T. P. Hytönen, A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa. Publ. Math. 54.2 (2010), 485-504. [7] T. P. Hytönen, The sharp weighted bound for general Calder´ on-Zygmund operators. Ann. of Math. (2) 175 (2012), no. 3, 1473-1506. 42B20 (42B25) [8] T. P. Hytönen; D. Yang; D. yang The Hardy space H 1 on non-homogeneous metric spaces. Math. Proc. Cambridge Philos. Soc. 153 (1) (2012), 9-31. [9] T. P. Hytönen; S. Liu; D. Yang; D. yang Boundedness of Calder´ on-Zygmund operators on nonhomogeneous spaces. Canad. J. Math. 64 (4) (2012), 892-923. [10] M. T. Lacey, An elementary proof of the A2 bound, Israel J. Math., 2015 to appear. [11] A. K. Lerner, A simple proof of the A2 conjecture, Int. Math. Res. Not. (IMRN) 14 (2013), 3159-3170. [12] A.K. Lerner, On pointwise estimates involving sparse operators. New York J. Math. 22 (2016), 341349. 42B20 (42B25) [13] A. K. Lerner; S. Ombrosi; C. Pérez; R. H. Torres; R. Trujillo-González, New maximal functions and multiple weights for the multilinear Calder´ on-Zygmund theory. Adv. Math. 220 (2009), no. 4, 1222-1264. 42B20 [14] Li, Kangwei; Moen, Kabe; Sun, Wenchang The sharp weighted bound for multilinear maximal functions and Calder¨ on-Zygmund operators. J. Fourier Anal. Appl. 20 (2014), no. 4, 751-765. 42B20 (42B25 42B30 47H60)

12


[15] A. Volberg; P. Zorin-Kranich, Sparse domination on non-homogeneous spaces with an application to Ap weights. [16] R. R. Coifman; Y. Meyer, Wavelets: Calder´ on-Zygmund and multilinear operators. Cambridge Univ. Press, Cambridge, UK, 1997. [17] J. M. Conde Alonso; J. Parcet, Nondoubling Calder´ on-Zygmund theory- A Dyadic Approach. arxiv:1604.03711. [18] Lin, H.; Meng, Y. Maximal multilinear Calder´ on-Zygmund operators with non-doubling measures. Acta Math. Hungar. 124 (2009), no. 3, 263-287. 42B20 (42B25) [19] Orobitg, Joan; Pérez, Carlos Ap weights for nondoubling measures in Rn and applications. Trans. Amer. Math. Soc. 354 (2002), no. 5, 2013-2033. 42B25 (42B20) [20] Xu, Jingshi, Boundedness of multilinear singular integrals for non-doubling measures. J. Math. Anal. Appl. 327 (2007), no. 1, 471-480. 42B20 (47G10) Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur-208 016 E-mail address: [email protected], [email protected] Department of Mathematics, Indian Institute of Science Education and Research, Bhopal-462 066 E-mail address: [email protected]