Singular Integrals and weights: a modernintroduction

Carlos PerezUniversity of Seville 1 2

Spring School on Function Spaces and InequalitiesPaseky, June 2013

1The author would like to thank Professors Jaroslav Lukes, Lubos Pick and Petr Posta for theinvitation to deliver these lectures.

2 The author would like to acknowledge the support of the Spanish grant MTM2012-30748 andthe grant from the Junta de Andalucıa, “proyecto de excelencia” FQM-4745

Contents

1 Improving the Ap theorem: the mixed Ap − A∞ approach 31.1 Improving the Ap theorem: the mixed Ap − A∞ approach . . . . . . . 31.2 Sharpenes of the exponents, Yano’s condition and the Rubio de Fran-

cia’s algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Two weight problem: sharp Sawyer’s theorem . . . . . . . . . . . . . . 12

2 Some applications 142.1 The factorization theorem with control on the bounds . . . . . . . . . . 142.2 Commutators of operators with BMO functions: quadratic estimates . 162.3 A preliminary result: a sharp connection between the John-Nirenberg

theorem and the A2 class . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Results within the Ap context . . . . . . . . . . . . . . . . . . . . . . . 18

3 Singular Integrals: The A1 theory 213.1 Estimates involving A∞ weights . . . . . . . . . . . . . . . . . . . . . . 233.2 The main lemma and the proof of the linear growth theorem . . . . . . 273.3 Proof of the logarithmic growth theorem. . . . . . . . . . . . . . . . . . 28

4 Singular Integrals: The Ap theory 314.1 The dyadic Ap theory of Singular Integrals . . . . . . . . . . . . . . . . 32

2

Chapter 1

Improving the Ap theorem: themixed Ap − A∞ approach

1.1 Improving the Ap theorem: the mixed Ap−A∞approach

Most of the basic definitions and notations can be found from the lectures delivered byProf. Javier Duoandikoetxea Paseky-2013, Lecture notes [13]. These lectures startedfrom the celebrated paper [37] by B. Muckenhoupt where it is proved the fundamentalresult characterizing all the weights for which the Hardy–Littlewood maximal operatoris bounded on Lp(w); the surprisingly simple necessary and sufficient condition is thecelebrated Ap condition of Muckenhoupt:

[w]Ap := supQ

(−∫Q

w(x) dx

)(−∫Q

w(x)1−p′ dx

)p−1

.

Of course the operator norm ‖M‖Lp(w) will depend on the Ap condition of w but thefirst quantitative estimate was proved by S. Buckley [4] as part of his Ph.D. thesis:

Let 1 < p < ∞ and let w ∈ Ap, then the Hardy-Littlewwod maximalfunction satisfies the following operator estimate:

‖M‖Lp(w) ≤ cn p′ [w]

1p−1

Ap

namely,

supw∈Ap

1

[w]1p−1

Ap

‖M‖Lp(w) ≤ cn p′. (1.1.1)

Furthermore the result is sharp in the sense that: for any ε > 0

supw∈Ap

1

[w]1p−1−ε

Ap

‖M‖Lp(w) =∞ (1.1.2)

3

See the lecture notes [13] where a proof of this result can be found. Here we willpresent an improvement of this result based on an argument closely related to theoriginal one of Muckenhoupt with better bounds. To do this we need to recall the A∞class of weights. This class of weights is defined in a natural way by the union of allAp class

A∞ := ∪p>1Ap.

The A∞ class of weights shares a lot of interesting properties, we remit to [13] for moreinformation.

It is known that another possible way of defining a constant for this class is bymeans of the quantity

[w]expA∞:= sup

Q−∫Q

w exp(−∫Q

− logw),

as can be found in for instance in [19]. This constant was introduced by Hruscev in[22]. This definition is natural because is obtained letting p → ∞ in the definition ofthe Ap constant. In fact we have by Jensen’s inequality:

[w]expA∞≤ [w]Ap .

On the other hand in [25] the authors use a “new” A∞ constant (which was originallyintroduced by Fujii in [17] and rediscovered later by Wilson in [49]) which is moresuitable.

Definition 1.1.1.

[w]A∞ := supQ

1

w(Q)

∫Q

M(wχQ) dx.

Observe that [w]A∞ ≥ 1 by the Lebesgue differentiation theorem.It is not difficult to show using the logarithmic maximal function

M0f := supQ

exp(−∫Q

log|f |)χQ

that[w]A∞ ≤ cn [w]expA∞

See [25] for details. In fact it is shown in the same paper with explicit examples that[w]A∞ is much smaller than [w]expA∞

(actually exponentially smaller).We also refer the reader to the forthcoming work of Duoandikoetxea, Martin-Reyes

and Ombrosi [14] for a discussion regarding different definitions of A∞ classes (also[13]).

The following result is an improvement of Buckley’s estimate (1.1.1).

4

Theorem 1.1.1. Let 1 < p <∞ and let σ = w−1/(p−1), then

‖M‖Lp(w) ≤ cnp′([w]Ap [σ]A∞

)1/p. (1.1.3)

Using the duality relationship

[σ]1p′

Ap′= [w]

1p

Ap

we see immediately that ([w]Ap [σ]A∞

)1/p ≤ [w]1p−1

Ap

yielding (1.1.1).This result and Theorem 1.1.2 below were first proved in [25] but the approaches

we present here for both results are simpler and taken from [28].

Theorem 1.1.2 (An optimal reverse Holder inequality). Define rw := 1 +1

cn [w]A∞,

where cn is a dimensional constant. Note that r′w ≈ [w]A∞.

a) If w ∈ A∞, then (−∫Q

wrw)1/rw

≤ 2−∫Q

w.

b) Furthermore, the result is optimal up to a dimensional factor: If a weight wsatisfies the RHI, i.e., there exists a constant K such that(

−∫Q

wr)1/r

≤ K−∫Q

w,

then there exists a dimensional constant c = cn, such that [w]A∞ ≤ cnK r′.

We mention that some similar results for the real line have been independentlyobtained by O. Beznosova and A. Reznikov in [3] by means of the Bellman functiontechnique.

We postpone the proof from now deriving first the following corollary which isusually called the open property of the Ap condition.

Corollary 1.1.1 (The Precise Open property). Let 1 < p <∞ and let w ∈ Ap. Thenw ∈ Ap−ε where

ε =p− 1

r(σ)′=

p− 1

1 + τn[σ]A∞

where as usual σ = w1−p′ . Furthermore

[w]Ap−ε ≤ 2p−1 [w]Ap

5

Proof. Since w ∈ Ap, σ ∈ Ap′ ⊂ A∞, and hence

−∫Q

w(−∫Q

σr(σ)) p−1r(σ) ≤ −

∫Q

w(

2−∫Q

σ)p−1

.

Choose ε so that p−1r(σ)

= p−ε−1, namely ε = p−1r(σ)′

and observe that ε > 0 and p−ε > 1.This yields that w ∈ Ap−ε.

Proof of Theorem 1.1.1. We will use

‖M‖Lq,∞(w)

≈ [w]1q

Aq1 < q <∞

(with dimensional constant) and

x ∈ Rn : Mf(x) > 2t ⊂ x ∈ Rn : Mft(x) > t

where ft = f χf>t.Now, since w ∈ Ap we use the ”The Precise Open property” above: w ∈ Ap−ε,

ε = p−1r(σ)′

and [w]Ap−ε ≤ 2p−1 [w]Ap .∫RnMf(y)pw(y)dy = p

∫ ∞0

tpwy ∈ Rn : Mf(x) > t dtt

= p2p∫ ∞

0

tpwy ∈ Rn : Mf(x) > 2t dtt≤ p 2p

∫ ∞0

tpwy ∈ Rn : Mft(x) > t dtt

≤ p cp−εn [w]Ap−ε 2p∫ ∞

0

tp∫Rn

ft(y)p−ε

tp−εw(y)dy

dt

t≤

≤ p cp−εn 2p−1 [w]Ap 2p∫Rn

∫ f(y)

0

tεdt

tf(y)p−εw(y)dy

=1

ε22p−1 p cp−εn [w]Ap

∫Rnf(y)pw(y)dy = p′r(σ)′ 22p−1 cp−εn [w]p

∫Rnf(y)pw(y)dy

= p′(1 + τn[σ]A∞) p 22p−1 cp−εn [w]p

∫Rnf(y)pw(y)dy

and this yields (1.1.3).

To prove Theorem 1.1.2 we begin with the following lemma. It it interesting on itsown, since it can be viewed as a self-improving property of the maximal function whenrestricted to A∞ weights.

6

Lemma 1.1.3. Let w be any A∞ weight and let Q0 be a cube. Then for any 0 < ε ≤1

2n+1[w]A∞, we have that

−∫Q0

(M(wχQ0))1+ε dx ≤ 2[w]A∞

(−∫Q0

w dx

)1+ε

, (1.1.4)

where M denotes the dyadic maximal function associated to the cube Q0.

Proof. Let w = wχQ0 and define Ωλ := Q0 ∩ Mw > λ. We start with the layer cakeformula: ∫

Q0

(Mw)1+ε dx=

∫ ∞0

ελε−1Mw(Ωλ) dλ

=

∫ wQ0

0

ελε−1

∫Q0

Mw dλ+

∫ ∞wQ0

ελε−1Mw(Ωλ) dλ

Now, for λ ≥ wQ0 , by the Calderon-Zygmund decomposition (see [13]) there is a familyof maximal nonoverlapping dyadic cubes Qjj for which

Ωλ =⋃j

Qj and −∫Qj

w dx > λ.

Therefore, by using this decomposition and the definition of the A∞ constant, we canwrite ∫

Q0

(Mw)1+ε dx ≤ wεQ0[w]A∞w(Q0) +

∫ ∞wQ0

ελε−1∑j

∫Qj

Mw dxdλ. (1.1.5)

By maximality of the cubes in Qjj, it follows that the dyadic maximal function Mcan be localized: Mw(x) = M(wχQj)(x) for any x ∈ Qj, for all j ∈ N. Now, if we

denote by Q to the dyadic parent of a given cube Q, we have that∫Qj

Mw dx =

∫Qj

M(wχQj)≤ [w]A∞w(Qj) ≤ [w]A∞w(Qj)

= [w]A∞wQj |Qj|≤ [w]A∞λ2n|Qj|

Therefore, ∑j

∫Qj

Mw dx ≤∑j

[w]A∞λ2n|Qj| ≤ [w]A∞λ2n|Ωλ|,

7

and then (1.1.5) becomes∫Q0

(Mw)1+ε dx ≤ wεQ0[w]A∞w(Q0) + ε[w]A∞2n

∫ ∞wQ0

λε|Ωλ|dλ.

Averaging over Q0, we obtain that

−∫Q0

(Mw)1+ε dx ≤ w1+εQ [w]A∞ +

ε2n[w]A∞1 + ε

−∫Q0

(Mw)1+ε dx. (1.1.6)

So the desired inequality follows for any 0 < ε ≤ 12n+1[w]A∞

by absorbing the last term

into the left.

Proof of Theorem 1.1.2. Denoting again w = wχQ0 we have that∫Q0

w1+ε dx ≤∫Q0

(Mw)ε wdx.

Now we argue in a similar way as in the previous lemma to obtain that∫Q0

(Mw)ε wdx=

∫ ∞0

ελε−1w(Ωλ) dλ

=

∫ wQ0

0

ελε−1w(Q0) dλ+

∫ ∞wQ0

ελε−1w(Ωλ) dλ

≤wεQ0w(Q0) +

∫ ∞wQ0

ελε−1∑j

w(Qj) dλ,

where the cubes Qjj are from the decomposition of Ωλ above. Therefore,∫Q0

(Mw)ε wdx≤wεQ0w(Q0) + ε2n

∫ ∞wQ0

λε∑j

|Qj| dλ

≤wεQ0w(Q0) + ε2n

∫ ∞wQ0

λε|Ωλ| dλ

≤wεQ0w(Q0) +

ε2n

1 + ε

∫Q0

(Mw)1+ε dx.

Averaging over Q0 we obtain

−∫Q0

w1+ε dx ≤ w1+εQ0

+ε2n

1 + ε−∫Q0

(Mw)1+ε dx.

8

Now we use Lemma 1.1.3 to conclude with the proof:

−∫Q0

w1+ε dx≤w1+εQ0

+ε2n

1 + ε−∫Q0

(Mw)1+ε dx

≤w1+εQ0

+ε2n+1[w]A∞

1 + ε

(−∫Q0

w dx

)1+ε

≤ 2

(−∫Q0

w dx

)1+ε

since, by hypothesis,ε2n[w]A∞

1+ε≤ 1

2.

1.2 Sharpenes of the exponents, Yano’s condition

and the Rubio de Francia’s algorithm.

In this section we will prove the sharpness of the exponent in (1.1.2). However we willdo it from a very general point of view showing that the sharpness is intimately relatedto the failure of the boundedness of the operator when p = 1. More precisely is relatedto the way it blows up the bound norm of the operator as p → 1 which in turn isrelated to the well known Yano’s classical condition. In fact the method is so generalthat also shows that the exponents in Theorem 1.1.1 are sharp but we will leave thesedetails aside and remit to the work [34].

T will denote in this section an operator (not necessarily linear) bounded on Lp(Rn),1 < p <∞. We will assume that for some 0 < α <∞

‖T‖Lp(Rn) = O(1

(p− 1)α) p→ 1

and we will denote by αT to the smallest of these exponents α > 0, namely for anyε > 0

lim supp→1

(p− 1)αT−ε‖T‖Lp(Rn) =∞ (1.2.1)

This is satisfied in the usual cases with appropriate exponents by: maximal opera-tors, singular integrals (αT = 1), commutators of singular integrals with BMO functionsαT = 2, generalized commutators αT = k ((see Chapter 2) for definitions).

Theorem 1.2.1. Let T be an operator as above with exponent αT ∈ (0,∞). Supposefurther that for some 1 < p0 <∞

‖T‖Lp0 (w) ≤ cp0,αT [w]αTp0−1

Ap0w ∈ Ap0 . (1.2.2)

Then the inequality is sharp in the sense the exponent αT cannot be replaced by αT − ε,ε > 0.

9

This result can be applied to many operators: T = Mk, T = [b,H], b ∈ BMO, ormore generally T kb . In each case an explicit example was built to show the sharpnessof the exponent.

Corollary 1.2.1. Let M be Hardy-Littlewood maximal function and let 1 < p < ∞and w ∈ Ap. Then

‖Mk‖Lp(w) ≤ cn,,k,p[w]kp−1

Ap. (1.2.3)

and the exponent is sharp.

Te proof of the theorem is based on the Rubio de Francia’s algorithm which is themain tool to prove the extrapolation theorem and in particular some of the new ideasfrom [10] (also [12])

Proof of Theorem 1.2.1. To prove the theorem we let α > 0 be a positive parametersatisfying (1.2.2) instead of αT :

‖T‖Lp0 (w) ≤ cp0,α [w]α

p0−1

Ap0w ∈ Ap0 . (1.2.4)

First we claim first the following estimate which follows from (1.2.4) and is the keyinequality:

‖T‖Lp(Rn) ≤ 21+α cp0,α ‖M‖αp0−pp0−1

Lp(Rn) 1 < p < p0. (1.2.5)

Suppose then 1 < p < p0 and perform the iteration technique R as follows:

R(h) =∞∑k=0

1

2kMk(h)

‖M‖kLp(Rn)

Then we have(A) h ≤ R(h)(B) ‖R(h)‖Lp(Rn) ≤ 2 ‖h‖Lp(Rn)

(C) [R(h)]A1 ≤ 2 ‖M‖Lp(Rn)

Then, combining Holder’s inequality, the key hypothesis (1.2.4) together with prop-erties of (A) and (B)

‖T (f)‖Lp(Rn) =(∫

Rn(Tf)p (Rf)

−(p0−p) pp0 (Rf)(p0−p) pp0 dx

)1/p

≤(∫

Rn|Tf |p0 (Rf)−(p0−p) dx

)1/p0 (∫Rn

(Rf)p dx) p0−p

pp0

≤ 2 cp0,α [R(f)−(p0−p)]α

p0−1

Ap0

(∫Rn|f |p0 (Rf)−(p0−p) dx

)1/p0‖f‖

p0−pp0

Lp(Rn)

10

≤ 2 cp0,α [R(f)−(p0−p)]α

p0−1

Ap0

(∫Rn|f |p dx

)1/p0‖f‖

1− pp0

Lp(Rn)

= 2 cp0,α [R(f)−(p0−p)]α

p0−1

Ap0‖f‖Lp(Rn) = 2 cp0,α [R(f)

p0−pp0−1 ]αAp′0

‖f‖Lp(Rn)

since [w]1q

Aq= [w1−q′ ]

1q′

Aq′. Now, since p0−p

p0−1< 1 we can use Jensen’s inequality to

continue with

≤ 2 cp0,α [R(f)]αp0−pp0−1

Ap′0‖f‖Lp(Rn) ≤ 2 cp0,α [R(f)]

αp0−pp0−1

A1‖f‖Lp(Rn)

≤ 21+α cp0,α ‖M‖αp0−pp0−1

Lp(Rn) ‖f‖Lp(Rn)

by making use of property (C). Since this holds for any f we get the key inequality

‖T‖Lp(Rn) ≤ 21+α cp0,α ‖M‖αp0−pp0−1

Lp(Rn) 1 < p < p0.

To finish the proof we let α = αT in this argument and assume that (1.2.4) holdswith αT replace by αT − ε, for a tiny ε. Then (1.2.5) holds with αT − ε:

‖T‖Lp(Rn) ≤ 21+αT−ε cp0,αT−ε ‖M‖(αT−ε)

p0−pp0−1

Lp(Rn) 1 < p < p0.

and distributing the exponents:

‖M‖(αT−ε) p−1

p0−1

Lp(Rn) ‖T‖Lp(Rn) ≤ cp0,αT ,ε ‖M‖αT−εLp(Rn) 1 < p < p0

and using that

‖M‖p ≈1

p− 1p ∈ (1,∞)

we have as when p is close to 1,

(1

p− 1)(αT−ε) p−1

p0−1 (p− 1)αT−ε ‖T‖Lp(Rn) ≤ cp0,αT ,ε.

Finally letting p→ 1 and since α = α(T ) we assume (1.2.1), namely

lim supp→1

(p− 1)αT−ε‖T‖Lp(Rn) =∞

we conclude the proof of the theorem.

11

1.3 Two weight problem: sharp Sawyer’s theorem

Another highlight of the theory of weights is the two weight characterization of themaximal theorem due to E. Sawyer [48] (see also [19]):

Let 1 < p < ∞, and let u, σ two unrelated weights, then there is a finiteconstant C such that

‖M(fσ)‖Lp(u) ≤ C ‖f‖Lp(σ) (1.3.1)

if and only if there is a finite constant K such that for any cube Q(∫Q

M(σ χQ)p udx

)1/p

≤ K σ(Q)1/p <∞

K. Moen proved in [35] a quantitative version of Sawyer’s theorem as follows: if welet

[u, σ]Sp = supQ

(∫QM(σ χQ)p udx

)1/p

σ(Q)1/p

then we have.

Theorem 1.3.1. Let 1 < p <∞ and let u, σ and ‖M‖ as above. Then

[u, σ]Sp ≤ ‖M‖ ≤ cn p′ [u, σ]Sp

Very recently and in a joint work with E. Rela [45] an application of this resultwas found obtaining an improvement of the mixed Ap − A∞ Theorem 1.1.1 as well assome other two weight estimates with “bumps” conditions providing new quantitativeestimates of results derived in [41].

We present now the two weight version of Theorem 1.1.1 as a corollary of Theorem1.3.1.

Corollary 1.3.1. Under the same condition as above

‖M‖ ≤ cn p′ ([u, σ]Ap [σ]A∞)1/p

where [u, σ]Ap is the two weight Ap constant:

[u, σ]Ap := supQ

(−∫Q

w(x) dx

)(−∫Q

σ(x) dx

)p−1

.

12

Proof. By Theorem 1.3.1 it is enough to prove that

[u, σ]Sp ≤ cn ([u, σ]Ap [σ]A∞)1/p. (1.3.2)

Now, let a > 1 a universal parameter to be chosen. Fix a cube Q and let k0 the integersuch that ak0 ≤ −

∫Qσ < ak0+1. Then∫

Q

|M(σ χQ)|pu dx =

∫M(σ χQ)≤ak0

|M(σ χQ)|pu dx+∞∑

k=k0

∫ak<M(σ χQ)≤ak+1

|M(σ χQ)|pu dx

≤ (−∫Q

σ)pu(Q) + ap∞∑

k=k0

akpu(x ∈ Q : M(σ χQ) > ak)

we know use the Calderon-Zygmund cubes locally and at each level k. Indeed, since

ak > −∫Q

σ k ≥ k0

we can find a family of dyadic cubes with respect to Q Qk,j such that for each ofthese k

x ∈ Q : M(σ χQ) > ak = ∪k,jQkj

with

ak < −∫Qkj

σ ≤ 2n ak j ∈ Z

Furthermore the family of cubes satisfies the so called sparseness property (see Chapter4). Then using the Ap condition we continue with

≤ [u, σ]Apσ(Q) + ap∑k,j

(−∫Qkj

σ dx

)p

u(Qkj )

≤ [u, σ]Apσ(Q) + ap∑k,j

(−∫Qkj

σ dx

)p−1

−∫Qkj

u dx σ(Qkj )

≤ [u, σ]Apσ(Q) + cn [u, σ]Ap∑j,k

σ(Qkj )

Now by the sparseness property of the cubes we can continue last sum with

≤ cn∑j,k

−∫Qkj

σ dx |Ekj | ≤ cn

∑j,k

∫Ekj

M(χQσ) dx ≤ cn

∫Q

M(χQσ) dx ≤ cn [σ]A∞ σ(Q)

since the family Ekj is pairwise disjoint and all of them are contained in Q. Finally,

dividing by σ(Q) and taking the supremum over all the cubes Q we obtain the desiredestimate (1.3.2).

13

Chapter 2

Some applications

2.1 The factorization theorem with control on the

bounds

Muckenhoupt already observed in [37] that it follows from the definition of the A1

class of weights that if w1, w2 ∈ A1, then the weight

w = w1w1−p2

is an Ap weight. Furthermore we have

[w]Ap ≤ [w1]A1 [w2]p−1A1

(2.1.1)

He conjectured that any Ap weight can be written in this way. This conjecture wasproved by P. Jones’s showing that if w ∈ Ap then there are A1 weights w1, w2 such thatw = w1w

1−p2 .

It is also well known that the modern approach to this question uses completelydifferent path and it is due to J. L. Rubio de Francia as can be found in [19] and alsoin [13]. Here we present a variation of these ideas which appeared in [21] obtainingresults with quantitative control on the constants. To be more precise we have thefollowing version of the factorization theorem.

Theorem 2.1.1. Let 1 < p < ∞ and let w ∈ Ap, then there are A1 weightsu, v ∈ A1, such that

w = u v1−p

in such a way that

[u]A1 ≤(

max‖M‖1/p′

Lp(w), ‖M‖1/p

Lp′ (σ))p

& [v]A1 ≤(

max‖M‖1/p′

Lp(w), ‖M‖1/p

Lp′ (σ))p′

(2.1.2)

and hence [w]Ap ≤(

max‖M‖1/p′

Lp(w), ‖M‖1/p

Lp′ (σ))2p ≤ cn,p[w]2Ap

14

Proof. We use Rubio de Francia’s iteration scheme or algorithm to our situation. Define

S1(f)p′:= w1/pM(

|f |p′

w1/p),

and

S2(f)p :=1

w1/pM(|f |pw1/p),

Observe that Si : Lpp′(Rn)→ Lpp

′(Rn) with constants

‖S1‖Lpp′ (Rn) ≤ ‖M‖1/p′

Lp(w) & ‖S2‖Lpp′ (Rn) ≤ ‖M‖1/p

Lp′ (σ)

where as usual σ = w1−p′

Hence, the operator S = S1 + S2 is also bounded on

Lpp′(Rn) with

‖S‖Lpp′ (Rn) ≤ 2 max‖M‖1/p′

Lp(w), ‖M‖1/p

Lp′ (σ) (≤ cn,p[w]

1/pAp

)

Now, define the Rubio de Francia algorithm R as

R(h) :=∞∑k=0

1

2kSk(h)

(‖S‖Lpp′ (Rn))k.

Observe that R is also bounded on Lpp′(Rn) . Now, if h ∈ Lpp

′(Rn) is fixed,

R(h) ∈ A1(S) ∩ Lpp′(Rn). More precisely

S(R(h)) ≤ 2 ‖S‖Lpp′ (Rn)R(h),

and hence R(h) ∈ A1(Si) i = 1, 2, with

Si(R(h)) ≤ 2 ‖S‖Lpp′ (Rn) R(h) i = 1, 2

HenceM(R(h)p

′w−1/p) ≤ (2 ‖S‖Lpp′ (Rn))

p′ R(h)p′w−1/p

andM(R(h)pw1/p) ≤ (2 ‖S‖Lpp′ (Rn))

p R(h)pw1/p.

If we let v := R(h)p′w−1/p & u := R(h)pw1/p, then we have

u, v ∈ A1, with w = uv1−p and

[v]A1 ≤ (2 ‖S‖Lpp′ (Rn))p′ & [u]A1 ≤ (2 ‖S‖Lpp′ (Rn))

p

and hence

[v]A1 ≤(

max‖M‖1/p′

Lp(w), ‖M‖1/p

Lp′ (σ))p′

& [u]A1 ≤(

max‖M‖1/p′

Lp(w), ‖M‖1/p

Lp′ (σ))p

15

(observe that

max[v]1p′

A1, [u]

1p

A1 ≤ max‖M‖1/p′

Lp(w), ‖M‖1/p

Lp′ (σ) ≤ cn[w]

1p

Ap

≈ ‖M‖Lp(w)→Lp,∞(w)

≈ ‖M‖Lp′ (σ)→Lp′,∞(σ)

)

which produces

[v]A1 ≤ cn[w]1p−1

Ap& [u]A1 ≤ cn[w]Ap

namely

max[u]A1 , [v]A1 ≤ [w]max1, 1

p−1

Ap.

We finish the section by remarking that there is no good extrapolation theoremwith mixed Ap−A∞ bounds. A first result can be found in [25] but it seems not to besharp.

2.2 Commutators of operators with BMO func-

tions: quadratic estimates

In this section we consider commutators of linear operators T with BMO functionsdefined by appropriate functions by

[b, T ]f(x) = b(x)T (f)(x)− T (b f)(x)

When T is a singular integral operator, these operators were considered by Coifman,Rochberg and Weiss in [8]. If the kernel of the operator is K then formally we have

[b, T ]f(x) =

∫Rn

(b(x)− b(y))K(x, y)f(y) dy,

where K is a kernel satisfying the standard Calderon-Zygmund estimates. Althoughthe original interest in the study of such operators was related to generalizations ofthe classical factorization theorem for Hardy spaces many other applications have beenfound. The main result from [8] states that [b, T ] is a bounded operator on Lp(Rn),1 < p < ∞, when b is a BMO function. In fact, the BMO condition of b is also anecessary condition for the Lp-boundedness of the commutator when T is the Hilberttransform. We may think that these operators behave as Calderon-Zygmund operators,however there are some differences. For instance, simple examples show that in general[b, T ] fails to be of weak type (1, 1) when b ∈ BMO. This was observed in [42] where it

16

is also shown that there is an appropriate weak-L(logL) type estimate replacement. Tostress this point of view it is also shown in [43] that the right operator controlling [b, T ]is M2 = M M , instead of the Hardy-Littlewood maximal function M explaining thereason of why these operators are more singular. . We pursue in this way by showingthat commutators have an extra “bad” behavior from the point of view of Ap weightswhen comparing with Theorem 4.1.1. In particular we will prove weighted estimates forthese commutators with quantitate control on the bounds using a very general method.

2.3 A preliminary result: a sharp connection be-

tween the John-Nirenberg theorem and the A2

class

In this section we outline the main results from [5]. We want to stress the relevance ofthe reverse Holder property of the A2 weights in conjunction with the following sharpversion of the classical and well known John-Nirenberg theorem.

For a locally integrable b : Rn → R we define

‖b‖BMO = supQ

1

|Q|

∫Q

|b(y)− bQ| dy <∞

where the supremum is taken over all cubes Q ∈ Rn with sides parallel to the axes,and

bQ =1

|Q|

∫Q

b(y) dy

The main relevance of BMO is due to the fact that has an exponential self-improving property, namely the celebrated John-Nirenberg’s Theorem. We need avery precise version of it, as follows:

Theorem 2.3.1. [Sharp John-Nirenberg inequality] There are dimensional constants0 ≤ αn < 1 < βn such that

supQ

1

|Q|

∫Q

exp

(αn

‖b‖BMO

|b(y)− bQ|)dy ≤ βn (2.3.1)

In fact we can take αn = 12n+2 .

For the proof of this we remit to the Lecture Notes by J.L. Journe [29] p. 31-32.The result there is not so explicit but it follows from the proof which is very interestingand different from the usual ones that can be found in many references.

We derive from Theorem 2.3.1 the following relationship between BMO and theA2 class of weights Lemma 2.3.2 that will be used in the proof of the main theorem of

17

this section. Indeed, it is well known that if w ∈ A2 then b = logw ∈ BMO. A partialconverse also holds, if b ∈ BMO there is an s0 > 0 such that w = esb ∈ Ap, |s| ≤ s0.We need a more precise version of this.

Lemma 2.3.2. Let b ∈ BMO and let αn < 1 < βn be the dimensional constantsfrom (2.3.1). Then

s ∈ R, |s| ≤ αn‖b‖BMO

=⇒ es b ∈ A2 and [es b]A2 ≤ β2n

Proof. By Theorem 2.3.1, if |s| ≤ αn‖b‖BMO

and if Q is fixed

1

|Q|

∫Q

exp(|s||b(y)− bQ|) dy ≤1

|Q|

∫Q

exp(αn

‖b‖BMO

|b(y)− bQ|) dy ≤ βn

and then1

|Q|

∫Q

exp(s(b(y)− bQ)) dy ≤ βn

and1

|Q|

∫Q

exp(−s(b(y)− bQ)) dy ≤ βn.

If we multiply the inequalities, the bQ parts cancel out:(1

|Q|

∫Q

exp(s(b(y)− bQ)) dy

) (1

|Q|

∫Q

exp(s(bQ − b(y))) dy

)

=

(1

|Q|

∫Q

exp(sb(y)) dy

) (1

|Q|

∫Q

exp(−sb(y)) dy

)≤ β2

n

namely es b ∈ A2 with[es b]A2 ≤ β2

n

2.4 Results within the Ap context

In this section we use the extrapolation theorem with sharp bounds (see [13] or [10]) to get the optimal estimates for commutators. We emphasize that next results arestated for very general operators.

18

Theorem 2.4.1. Let T be a linear operator such that

‖T‖L2(w) ≤ c [w]A2 w ∈ A2. (2.4.1)

Then there is a constant c independent of w and b such that

‖[b, T ]‖L2(w) ≤ c [w]2A2‖b‖BMO. (2.4.2)

These results can be found in [5] as well some generalizations. Observe the quadraticexponent which makes it different from the non commutator case. As an easy conse-quence of the extrapolation theorem with sharp bounds we have the following.

Corollary 2.4.1. Let T be a linear operator satisfying (2.4.1). Let 1 < p < ∞, thethere is a constant cn,p such that

‖[b, T ]‖Lp(w) ≤ cn,p [w]2 max1, 1

p−1

Ap‖b‖BMO. (2.4.3)

These results have been improved in [25] by replacing the Ap constant by appropri-ate mixed Ap − A∞ constants. As a sample if T is as above we have

‖[b, T ]‖L2(w) ≤ c [w]1/2A2

([w]A∞ + [w−1]A∞

)3/2 ‖b‖BMO.

See Chapter 4 for similar results for Caldern-Zygmund operators.

Proof. We use a beautiful idea that goes back to the original paper [8]. The methodis very general and goes beyond the case of linear operators as shown in [1].

We “conjugate” the operator as follows T : if z is any complex number we define

Tz(f) = ezbT (e−zbf).

Then, a computation gives (for instance for ”nice” functions),

[b, T ](f) =d

dzTz(f)|z=0 =

1

2πi

∫|z|=ε

Tz(f)

z2dz , ε > 0

by the Cauchy integral theorem.Now, by Minkowski’s inequality

‖[b, T ](f)‖L2(w) ≤1

2π ε2

∫|z|=ε‖Tz(f)‖L2(w)|dz| ε > 0. (2.4.4)

The key point is to find the appropriate radius ε. To do this we look at the innernorm ‖Tz(f)‖L2(w)

‖Tz(f)‖L2(w) = ‖T (e−zbf)‖L2(we2Rez b) ,

19

and try to find appropriate bounds on z. To do this we use the main hypothesis,namely that T is bounded on L2(w) if w ∈ A2 with

‖T‖L2(w) ≤ c [w]A2 .

Hence we should estimate [we2Rez b]A2 . But since w ∈ A2 we can use the sharp reverseHolder inequality to obtain:

[we2Rez b]A2 ≤ 4 [w]A2 [e2Rez r′ b]1r′A2

where r = rw = 1+ 1cn[w]A2

≤ 1+ 1cn[w]A∞

< 2. Now, since b ∈ BMO we are in a position

to apply Lemma 2.3.2,

if |2Rez r′| ≤ αn‖b‖BMO

then [e2Rez r′ b]A2 ≤ β2n.

Hence for these z,

[we2Rez b]A2 ≤ 4 [w]A2 β2r′n ≤ 4 [w]A2 βn,

since 1 < r < 2.Using this estimate and for these z we have

‖Tz(f)‖L2(w) ≤ 4βn [w]A2 ‖f‖L2(w).

Finally, choosing the radius

ε =αn

2r′‖b‖BMO

,

finish the proof of the theorem.

20

Chapter 3

Singular Integrals: The A1 theory

The Hardy-Littlewood maximal function is not a linear operator but it is a sort ofself-dual operator since the following inequality holds

‖Mf‖L1,∞(w) ≤ c

∫Rn|f |Mwdx, (3.0.1)

for any nonnegative functions f and w and where the constant c is independent ofboth functions f and w. The inequality (3.0.1) it is much more than an interestingimprovement of the classical weak-type (1, 1) property of M . M is also “self-dual”from the Lp point of view:∫

Rn(Mf)pw dx ≤ cp

∫Rn|f |pMwdx f, w ≥ 0. (3.0.2)

This estimate follows from the classical interpolation theorem of Marcinkiewicz. Bothresults (3.0.1) and (3.0.2) were proved by C. Fefferman and E.M. Stein in [16] toderive the following vector-valued extension of the classical Hardy-Littlewood maximaltheorem: for every 1 < p, q <∞, there is a finite constant c = cp,q such that∥∥∥∥(∑

j

(Mfj)q) 1q

∥∥∥∥Lp(Rn)

≤ c

∥∥∥∥(∑j

|fj|q) 1q

∥∥∥∥Lp(Rn)

. (3.0.3)

This is a very deep theorem and has been used a lot in modern harmonic analysisexplaining the central role of inequality (3.0.1).

One of the main purposes of these lectures is to find corresponding estimates forCalderon-Zygmund operators T instead of M . Here we use the standard concept ofCalderon-Zygmund operator as can be found in many places as for instance in [20] or[11].

B. Muckenhoupt and R. Wheeden during the 70’s he Muckenhoupt-Wheeden con-sidered the following problem:

‖Tf‖L1,∞(w) ≤ cT

∫Rn|f |Mwdx. (3.0.4)

21

when T = H is the the Hilbert transform:

Hf(x) = pv

∫R

f(y)

x− ydy.

It is very unfortunate that this inequality is false as shown first in [46] in the case of avery special dyadic singular integral operator and then in [47] for the Hilbert transform.

We remit to [44] for a discussion on this estimate and some other related for otheroperators.

It seems that the best result can be found in [40] where M is replaced by ML(logL)ε

is a “ε-logarithmically” bigger maximal type operator than M . If ε = 0 we recover Mbut the constant cε blows up as ε→ 0. The precise result is the following.

Theorem 3.0.2 (The L(logL)ε theorem). There exists a constant c depending on Tsuch that for any ε > 0, any function f and any weight w

supλ>0

λwx ∈ Rn : |Tf(x)| > λ ≤ c

ε

∫Rn|f |ML(logL)ε(w)dx w ≥ 0. (3.0.5)

We remark that the operator ML(logL)ε is pointwise smaller than Mr = MLr , r > 1.

Remark 3.0.3. We also remark that in [40] the constant 1ε

in front does not appearexplicitly. However, the way that estimate (3.0.5) blows up is relevant these days sinceits connection with questions related to the precise quantitate two weight estimates forCalderon-Zygmund operators (see [26]). In fact, in this work this theorem has beenimproved by replacing T by T∗, the maximal singular integral operator. The maindifficulty is that T∗ is not a linear operator and the approach used in these notes cannotbe applied.

Observe that the A1 class of weights can be defined immediately from Fefferman-Stein’s inequality (3.0.1) and in fact this is what the authors did in that paper: if theweight w ∈ A1 then

‖Mf‖L1,∞(w) ≤ cn [w]A1

∫Rn|f |wdx (3.0.6)

and it is natural to ask wether the corresponding inequality holds for singular integrals(say for the Hilbert transform). Then for a while there was a conjecture called the A1

conjecture: if w ∈ A1, then

‖Tf‖L1,∞(w) ≤ c [w]A1

∫Rn|f |wdx. (3.0.7)

However, this inequality seems to be false too (see [38]) for T = H, the Hilbert trans-form.

In this chapter we will present some recent progress in connection with this conjec-ture exhibiting an extra logarithmic growth in (3.0.7) which in view of [38] could be

22

the best possible result. To prove this result we have to study first the correspondingweighted Lp(w) estimates with 1 < p <∞ and w ∈ A1 being the result this time fullysharp. The final part of the proofs of both theorem can be found in Sections 3.2 and3.3 and are essentially taken from [33] and [32] and the recent improvement in [25].Again we remit to [44] for a more complete discussion about these estimates and theirvariants.

We state now the main theorems of this chapter. From now on T will always denoteany Calderon-Zygmund operator and we assume that the reader is familiar with theclassical unweighted theory.

Theorem 3.0.4. Let T be a Calderon-Zygmund operator and let 1 < p <∞. Then

‖T‖Lp(w) ≤ cT pp′ [w]

1/pA1

[w]1/p′

A∞, (3.0.8)

and hence‖T‖Lp(w) ≤ cT pp

′ [w]A1 . (3.0.9)

The result (3.0.8) is best possible as well as (3.0.9) in a sense similar to (1.1.1).As an application of this result we obtain the following endpoint estimate.

Theorem 3.0.5. [The logarithmic growth theorem] Let T be a Calderon-Zygmund op-erator. Then

‖T‖L1,∞(w) ≤ cT [w]A1 log(e+ [w]A∞). (3.0.10)

As mentioned, in view of [38], (3.0.10) seems to be the best possible result.

Remark 3.0.6. As in remark 3.0.3, these two theorems can be further improved byreplacing T by T∗ the maximal singular integral operator. Again, the method presentedin these notes cannot be applied because is based on the fact that T is linear while T∗is not. See [26].

3.1 Estimates involving A∞ weights

In harmonic analysis, there are a number of important inequalities of the form∫Rn|Tf(x)|pw(x) dx ≤ C

∫Rn|Sf(x)|pw(x) dx, (3.1.1)

where T and S are operators. Typically, T is an operator with some degree of singu-larity (e.g., a singular integral operator), S is an operator which is, in principle, easierto handle (e.g., a maximal operator), and w is in some class of weights.

23

The standard technique for proving such results is the so-called good-λ inequalityof Burkholder and Gundy. These inequalities compare the relative measure of the levelsets of S and T : for every λ > 0 and ε > 0 small,

w(y ∈ Rn : |Tf(y)| > 2λ, |Sf(y)| ≤ λε) ≤ C εw(y ∈ Rn : |Sf(y)| > λ). (3.1.2)

Here, the weight w is usually assumed to be in the class A∞ = ∪p>1Ap. Given inequality(3.1.2), it is easy to prove the strong-type inequality (3.1.1) for any p, 0 < p < ∞, aswell as the corresponding weak-type inequality

‖Tf‖Lp,∞(w) ≤ C ‖Sf‖Lp,∞(w). (3.1.3)

In these notes the special case of

‖Tf‖Lp(w) ≤ c ‖Mf‖Lp(w) (3.1.4)

where T is a Calderon-Zygmund operator and M is the maximal function, will play acentral role in the proof of Theorem 3.0.4. Estimate (3.1.4) was proved by Coifman-Fefferman in the celebrated paper [6]. In our context the weight w will also satisfy theA∞ condition but the problem is that the behavior of the constant is too rough. Weneed a more precise result for very specific weights.

Lemma 3.1.1. Let w be any weight and let 1 ≤ p, r < ∞. Then, there is aconstant c = c(n, T ) such that:

‖Tf‖Lp(Mrw)1−p) ≤ cp ‖Mf‖Lp(Mrw)1−p)

This is the main improvement in [33] of [32] where we had obtained logarithmicgrowth on p. It is an important step towards the proof of the the linear growth Theorem3.0.4.

The above mentioned good λ of Coifman-Fefferman is not sharp because instead ofc p gives C(p) ≈ 2p because

[(Mrw)1−p)]Ap ≈ (r′)p−1

The proof of this lemma is tricky and it combines another variation the of Rubiode Francia algorithm together with a sharp L1 version of (3.1.4):

‖Tf‖L1(w) ≤ c[w]Aq‖Mf‖L1(w) w ∈ Aq, 1 ≤ q <∞ (3.1.5)

The original proof given in [33] of this estimate was based on an idea of R. Fefferman-Pipher from [15] which combines a sharp version of the good-λ inequality of S. Buckleytogether with a sharp reverse Holder property of the weights. The result of Buckley

24

establishes a very interesting exponential improvement of the good-λ estimate of abovementioned Coifman-Fefferman estimate as can be found in [4]:

|x ∈ Rn : T ∗(f) > 2λ,Mf < γλ| ≤ c1e−c2/γ|T ∗(f) > λ| λ, γ > 0 (3.1.6)

where T ∗ is the maximal singular integral operator. This approach is interesting on itsown but we will present in these lecture notes a ‘more efficient approach based on thefollowing estimate: Let 0 < p <∞, 0 < δ < 1 and let w ∈ Aq, 1 ≤ q <∞, then

‖f‖Lp(w) ≤ c p[w]Aq ‖M#δ (f)‖Lp(w) (3.1.7)

for any function f such that |x : |f(x)| > t| <∞. Here,

M#δ f(x) = M#(|f |δ)(x)1/δ

and M# is the usual sharp maximal function of Fefferman-Stein:

M#(f)(x) = supQ3x

1

|Q|

∫Q

|f(y)− fQ| dy.

This result can be found in [39] and the proof is based on properties of rearrangementof functions and corresponding local sharp maximal operator.

To prove (3.1.5) we combine (3.1.7) with the following pointwise estimate [2]:

Lemma 3.1.2. Let T be any Calderon-Zygmund operator and let 0 < δ < 1, then thereis a constant cT,δ such that

M#δ (T (f ))(x) ≤ cMf(x)

and in fact we have.

Corollary 3.1.1. Let 0 < p <∞ and let w ∈ Aq. Then

‖Tf‖Lp(w) ≤ c [w]Aq ‖Mf‖Lp(w)

for any f such that |x : |Tf(x)| > t| <∞.

We now finish this section by proving the “tricky” Lemma 3.1.1. The proof is basedon the following lemma which is another variation of the Rubio de Francia algorithm.

Lemma 3.1.3. Let 1 < s < ∞ and let w be a weight. Then there exists anonnegative sublinear operator R satisfying the following properties:(a) h ≤ R(h)(b) ‖R(h)‖Ls(w) ≤ 2‖h‖Ls(w)

(c) R(h)w1/s ∈ A1 with[R(h)w1/s]A1 ≤ cs′

25

Proof. We consider the operator

S(f) =M(f w1/s)

w1/s

Since ‖M‖Ls ∼ s′, we have

‖S(f)‖Ls(w) ≤ cs′‖f‖Ls(w).

Now, define the Rubio de Francia operator R by

R(h) =∞∑k=0

1

2kSk(h)

(‖S‖Ls(w))k.

It is very simple to check that R satisfies the required properties.

Proof of Lemma 3.1.1. We are now ready to give the proof of the “tricky” Lemma,namely to prove ∥∥∥∥ Tf

Mrw

∥∥∥∥Lp(Mrw)

≤ cp

∥∥∥∥ Mf

Mrw

∥∥∥∥Lp(Mrw)

By duality we have,∥∥∥∥ Tf

Mrw

∥∥∥∥Lp(Mrw)

= |∫RnTf h dx| ≤

∫Rn|Tf |h dx

for some ‖h‖Lp′ (Mrw) = 1. By Lemma 3.1.3 with s = p′ and v = Mrw there existsan operator R such that(A) h ≤ R(h)(B) ‖R(h)‖Lp′ (Mrw) ≤ 2‖h‖Lp′ (Mrw)

(C) [R(h)(Mrw)1/p′ ]A1 ≤ cp.We want to make use of property (C) combined with the following two facts: First,

if w1, w2 ∈ A1, and w = w1w1−p2 ∈ Ap, then by (2.1.1)

[w]Ap ≤ [w1]A1 [w2]p−1A1

Second, if r > 1 then (Mf)1r ∈ A1 by the Coifman-Rochberg theorem from [7] but

we need a more precise estimate which follows from the proof:

[(Mf)1r ]A1 ≤ cn r

′.

Hence combining we obtain

[R(h)]A3 = [R(h)(Mrw)1/p′((Mrw)1/2p′

)−2]A3

≤ [R(h)(Mrw)1/p′ ]A1 [(Mrw)1/2p′ ]2A1

≤ cp.

26

Therefore, by Corollary 3.1.1 and by properties (A) and (B),∫Rn|Tf |h dx≤

∫Rn|Tf |R(h) dx

≤ c[R(h)]A3

∫RnM(f)R(h) dx

≤ cp∥∥∥∥ Mf

Mrw

∥∥∥∥Lp(Mrw)

‖h‖Lp′ (Mrw).

3.2 The main lemma and the proof of the linear

growth theorem

In this section we combine all the previous information to finish the proof of thelinear growth Theorem 3.0.4. We need a a lemma which immediately gives the proof.

Lemma 3.2.1. Let T be any Calderon-Zygmund singular integral operator and let wbe any weight. Also let 1 < p <∞ and 1 < r < 2.

Then, there is a c = cn such that:

‖Tf‖Lp(w) ≤ cp′( 1

r − 1

)1−1/pr

‖f‖Lp(Mrw)

In applications we will use the following consequence

‖Tf‖Lp(w) ≤ cp′ (r′)1/p′‖f‖Lp(Mrw)

since t1/t ≤ 2, t ≥ 1.We are now ready to finish the proof of the linear growth theorem 3.0.4.

Proof of Theorem 3.0.4. Indeed, since w ∈ A1 ⊂ A∞ we apply the sharp ReverseHolder’s exponent r = rw = 1 + 1

2n+1[w]A∞obtaining

‖Tf‖Lp(w) ≤ cT pp′ [w]

1/p′

A∞‖f‖Lp(Mrw)

and then if further w ∈ A1

‖T‖Lp(w) ≤ cT pp′ [w]

1/pA1

[w]1/p′

A∞.

27

Proof of the lemma. We consider to the equivalent dual estimate:

‖T ∗f‖Lp′ (Mrw)1−p′ ) ≤ cp′( 1

r − 1

)1−1/pr

‖f‖Lp′ (w1−p′ )

Then use the “tricky” Lemma 3.1.1 since T ∗ is also a Calderon-Zygmund operator

‖ T∗f

Mrw‖Lp′ (Mrw)) ≤ p′ c ‖ Mf

Mrw‖Lp′ (Mrw))

Next we note that by Holder’s inequality with exponent pr,

1

|Q|

∫Q

fw−1/pw1/p ≤(

1

|Q|

∫Q

wr)1/pr (

1

|Q|

∫Q

(fw−1/p)(pr)′)1/(pr)′

and hence,

(Mf)p′ ≤ (Mrw)p

′−1M(

(fw−1/p)(pr)′)p′/(pr)′

From this, and by the classical unweighted maximal theorem with the sharp con-stant, ∥∥∥∥ Mf

Mrw

∥∥∥∥Lp′ (Mrw)

≤ c( p′

p′ − (pr)′

)1/(pr)′∥∥∥∥ fw∥∥∥∥Lp′ (w)

= c(rp− 1

r − 1

)1−1/pr∥∥∥∥ fw∥∥∥∥Lp′ (w)

≤ cp( 1

r − 1

)1−1/pr∥∥∥∥ fw∥∥∥∥Lp′ (w)

.

3.3 Proof of the logarithmic growth theorem.

Proof of Theorem 3.0.5. The proof is based on ideas from [40]. Applying the Calderon-Zygmund decomposition to f at level λ, we get a family of pairwise disjoint cubesQj such that

λ <1

|Qj|

∫Qj

|f | ≤ 2nλ

Let Ω = ∪jQj and Ω = ∪j2Qj . The “good part” is defined by

g =∑j

fQjχQj(x) + f(x)χΩc(x)

28

and the “bad part” b as

b =∑j

bj

wherebj(x) = (f(x)− fQj)χQj(x)

Then, f = g + b.However, it turns out that b is “excellent” and g is really “ugly”. It is so good the

b part that we obtain the maximal function on the right hand side:

wx ∈ (Ω)c : |Tb(x)| > λ ≤ c

λ

∫Rn|f |Mwdx

by a well known argument using the cancellation of the bj and that we omit. Also

the term w(Ω) is the level set of the maximal function and the Fefferman-Stein applies(again we obtain the maximal function on the right hand side).

Combining we have

wx ∈ Rn : |Tf(x)| > λ≤w(Ω) + wx ∈ (Ω)c : |Tb(x)| > λ/2+wx ∈ (Ω)c : |Tg(x)| > λ/2.

and the first two terms are already controlled:

w(Ω) + wx ∈ (Ω)c : |Tb(x)| > λ/2 ≤ c

λ

∫Rn|f |Mwdx ≤ c[w]A1

λ

∫Rn|f |wdx

Now, by Chebyschev and the Lemma, for any p > 1 and r > 1 we have

λwx ∈ (Ω)c : |Tg(x)| > λ/2

≤ λ c(p′)p(r′)p−1 1

λp

∫Rn|g|pMr(wχ(Ω)c)dx

≤ c(p′)p(r′)p−1

∫Rn|g|Mr(wχ(Ω)c)dx.

By known standard geometric arguments we have∫Rn|g|Mr(wχ(Ω)c)dx ≤ c

∫Rn|f |Mrwdx.

Now if we choose p = 1 + 1log(r′)

we evan continue with

≤ cT log(r′)

∫Rn|f |Mrwdx r > 1.

29

Finally if we particularize r with the sharp reverse Holder’s exponent r = 1 + 1cn[w]A∞

,

we obtain

λwx ∈ (Ω)c : |Tg(x)| > λ/2 ≤ cT [w]A1(1 + log[w]A∞)

∫Rn|f |wdx.

This estimate combined with the previous one completes the proof.

30

Chapter 4

Singular Integrals: The Ap theory

In this chapter we plan to prove the A2 theorem: if T is a Calderon-Zygmund operator,then

‖T‖Lp(w) ≤ cp,n[w]max1, 1

p−1

Ap. (4.0.1)

Now, by the sharp Rubio de Francia extrapolation theorem it is enough to prove thisresult only for the special case p = 2:

‖T‖L2(w) ≤ cT [w]A2 . (4.0.2)

Observe that in this case the growth of the constant is simply linear. This is the reasonwhy the Ap result was called the A2 conjecture. After several previous results assumingsome extra regularity condition on the kernel stronger than the usual Holder-Lipschitzcondition, the theorem was finally proved by T. Hytonen in [23]. The proof that wewill present here is based on A. Lerner’s approach from [31]. The author found ahighly interesting method of passing from the continuous case to the dyadic discretecase by means of Theorem 4.1.2 below. We remit the reader to [31] for a more completeaccount of the history of (4.0.1). We plan to combine Lerner’s reduction together withsome nice ideas taken from Moen’s work [36] which avoids the use of the extrapolationtheorem and related to the main idea from [9].

An interesting point is that Lerner’s approach seems not to be sharp enough tolead to the following recent improvement of the A2 theorem that can be found in [25].Following the same circle of ideas presented in the previous chapters, the A2 constantin (4.0.2) is replaced by a mixed A2−A∞ constant. We will state the theorem withoutproof.

Theorem 4.0.1. Let T be a Calderon-Zygmund operator. Then there is a constantdepending cT depending such that

‖T‖L2(w) ≤ cT [w]1/2A2

max[w]A∞ , [w−1]A∞1/2.

31

Later on this estimate was improved in [24] for general p an even more recently theHolder-Lipschitz condition of the kernel Calderon-Zygmund operator has been replacedby the Dini-type condition in [27] and for other type of singular integral type operators.The result is the following.

Theorem 4.0.2. Let T be a Calderon-Zygmund operator with kernel satisfying a Dini-type condition. Then there is a constant depending on T such that

‖T‖Lp(w) ≤ cT [w]1/pAp

max[w]1/p′

A∞, [w1−p′ ]

1/pA∞.

It would be very nice to derive this result from the previous by means of an appro-priate extrapolation theorem. In fact, an extrapolation theorem in this direction canbe found in [25] but the output it is not sharp when considering the case of singularintegrals with exponents p 6= 2.

4.1 The dyadic Ap theory of Singular Integrals

In this section we will prove the following theorem.

Theorem 4.1.1. Let 1 < p < ∞ and let T be a Calderon-Zygmund singular integraloperator. Then, there is a constant c = cT such that for any Ap weight w,

‖T‖Lp(w)

≤ c p p′ [w]max

1, 1p−1

Ap

. (4.1.1)

To do this we will introduce some sort of dyadic singular integrals. To do this wesay that a dyadic grid, denoted D, is a collection of cubes in Rn with the followingproperties:1) each Q ∈ D satisfies |Q| = 2nk for some k ∈ Z;2) if Q,P ∈ D then Q ∩ P = ∅, P, or Q;3) for each k ∈ Z, the family Dk = Q ∈ D : |Q| = 2nk forms a partition of Rn.

We say that a family of dyadic cubes S ⊂ D is sparse if for each Q ∈ S,∣∣∣ ⋃Q′∈SQ′(Q

Q′∣∣∣ ≤ 1

2|Q|.

Given a sparse family, S, if we define

E(Q) := Q \⋃Q′∈SQ′(Q

Q′,

then the family E(Q)Q∈S is pairwise disjoint, E(Q) ⊂ Q, and |Q| ≤ 2|E(Q)|. Sparsefamilies have long been used in Calderon-Zygmund theory.

32

The sparseness property has already appeared in the proof of Corollary 1.3.1. Infact the family of cubes that appear in the proof, they are Calderon-Zygmund cubesasocciated to a fix cube, satisfies this property and plays a crucial role in the proof.

If S ⊂ D is a sparse family we define the sparse Calderon-Zygmund operator asso-ciated to S as

T Sf :=∑Q∈S

−∫Q

f dx · χQ.

As already mentioned the key idea is to “transplant” the continuous case to thediscrete version by means of the following theorem due to A. Lerner [31].

Theorem 4.1.2. Suppose that X is a Banach function spaces on Rn and T is aCalderon-Zygmund operator, then there exists a constant cT

‖T‖X ≤ cT supS⊂D‖T S‖X .

We will not prove this theorem, we will simply mention that a key tool is thedecomposition formula for functions found previously by Lerner in [?] using the localmean average (see also the Paseky lectures notes of 2011). The main idea of thisdecomposition goes back to the work of Fujii [18] where the standard average is usedinstead.

There is here a very interesting open problem related to this theorem, namely togo beyond the class of Banach function spaces as for instance the Marcinkiewicz spaceL1,∞. The “duality” property of Banach function spaces plays a central role in theproof.

In view of Theorem 4.1.2 the proof of Theorem 4.1.1 is reduced to prove the weightedestimates for the sparse operators T S .

The following properties of the family E(Q)Q∈S will be used: disjointness, E(Q) ⊂Q, and |Q| ≤ 2|E(Q)|.

Theorem 4.1.3. Suppose D is a dyadic grid, S ⊂ D is a sparse family, 1 < p < ∞,and w ∈ Ap. Then the following estimate holds

‖T S‖Lp(w) ≤ cp[w]max1, 1

p−1

Ap

wherecp = pp′2maxp,p′.

The main idea behind the proof of this theorem can be found in [9] but just for thecase p = 2.

Proof. Since T S is a positive operator we may assume f ≥ 0. We first consider thecase p ≥ 2, and let σ = w1−p′ . A simple argument shows that

‖T S‖Lp(w) = ‖T S( ·σ)‖Lp(σ),Lp(w).

33

If g ≥ 0 belongs to Lp′(w), then by duality it suffices to estimate∫RnT S(fσ)gw dx =

∑Q∈S

−∫Q

fσ dx ·∫Q

gw dx.

Now, by definition of the Ap constant, we have∑Q∈S

−∫Q

fσ dx ·∫Q

gw dx

=∑Q∈S

w(Q)σ(Q)p−1

|Q|p|Q|p−1

w(Q)σ(Q)p−1

∫Q

fσ dx ·∫Q

gw dx

≤ [w]Ap∑Q∈S

|Q|p−1

w(Q)σ(Q)p−1

∫Q

fσ dx ·∫Q

gw dx

= [w]Ap∑Q∈S

1

σ(Q)

∫Q

fσ dx · 1

w(Q)

∫Q

gw dx · |Q|p−1σ(Q)2−p

≤ 2p−1[w]Ap∑Q∈S

Aσ(f,Q)Aw(g,Q) · |E(Q)|p−1σ(Q)2−p

where

Aσ(f,Q) =1

σ(Q)

∫Q

fσ dx and Aw(g,Q) =1

w(Q)

∫Q

gw dx,

and in the last inequality we have used |Q| ≤ 2|E(Q)|. Observe that we cannot usethe [w]Ap constant again. However, since p ≥ 2 and E(Q) ⊂ Q we have

σ(Q)2−p ≤ σ(E(Q))2−p,

(note: |E(Q)| ≥ |Q|/2 > 0 so σ(E(Q)) > 0) which in turn yields∫RnT S(fσ)gw dx

≤ 2p−1[w]Ap∑Q∈S

Aσ(f,Q)Aw(g,Q) · |E(Q)|p−1σ(E(Q))2−p. (4.1.2)

By Holder’s inequality we have

|E(Q)| ≤ w(E(Q))1pσ(E(Q))

1p′ ,

so|E(Q)|p−1σ(E(Q))2−p ≤ σ(E(Q))

1pw(E(Q))

1p′ . (4.1.3)

34

Utilizing inequality (4.1.3) in the sum in (4.1.2), followed by a discrete Holder inequal-ity, followed by standard dyadic type maximal function bounds. Then, we arrive atthe desired estimate:∑

Q∈S

Aσ(f,Q)Aw(g,Q) · |E(Q)|p−1σ(E(Q))2−p

≤∑Q∈S

Aσ(f,Q)Aw(g,Q)σ(E(Q))1pw(E(Q))

1p′

≤(∑Q∈S

Aσ(f,Q)pσ(E(Q))) 1p ·(∑Q∈S

Aw(g,Q)p′w(E(Q))

) 1p′

≤ ‖MDσ f‖Lp(σ)‖MD

w g‖Lp′ (w)

≤ pp′‖f‖Lp(σ)‖g‖Lp′ (w).

The case 1 < p < 2 follows from duality, since (T S)∗ = T S , we have

‖T S‖Lp(w) = ‖T S‖Lp′ (σ) ≤ pp′2p′−1[σ]Ap′ = pp′2p

′−1[w]1p−1

Ap.

35

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