A new Ap –A∞ estimate for Calderón–Zygmund operators in spaces of homogeneous type

Accepted Manuscript A new Ap -A∞ estimate for Calderón-Zygmund operators in spaces of homogeneous type Kangwei Li

PII: DOI: Reference:

S0022-247X(15)00312-1 http://dx.doi.org/10.1016/j.jmaa.2015.03.076 YJMAA 19365

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Journal of Mathematical Analysis and Applications

Received date: 24 January 2015

Please cite this article in press as: K. Li, A new Ap -A∞ estimate for Calderón-Zygmund operators in spaces of homogeneous type, J. Math. Anal. Appl. (2015), http://dx.doi.org/10.1016/j.jmaa.2015.03.076

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A New Ap -A∞ Estimate for Calder´on-Zygmund Operators in Spaces of Homogeneous Type $ Kangwei Li School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

Abstract on-Zygmund operators In this note, we study the Ap -A∞ estimate for Calder´ in terms of the weak A∞ characteristics in spaces of homogeneous type. The weak A∞ class was introduced recently by Anderson, Hyt¨onen and Tapiola. Our estimate is new even in Euclidean space. Keywords: two weight inequalities; bump conditions; spaces of homogeneous type; sparse operators

1. Introduction and Main Results Weighted norm inequalities and spaces of homogeneous type are hot topics in harmonic analysis, both of them play important roles in partial differential equations. Let T be a Calder´ on-Zygmund operator and (w, σ) be a pair of weights. In the Euclidean setting, Hyt¨onen and Lacey [9] proved that if w(Q)  σ(Q) p−1 sup <∞ [w, σ]Ap := |Q| Q:cubes in Rn |Q| and w, σ ∈ A∞ , then the following estimate holds 1

1 

1

T (·σ)Lp (σ)→Lp (w) ≤ Cn,p,T [w, σ]Ap p ([w]Ap ∞ + [σ]Ap ∞ ).

(1.1)

When p = 2, (1.1) was proved by Hyt¨onen and P´erez [11]. It is well known that (1.1) extends the A2 theorem, which was first proved by Hyt¨onen [6], $ This work was partially supported by the National Natural Science Foundation of China(11371200), the Research Fund for the Doctoral Program of Higher Education (20120031110023) and the Ph.D. Candidate Research Innovation Fund of Nankai University. Email address: [email protected] (Kangwei Li)

Preprint submitted to Elsevier

March 31, 2015

see also in [14] for a simple proof by Lerner, and in [3], Anderson and Vagharshakyan also gave a proof in spaces of homogeneous type. Our goal is to extend (1.1) with the weak A∞ characteristics (which will be introduced below) to spaces of homogeneous type. Now let us recall some definitions. By a space of homogeneous type (SHT ) we mean an ordered triple (X, ρ, μ), where X is a set, ρ is a quasimetric on X, i.e., (i). ρ(x, y) = 0 if and only if x = y; (ii). ρ(x, y) = ρ(y, x) for all x, y ∈ X; (iii). ρ(x, z) ≤ κ(ρ(x, y) + ρ(y, z)) for some κ ≥ 1 and all x, y, z ∈ X; and μ is a nonnegative Borel measure on X which satisfies the following doubling condition μ(B(x, 2r)) ≤ Dμ(B(x, r)), where B(x, r) := {y ∈ X : ρ(x, y) < r} and the dilation of a ball B := B(x, r) denoted by λB will be understood as B(x, λr). We point out that the doubling property implies that any ball B(x, r) can be covered by at most N := ND,κ balls of radius r/2. Next let us introduce the weak A∞ class, which was first introduced by Anderson, Hyt¨ onen and Tapiola in [2]. For every δ > 1, we say w belongs to δ-weak A∞ class Aδ∞ if  1 M (1B w)(y)dμ(y) < ∞, [w]Aδ∞ := sup B w(δB) B where the supremum is taken over all balls B ⊂ X. We collect some properties of this weak A∞ class and refer the readers to [2] for a proof. 

Proposition 1.1. (i). Aδ∞ = Aδ∞ for all δ, δ  > κ. So hereafter, we denote by Aweak := A2κ ∞ ∞ the weak A∞ class; (ii). For any w ∈ Aweak ≥ 1/(2(2κ)log2 N ); ∞ , we have [w]Aweak ∞ weak (iii). Let w ∈ A∞ . Then there exists a constant α := α(κ, D) such that for every 0 <  ≤ α[w]1weak , A∞

  1  1+ − w1+ dμ − B

2κB

wdμ.

(1.2)

Now we are ready to state the main result in this paper. Theorem 1.2. Given p, 1 < p < ∞ and an SHT (X, ρ, μ). Let T be any Calder´ on-Zygmund operator and (w, σ) be a pair of weights. Then we have 1

1

1

weak p p T (·σ)Lp (σ)→Lp (w) ≤ C[w, σ]Ap p (([w]weak A∞ ) + ([σ]A∞ ) ),

2

where [w, σ]Ap :=

sup

B:balls in X

  p−1 − wdμ − σdμ B

B

and the constant C is independent of the weights (w, σ). Remark 1.3. Note that the result is new already in the case that X = Rn with Euclidean distance and Lebesgue measure, since the weak A∞ class is strictly larger than classical A∞ already in this setting. 2. Proof of the Main result In this section, we will give a proof for Theorem 1.2. First, we introduce the bump conditions. By a Young function φ, we mean that φ : [0, ∞) → [0, ∞) is continuous, convex and increasing satisfying φ(0) = 0 and φ(t)/t → ¯ ∞ as t → ∞. Recall that the complementary function of φ, denoted by φ, is defined by ¯ := sup{st − φ(s)}. φ(t) s>0

Given two Young functions Φ, Ψ, define [w, σ]Φ,p

 1 p 1 := sup − wdμ σ p Φ,B , B

B

  1 p 1 [σ, w]Ψ,p := sup w p Ψ,B − σdμ , B

B

where the supremum is taken over all balls in X and the Luxemburg norm is defined by  f  dμ ≤ 1}. f Φ,B := inf{λ > 0, − Φ λ B There is a famous problem named the separated bump conjecture, which states that for any Calder´ on-Zygmund operator T , if [w, σ]Φ,p + [σ, w]Ψ,p < ∞,

(2.1)

¯ ∈ Bp (the Bp condition is recalled in (2.2) below), ¯ ∈ Bp and Ψ where Φ then T (·σ) is bounded from Lp (σ) to Lp (w). For the so-called log-bumps, namely, when 



Φ(t) = tp log(e + t)p −1+δ and Ψ(t) = tp log(e + t)p−1+δ ,

3

this conjecture has been verified in [5] in the Euclidean setting and in [1] for spaces of homogeneous type. For more about the separated bump conjecture, see [12, 17] and the references therein. In the rest of this paper, by carefully calculating the constants, we will show the following quantitative estimate for the power bumps. Theorem 2.1. Given p, 1 < p < ∞ and an SHT (X, ρ, μ). Let T be a Calder´ on-Zygmund operator and (w, σ) is a pair of weights satisfying (2.1)  for Φ(t) = tp r and Ψ(t) = tps , where 1 < r, s ≤ 1 + 2(2κ)log2 N /α(κ, D). Then we have 

¯ 1/p + [σ, w]Ψ,p [Ψ] ¯ 1/p ), T (·σ)Lp (σ)→Lp (w) ≤ CT,p,D,κ ([w, σ]Φ,p [Φ] B  Bp p

where recall that for a Young function φ ∈ Bp ,  ∞ φ(t) dt [φ]Bp := . p t 1/2 t

(2.2)

Now with Theorem 2.1 we are ready to prove Theorem 1.2. Proof of Theorem 1.2. In fact, taking r =1+

1 . α[σ]weak A∞

By the reverse H¨older’s inequality (1.2), we have   1   1 p p r − σ r dμ [w, σ]Φ,p = sup − wdμ B

B

B

  1  p −  sup − wdμ B

≤

B

2κB

 1 σdμ

p

1/p Cp,D,κ [w, σ]Ap .



For Φ(t) = tp r , by definition, we know   ¯ Φ(t) = t(p r)

 1   1 p r − 1   p r−1 p,κ,D t(p r) . p r p r

Hence  ¯ Bp p,κ,D [Φ]

∞ (p r) t 1/2

tp

(r−1)p p r − 1 dt = 2 p r−1 ≤ Cp,κ,D [σ]weak A∞ . t (r − 1)p

4

Then by taking similar value of s we can get the result as desired.  In the rest of this paper, we will focus on the proof of Theorem 2.1. We will reduce the estimates for Calder´on-Zygmund operators to the so-called sparse operators. So first let us introduce the following result, which can be found in [7], see also in [4]. Here we follow the version used in [2]. Theorem 2.2. Let 0 < η < 1 satisfy 96κ6 η ≤ 1. Then there exists countable sets of points {zαk,t : α ∈ Ak }, k ∈ Z, t = 1, 2, · · · , K = K(κ, N, η), and a finite number of dyadic systems D t := {Qk,t α : α ∈ Ak , k ∈ Z}, such that (i). for every t ∈ {1, 2, · · · , K} we have (a) X = ∪α∈Ak Qk,t α (disjoint union) for every k ∈ Z; (b) Q, P ∈ D t ⇒ Q ∩ P = {∅, Q, P }; k,t k,t k,t t k k (c) Qk,t α ∈ D ⇒ B(zα , c1 η ) ⊆ Qα ⊂ B(zα , C1 η ), where c1 := (12κ4 )−1 and C1 := 4κ2 ; (ii). for every ball B = B(x, r) there exists a cube QB ∈ ∪t D t such that B ⊆ QB and l(QB ) = η k−1 , where k is the unique integer such that η k+1 < r ≤ η k and l(QB ) = η k−1 means that QB = Qk−1,t for some α indices α and t. By the doubling property, we know that μ(B(x, r))  μ(QB ). And if k−1,t Qk,t , by the doubling property we also know that there exists α ⊂ Qβ some constant Cκ,D such that μ(Qk−1,t ) ≤ μ(B(zβk−1,t , C1 η k−1 )) ≤ Cκ,D μ(B(zαk,t , c1 η k )) ≤ Cκ,D μ(Qk,t α ). β Theorem 2.2 characterizes the structure of dyadic system in spaces of homogeneous type. See also in [8] for an exact characterization of which kinds of sets can be dyadic cubes. Now with Theorem 2.2 we can get the following result, which was proved in [1]. Lemma 2.3. Given a pair of weights (w, σ), and Young functions Φ and Ψ. Then [w, σ]Φ,p 

max

t

[w, σ]D Φ,p ,

[σ, w]Ψ,p 

t∈{1,2,··· ,K}

where t [w, σ]D Φ,p

max

 1 p 1 := sup − wdμ σ p Φ,Q Q∈D t

Q

Dt

and [σ, w]Ψ,p is defined similarly. 5

t

[σ, w]D Ψ,p ,

t∈{1,2,··· ,K}

Now for any fixed D t , t ∈ {1, 2, · · · , K}, we call a family S ⊂ D t sparse if for any Q ∈ S, μ(E(Q)) ≥ 12 μ(Q), where E(Q) = Q \ ∪Q ∈S,Q Q Q . Our purpose is to reduce the estimates for Calder´on-Zygmund operators to the following so-called sparse operators,  S − f (y)dμ(y)1Q (x), T (f )(x) := Q∈S Q

where S ⊂ D is a sparse family in some dyadic system D. In [14], Lerner gave a nice formula which reduces the norm of Calder´on-Zygmund operators to sparse operators. (In the recent book by Lerner and Nazarov [16], it has been shown that Calder´ on-Zygmund operators can be dominated pointwise by the sparse operators.) In [1], the authors showed that Lerner’s formula also holds in spaces of homogeneous type. Lemma 2.4. Given an SHT (X, ρ, μ) and a Calder´ on-Zygmund operator T , then for any Banach function space Y , T (f σ)Y ≤ CD,κ sup T S (f σ)Y , D t ,S

where the supremum is taken over every dyadic system D t , t = 1, 2, · · · , K and every sparse family S in D t . In the rest of this paper, we only need to prove Theorem 2.1 for sparse operators. We follow the strategy of [5]. We further reduce the problem to estimate testing conditions. To be precise, we have the following, see [13] for a proof. Lemma 2.5. For fixed t ∈ {1, 2, · · · , K}, suppose S is a sparse family in D t . Then   Q∈S −Q σdμ1Q (x)Lp (w) Q⊂R T S (·σ)Lp (σ)→Lp (w) ≤ sup (2.3) σ(R)1/p R   Q∈S −Q wdμ1Q (x)Lp (σ) Q⊂R . + sup w(R)1/p R Before we give further estimates, we introduce the following result. In the Euclidean case this is due to P´erez [18], and in spaces of homogeneous type, see P´erez and Wheeden [19] and Pradolini and Salinas [20]. We give the version used in [1].

6

Lemma 2.6. Given p, 1 < p < ∞ and an SHT (X, ρ, μ) and a Young function Φ such that Φ ∈ Bp , then 1/p

MΦD Lp (μ) ≤ Cκ,D [Φ]Bp f Lp (μ) , where D is some dyadic system in X and MΦD f (x) := sup f Φ,Q . Qx Q∈D

Then by Theorem 2.2 we immediately get 1/p

 [Φ]Bp f Lp (μ) . MΦ Lp (μ) ≤ Cκ,D

(2.4)

Now by symmetry we concentrate on the first term of (2.3). We follow the technique introduced in [9], see also in [5]. For convenience, set f Q = − f dμ and denote Q ≤ 2a+1 and Q ⊂ R}. Sa := {Q ∈ S : 2a < wQ σp−1 Q Denote by P0a the collection of maximal cubes in Sa . Now we define a Pna := {maximal cubes P  ⊂ P ∈ Pn−1 such that P  ∈ Sa , σP  > 2σP }.

Then denote P a := ∪n Pna . For any P ∈ P a , set Sa (P ) := {Q ∈ Sa : π(Q) = P }, where π(Q) is the minimal cube in P a which contains Q. We have the following Lemma, which was proved in [10] in the Euclidean case. Since we are working in spaces of homogeneous type, we give a proof here. The idea of the proof can also be found in [15]. Lemma 2.7. There exists a constant c such that w{x ∈ P : T Sa (P ) (σ) > tσP }  e−ct w(P ), where T Sa (P ) (f )(x) :=

  − f (y)dμ(y)1Q (x).

Q∈Sa (P ) Q

7

Proof. First, we claim that μ{x ∈ P :



1Q (x) > t}  2−t μ(P ).

(2.5)

Q∈Sa (P )

In fact, for any nonnegative integer τ , set

 1Q (x) > τ } := Qτj , {x ∈ P : j

Q∈Sa (P )

where {Qτj }j is a family of disjoint dyadic cubes in Sa (P ). The cubes {Qτj }j are defined as follows. Let {Q0j }j be the collection of all maximal cubes in Sa (P ) and define {Qτj +1 }j inductively as those cubes that are maximal with respect to inclusion in Sa (P ) and contained in some Qτj . Since Sa (P ) is sparse, actually Qτj is maximal with respect to the following property  1 > τ. Q∈Sa (P ) Q⊇Qτ j

So if Qτj +1 ⊂ Qτl , we must have Qτj +1  Qτl . Then by the sparsity condition again, we get   1 1Q (x) > τ + 1} ≤ 1Q (x) > τ }. μ{x ∈ P : μ{x ∈ P : 2 Q∈Sa (P )

Q∈Sa (P )

Now (2.5) follows immediately. For integers b ≥ 0, we define Sa,b (P ) to be the set consisting of Q ∈ Sa (P ) such that

Set K =



2−b σP ≤ σQ ≤ 2−b+1 σP . −b/2 . b≥0 2

We have

w{x ∈ P : T Sa (P ) (σ) > tσP }  ≤ w{x ∈ P : T Sa,b (P ) (σ) > t2−b/2 K −1 σP }, b≥0

where

T Sa,b (P ) (σ) =



σQ 1Q (x).

Q∈Sa,b (P )

Again, we can denote {x ∈ P : T Sa,b (P ) (σ) > t2−b/2 K −1 σP } :=

j

8

Rjb ,

where {Rjb }j is a family of disjoint dyadic cubes in Sa,b (P ). Since {x ∈ P : T Sa,b (P ) (σ) > t2−b/2 K −1 σP }  ⊆ {x ∈ P : 1Q (x) > t2b/2 K −1 /2} Q∈Sa (P )

By (2.5) we get



μ(Rjb )  2−t2

b/2 K −1 /2

μ(P ).

(2.6)

j

Without loss of generality, we can assume that t ≥ 1. Now we have  w(Rjb ) w{x ∈ P : T Sa (P ) (σ) > tσP } ≤ j

b≥0



  2a μ(Rjb ) j

b≥0

≤

j

  2a+b μ(Rjb ) 

σp−1 P

j

b≥0

≤

σp−1 Rb

2b−t2

b/2 K −1 /2

b≥0

≤ 2−tK

−1 /4



2a μ(P )

2b−2

σp−1 P

b/2 K −1 /4

(by (2.6)) w(P )

b≥0

 2−tK

−1 /4

w(P ).

 Now we are ready to estimate the first term on the right side of (2.3). We have          σQ 1Q (x) p ≤ σQ 1Q (x) p   L (w)

Q∈S Q⊂R

Laj (P ) :=



Q∈Sa

a

P ∈P a

    Sa (P )  T (σ) 

= Set

a

L (w)

Lp (w)

x : T Sa (P ) (σ)(x) ∈ [j, j + 1)σP .

9

.

By Lemma 2.7 we have      T Sa (P ) (σ)  P ∈P a

Lp (w)

≤ 

∞  j=0 ∞ 

     (j + 1) σP 1Laj (P ) (x) P ∈P a

(j + 1)

 

P ∈P a

j=0



  P ∈P a

σpP w(P )

σpP e−cj w(P )

Lp (w)

1

p

1

p

.

Therefore,     σQ 1Q (x)  Q∈S Q⊂R

Lp (w)



  a

P ∈P a

σpP w(P )

1

p

.



1 We follow the idea of [5]. Define Φ0 (t) = tp (r+1)/2 , set γ = 2(r+1) . Since r > 1 and it is dominated by some constant that depends only on the structure constant of X, it is easy to check that  ∞ ¯ Φ0 (t) dt ¯ [Φ0 ]Bp = tp t 1/2

p (r + 1) − 2 pp(r−1) 2 (r+1)−2 p(r − 1) p r − 1 (r−1)p ¯ Bp . ≤ cp,κ,D 2 p r−1 = cp,κ,D [Φ] (r − 1)p

≤ cp,κ,D

Now notice that

2r+1 4

1 3 1 + = + < 1. r 4 4 4r Set 1/q = 1/4 − 1/(4r), by H¨older’s inequality, we have    2r+1 r · 2r+1 r+1 4 4 − σ 2 ≤ − σ Q

It follows that

2r+1 4 r

Q

1

 1  1  4 q 1 ·4 q · − σ4 · − 1 . Q

1

Q

1

p γ σ p Φ0 ,Q ≤ σ p 1−γ Φ,Q · σ p ,Q .

10

Therefore,  σpP w(P ) ≤ P ∈P a

≤

 P ∈P a



1

1

w(P )σ p pΦ0 ,P σ p pΦ¯

0 ,P

1

p(1−γ)

wP σ p Φ,P

P ∈P a p(1−γ) (a+1)γ

≤ [w, σ]Φ,p

2

 P ∈P a



1

1

p p σ p pγ ¯ p ,P σ Φ

0 ,P

1

σ p pΦ¯

0 ,P

μ(P )

μ(P )

(by Lemma 2.3) 1



p(1−γ) [w, σ]Φ,p 2(a+1)γ



p(1−γ) ¯ 0 ]Bp σ(R) [w, σ]Φ,p 2(a+1)γ [Φ p(1−γ) ¯ Bp σ(R). [w, σ]Φ,p 2(a+1)γ [Φ]

MΦ¯D0 (1R σ p )(x)p dμ (by sparseness)



Consequently,      Q∈S σQ 1Q (x) Q⊂R

σ(R)1/p

Lp (w)

(1−γ)

1

¯ p  [w, σ]Φ,p [Φ] Bp

(by Lemma 2.6)



2(a+1)γ/p

a 1

(1−γ)

γ/p

¯ p [w, σ] ≤ Cp,D,κ [w, σ]Φ,p [Φ] Bp Ap 1

¯ p . ≤ Cp,D,κ [w, σ]Φ,p [Φ] Bp This completes the proof. Acknowledgements. This work was done while the author was visiting Department of Mathematics and Statistics, University of Helsinki. He thanks the Department of Mathematics and Statistics, University of Helsinki and Professor Tuomas P. Hyt¨onen for hospitality and support. He thanks Olli Tapiola for the nice talk on the weak A∞ class. Particular thanks go to Professor Tuomas P. Hyt¨ onen for carefully reading the paper and many helpful suggestions. [1] T. Anderson, D. Cruz-Uribe and K. Moen, Logarithmic bump conditions for Calder´on-Zygmund operators on spaces of homogeneous type, Publicacions Matematiques, to appear. [2] T. Anderson, T. Hyt¨ onen and O. Tapiola, Weak A∞ weights and weak Reverse H¨ older property in a space of homogeneous type, http://arxiv.org/abs/1410.3608. 11

[3] T. Anderson and A. Vagharshakyan, A simple proof of the sharp weighted estimate for Calder´ on-Zygmund operators on homogeneous spaces, J. Geom. Anal., 24(2014), 1276–1297. [4] M. Christ, A T (b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), no. 2, 601–628. [5] D. Cruz-Uribe, A. Reznikov and A. Volberg, Logarithmic bump conditions and the two-weight boundedness of Calder´ on-Zygmund operators, Adv. Math., 255(2014) 706–729. [6] T. Hyt¨onen, The sharp weighted bound for general Calder´ onZygmund operators, Ann. of Math., 175 (2012), 1473–1506. [7] T. Hyt¨ onen and A. Kairema, Systems of dyadic cubes in a doubling metric space, Colloq. Math., 126(2012), 1–33. [8] T. Hyt¨ onen and A. Kairema, What is a cube? Ann. Acad. Sci. Fenn., 38(2013), 405–412. [9] T. Hyt¨ onen and M. Lacey, The Ap -A∞ inequality for general Calder´on-Zygmund operators, Indiana Univ. Math. J. 61 (2012), 2041–2052. [10] T. Hyt¨onen, M. Lacey, H. Martikainen, T. Orponen, M. Reguera, E. Sawyer, I. Uriarte-Tuero, Weak and strong type estimates for maximal truncations of Calder´ on-Zygmund operators on Ap weighted spaces, J. Anal. Math. 118(2012), 177–220. [11] T. Hyt¨onen and C. P´erez, Sharp weighted bounds involving A∞ , J. Anal. & P.D.E. 6(2013), 777–818. [12] M. Lacey, On the separated bumps conjecture for CalderonZygmund operators, to appear in Hokkaido Math. J., available at http://arxiv.org/abs/1310.3507. [13] M. Lacey, E. Sawyer and I. Uriate-Tuero, Two weight inequalities for discrete positive operators, http://arxiv.org/abs/0911.3437. [14] A. Lerner, A simple proof of the A2 conjecture, Int. Math. Res. Not. 2012; doi: 10.1093/imrn/rns145. [15] A. Lerner and K. Moen, Mixed Ap -A∞ estimates with one supremum, Studia Math., 219 (2013), 247–267. 12

[16] A. Lerner and F. Nazarov, Intuitive dyadic calculus: the basics. [17] F. Nazarov, A. Reznikov and A. Volberg, Bellman approach to the one-sided bumping for weighted estimates of Calder´on-Zygmund operators, http://arxiv.org/abs/1306.2653. [18] C. P´erez, On sufficient conditions for the boundedness of the HardyLittlewood maximal operator between weighted Lp -spaces with different weights, Proc. Lond. Math. Soc., 71(1995), 135–157. [19] C. P´erez and R. Wheeden, Uncertainty principle estimates for vector fields, J. Funct. Anal., 181(2001), 146–188. [20] G. Pradolini and O. Salinas, Maximal operators on spaces of homogeneous type, Proc. Amer. Math. Soc., 132(2004), 435–441.

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