HarmonicAnalysis JUHAKINNUNEN

Harmonic Analysis

Department of Mathematics, Aalto University 2022

1 Calderón-Zygmund decomposition 1

1.1 Dyadic subcubes of a cube . . . 1

1.2 Dyadic cubes of_Rⁿ . . . 7

1.3 Calderón-Zygmund decomposition of a function . . . 12

1.4 Dyadic maximal function on_Rⁿ . . . 15

1.5 Dyadic maximal function on a cube. . . 19

2 Marcinkiewicz interpolation theorem 21 3 Bounded mean oscillation 29 3.1 Basic properties ofBMO . . . 29

3.2 Completeness ofBMO. . . 35

3.3 The John-Nirenberg inequality . . . 38

3.4 Alternative proofs for the John-Nirenberg inequality . . . 43

3.5 Consequences of the John-Nirenberg inequality . . . 47

3.6 The sharp maximal function . . . 50

3.7 BMOand interpolation . . . 55

4 Muckenhoupt weights 57 4.1 TheA_pcondition. . . 57

4.2 Properties ofApweights . . . 67

4.3 A weak type characterization ofA_p . . . 73

4.4 A strong type characterization ofAp . . . 74

4.5 A_∞and reverse Hölder inequalities . . . 76

4.6 Self-improving properties ofAp . . . 82

4.7 Strong type characterization revisited . . . 84

5 A_pandBMO 86 5.1 Characterizations ofA_p . . . 86

5.2 Characterizations ofBMO . . . 91

5.3 Maximal functions andBMO. . . 95

Calderón-Zygmund 1

decomposition

Dyadic cubes and the Calderón-Zygmund decomposition are very useful tools in harmonic analysis. The property of dyadic cubes, that either one is contained in the other or the interiors of the cubes are disjoint, is very useful in constructing coverings with pairwise disjoint cubes. The Calderón -Zygmund decomposition gives decompositions of sets and functions into good and bad parts, which can be considered separately using real variable and harmonic analysis techniques.

1.1 Dyadic subcubes of a cube

A closed cube is a bounded interval inRⁿ, whose sides are parallel to the coordinate axes and equally long, that is,

Q=[a1,b1]× · · · ×[an,bn]

withb₁−a₁=. . .=b_n−a_n. The side length of a cubeQis denoted byl(Q). In case we want to specify the center, we write

Q(x,l)=

½

y∈Rⁿ:|yi−xi| É l

2,i=1, . . . ,n

¾

for a cube with center atx∈Rⁿ and side lengthl>0. IfQ=Q(x,l), we denote αQ=Q(x,αl) forα>0. ThusαQthe cube with the same center asQ, but the side length multiplied by factorα. The integral average off∈L¹_loc(Rⁿ) in a cubeQis denoted by

f_Q= Z

Q

f(x)dx= 1

|Q| Z

Q

f(x)dx.

LetQ=[a₁,b₁]×...×[a_n,b_n] be a closed cube inRⁿwith side lengthl=b₁−a₁= . . .=bn−an. We decomposeQinto subcubes recursively. DenoteD0={Q}. Bisect

1

each interval [a_i,b_i],i=1, 2, . . . , and obtain 2ⁿcongruent subcubes ofQ. Denote this collection of cubes by D1. The cubes inD1is a partition ofQ into dyadic subcubes with pairwise disjoint interiors, that is, only the boundaries of the cubes may overlap. This does not matter, since the union of the boundaries of the cubesQ⁰∈D1is a set of measure zero. Bisect every cube inD1and obtain 2ⁿ subcubes. Denote this collection of cubes byD2. By continuing this way, we obtain generations of dyadic cubesDk,k=0, 1, 2, . . . . The dyadic subcubes inDkare of the form

·

a1+m₁l

2^k ,a1+(m₁+1)l 2^k

¸

× · · · ×

·

an+m_nl

2^k ,an+(m_n+1)l 2^k

¸ ,

wherek=0, 1, 2, . . . andm_j=0, 1, . . . , 2^k−1, j=1, . . . ,n. The collection of all dyadic subcubes ofQis

D(Q)=D= [∞ k=0

Dk.

A cubeQ⁰∈D is called a dyadic subcube ofQ. Sometimes it is convenient to consider half open cubes of the type [a₁,b₁)×· · ·×[a_n,b_n) withb₁−a₁=. . .=b_n−a_n. The corresponding dyadic subcubes are disjoint and cover the original half open cube.

TH E M O R A L: For many phenomena in harmonic analysis it is enough to consider dyadic cubes instead of all cubes. Dyadic cubes have a rigid recursive structure.

Figure 1.1:Collections of dyadic subcubes.

Remark 1.1. Dyadic subcubes ofQhave the following properties:

(1) EveryQ⁰∈Dis a subcube ofQ.

(2) Cubes inDk coverQ and the interiors of the cubes inDk are pairwise disjoint for everyk=0, 1, 2, . . . .

(3) IfQ⁰,Q⁰⁰∈D, either one is contained in the other or the interiors of the cubes are disjoint. This is called the nesting property, see Figure 1.2.

(4) IfQ⁰∈Dkand j<k, there is exactly one parent cube inDj, which contains Q⁰.

(5) Every cubeQ⁰∈Dkis a union of exactly 2ⁿ children cubesQ⁰⁰∈Dk+1with

|Q⁰| =2ⁿ|Q⁰⁰|.

(6) IfQ⁰∈Dk, thenl(Q⁰)=2^−kl(Q) and|Q⁰| =2^−nk|Q|.

Figure 1.2:Nestedness property.

Assume thatf∈L¹_loc(Rⁿ). By the Lebesgue differentiation theorem limr→0

Z

B(x,r)|f(y)−f(x)|d y=0 for almost every x∈Rⁿ. (1.1) A pointx∈Rⁿ, at which (1.1) holds, is called a Lebesgue point of f. For every Lebesgue pointxwe have

limr→0

Z

B(x,r)

f(y)d y=f(x),

since

¯

¯

¯

¯ Z

B(x,r)

f(y)d y−f(x)

¯

¯

¯

¯É Z

B(x,r)|f(y)−f(x)|d y→0 as r→0.

Moreover, every Lebesgue pointxoff is a Lebesgue point of|f|, since Z

B(x,r)||f(y)| − |f(x)||d yÉ Z

B(x,r)|f(y)−f(x)|d y→0 as r→0.

We shall need the following version of the Lebesgue differentiation theorem.

Lemma 1.2. Assume thatx∈Rⁿ is a Lebesgue point off∈L¹_loc(Rⁿ). Then

ilim→∞

1

|Q_i| Z

Qi

f(y)d y=f(x)

wheneverQ1,Q2,Q3, . . . is any sequence of cubes containingxsuch that lim_i→∞|Q_i| = 0.

TH E M O R A L: The Lebesgue differentiation theorem does not only hold for balls but also for cubes and dyadic cubes.

Proof. Let Qi=Q(xi,li), where xi∈Rⁿ is the center and li=l(Qi) is the side lenght of the cubeQ_ifor everyi=1, 2, . . . . We observe thatQ(x_i,l_i)⊂B(x,pnl_i) for everyi=1, 2, . . . .

Figure 1.3:Q(x_i,l_i)⊂B(x,p

nl_i) for everyi=1, 2, . . . .

This implies

¯

¯

¯

¯ Z

Q(x_i,l_i)

f(y)d y−f(x)

¯

¯

¯

¯É Z

Q(x_i,l_i)|f(y)−f(x)|d y É|B(x,p

nl_i)|

|Q(x_i,l_i)| Z

B(x,p

nl_i)|f(y)−f(x)|d y

= |B(0, 1)|nⁿ² Z

B(x,p

nli)|f(y)−f(x)|d y→0 as i→ ∞,

sincel_i→0 asi→ ∞. ä

The following Calderón-Zygmund decomposition will be extremely useful in harmonic analysis.

Theorem 1.3 (Calderón-Zygmund decomposition of a cube (1952)). Assume that f∈L¹_loc(Rⁿ) and letQbe a cube inRⁿ. Then for every

tÊ Z

Q|f(y)|d y

there are countably or finitely many dyadic subcubesQ_i,i=1, 2, . . . , ofQsuch that

(1) the interiors ofQ_i,i=1, 2, . . . , are pairwise disjoint, (2) t<

Z

Q_i|f(y)|d yÉ2ⁿtfor everyi=1, 2, . . . and (3) |f(x)| Étfor almost everyx∈Q\S_∞

i=1Qi.

The collection of cubesQ_i,i=1, 2, . . . , is called the Calderón-Zygmund cubes inQ at levelt.

TH E M O R A L: A cube can be divided into good and bad parts so that in the good part (complement of the Calderón-Zygmund cubes) the function is small and in the bad part (union of the Calderón-Zygmund cubes) the integral average of a function is in control. Note that the Calderón-Zygmund cubes cover the set {x∈Q:|f(x)| >t}, up to a set of measure zero, and thus the bad part contains the set where the function is unbounded.

Proof. The strategy of the proof is the following stopping time argument. For everyx∈Qsuch that|f(x)| >twe choose the largest dyadic cubeQ⁰∈Dcontaining xsuch that

Z

Q⁰|f(y)|d y>t.

Then we use the fact that for any collection of dyadic subcubes ofQthere is a subcollection of dyadic cubes with disjoint interiors and with the same union as the original cubes. These are the desired Calderón-Zygmund cubes.

Then we give a rigorous argument. Consider (possible empty) collectionQ⁰of dyadic subcubesQ⁰∈DofQ, that satisfy

Z

Q⁰|f(y)|d y>t. (1.2)

The cubes inD⁰are not necessarily pairwise disjoint, but we consider a collection of maximal dyadic cubes with respect to inclusion for which (1.2) holds true. For everyQ⁰∈D⁰ we consider all cubesQ⁰⁰∈D⁰ withQ⁰⊂Q⁰⁰. The maximal cube Q_i is the union of all dyadic subcubes ofQwhich satisfy (1.2) and containQ⁰. Nestedness property of the dyadic subcubes, see Remark1.1(3), implies that Qi∈D. LetD={Qi}ibe the collection of these maximal cubes. Maximality means thatR

R|f(y)|d yÉtfor everyR⊃Q_i,R∈D. SinceR

Q|f(y)|d yÉt, for every cube Q⁰∈D⁰there exists a maximal cubeQi∈D⁰. We show that this collection has the desired properties.

Figure 1.4:Collection of maximal subcubes.

(1) This follows immediately from maximality of the cubes inD⁰and nested- ness property of the dyadic subcubes, see Remark1.1(3). Indeed, if the interiors of two different cubes inQintersect then one is contained in the other, and hence one of them cannot be maximal, see Figure 1.4.

(2) By (1.2) we haveQ∉D⁰. IfQ_i∈D⁰∩Dkfor some k, then by properties (4) and (5) of the dyadic subcubes we conclude thatQ_iis contained in some cube Q⁰∈Dk−1with|Q⁰| =2ⁿ|Q_i|, see Figure 1.5. SinceQ_imaximal, cubeQ⁰does not satisfy (1.2). Thus

t< 1

|Qi| Z

Qi

|f(y)|d yÉ|Q⁰|

|Qi| 1

|Q⁰| Z

Q⁰|f(y)|d yÉ2ⁿt.

Figure 1.5:Q_iis contained in some cubeQ⁰∈Dk−1with|Q⁰| =2ⁿ|Q_i|. (3) Assume thatx∈Q\S_∞

i=1Qi. By the beginning of the proof, tÊ

Z

Q⁰|f(y)|d y

for every dyadic subcubeQ⁰∈Dcontaining pointx. Thus there existQ⁰_k∈Dksuch thatx∈Q⁰_kfor everyk=1, 2, . . . . Note thatQ⁰₁⊃Q⁰₂⊃Q⁰₃. . . andT_∞

i=1Q⁰_i={x}, see the Figure 1.6. Ifxis a Lebesgue point off, Lemma1.2implies

|f(x)| = lim

k→∞

1

|Q⁰

k| Z

Q⁰_k|f(y)|d yÉt.

ä Remark 1.4. The assumption tÊR

Q|f(y)|d y implies that, if the collection of the Calderón-Zygmund cubes {Q_i}_iis nonempty, the cubes are proper subcubes ofQ, that isQi⊂Q andQi6=Q for every i=1, 2, . . . . On the other hand, the collection of the Calderón-Zygmund cubes is nonempty in the caseR

Q⁰|f(y)|d yÉt for every dyadic subcube Q⁰ of Q. This occurs, if t>R

Q|f(y)|d y, since then

|{y∈Q:|f(y)| >t}| >0, compare to Theorem1.3(3).

1.2 Dyadic cubes of _R ⁿ

Next we consider the dyadic cubes inRⁿ and a global version of the Calderón- Zygmund decomposition. A half open dyadic interval inRis an interval of the

Figure 1.6:Q⁰_k∈Dksuch thatx∈Q⁰_kfor everyk=1, 2, . . . .

form

[m2^−k, (m+1)2^−k),

wherem,k∈Z. The advantage of considering half open intervals is that they are pairwise disjoint. A dyadic interval ofRⁿis a cartesian product of one-dimensional dyadic intervals

n

Y

j=1

[m_j2^−k, (m_j+1)2^−k),

wherem1, . . . ,mn,k∈Z. The collection of dyadic cubesDk,k∈Z, consists of the dyadic cubes with the side length 2^−k. The collection of all dyadic cubes inRⁿis

D(Rⁿ)=D=[

k∈ZDk.

Observe that Dk consist of cubes whose vertices lie on the lattice 2^−kZⁿ and whose side length is 2^−k. The dyadic cubes in thekth generation can be defined asDk=2^−k([0, 1)ⁿ+Zⁿ). The cubes inDkcover the wholeRⁿ and are pairwise disjoint, see Figure 1.7. Moreover, the dyadic cubes have the same properties (2)-(5) in Remark1.1as the dyadic subcubes of a given cube.

WA R N I N G: It is not true that every cube is a subcube of a dyadic cube. For example, consider [−1, 1]ⁿ. However, there is a substitute for this property: For every cubeQthere is a dyadic cubeQ⁰∈Dsuch thatQ⁰⊂Q⊂5Q⁰(exercise).

Figure 1.7:Dyadic cubes inRⁿ.

Remarks 1.5:

(1) For any subcollectionQ⊂D of dyadic cubes whose union is a bounded set, there is a subcollection of pairwise disjoint maximal cubes with the same union. A cubeQ⁰∈Qis called maximal, if there does not exist any strictly largerQ∈QwithQ⁰⊂Q, see Figure 1.8. A useful property is that the collection maximal cubes are always pairwise disjoint. This follows at once from nestedness property of the dyadic cubes. Indeed, if two different cubes inQsatisfyQ∩Q⁰6= ;, then one is contained in the other, and hence one of them cannot be maximal.

(2) Assume thatf∈L¹_loc(Rⁿ). Then E_kf(x)= X

Q∈Dk

µZ

Q

f(y)d y

¶ χQ(x)

is the conditional expectation of f with respect to the increasing collection ofσ-algebra generated byDk,k∈Z. Note that

Z

RⁿE_k(x)dx= Z

Rⁿf(x)dx

for everyk∈ZandE_kcan be considered as a discrete analog of an approx- imation of the identity.

In the one dimensional case every nonempty open set is a union of countably many disjoint open intervals and the Lebesgue outer measure of an open set is the

Figure 1.8:A collection of maximal cubes.

sum of volumes of these intervals. Next we consider this question in the higher dimensional case.

Lemma 1.6. Every nonempty open set inRⁿis a union of countably many pair- wise disjoint dyadic cubes.

Proof. LetΩbe a nonempty open set inRⁿ. Consider dyadic cubes inQ1that are contained inΩ^{and denote}Q1={Q∈D1:Q⊂Ω}. Then consider dyadic cubes in Q2that are contained inΩand do not intersect any of the cubes inQ1and denote

Q2={Q∈D2:Q⊂Ω^,Q∩J= ;for everyJ∈Q1}.

Recursively define Qk=

(

Q∈Dk:Q⊂Ω^,Q∩J= ;for everyJ∈

k−1[

i=1

Qi

)

for every k=2, 3, . . . . Then Q=S_∞

k=1Qk is a countable collection of pairwise disjoint dyadic cubes.

Claim:Ω=S

Q∈QQ.

Reason. It is clear from the construction thatS

Q∈QQ⊂Ω. For the reverse inclu- sion, letx∈Ω^{. Letk}be so large that the common diameter of the cubes inDkis smaller thanr, that is,p

n2^−k<r. SinceΩis open, there exists a ballB(x,r)⊂Ω

withr>0. Since the dyadicQkcubes coverRⁿ, there exists a dyadic cubeQ∈Qk

withx∈QandQ⊂B(x,r)⊂Ω. There are two possibilitiesQ∈QkorQ∉Qk. If Q∈Qk, thenx∈Q⊂S

Q∈QQ. IfQ∉Qk, there existsJ∈S_k−1

i=1QiwithJ∩Q6= ;. The nesting property of dyadic cubes impliesQ⊂J andx∈Q⊂J⊂S

Q∈QQ. _■

Remark 1.7. The Whitney decomposition of a nonempty proper open subsetΩ ofRⁿ states that it can be represented as a union of countably many pairwise disjoint dyadic intervals whose side lengths are comparable to their distance to the boundary of the open set. More precisely, there are pairwise disjoint dyadic cubesQ_i,i=1, 2, . . . , such that

• Ω=S_∞

i=1Qi,

• p

nl(Qi)Édist(Qi,Rⁿ\Ω)É4p nl(Qi),

• if the boundaries ofQ_iandQ_jtouch, then ¹₄É^l(Q_l(Qⁱ_j⁾₎É4,

• for everyQ_ithere exist at most 12ⁿ cubes in the collection that touch it.

See [9, Proposition 7.3.4].

Theorem 1.8 (Global Calderón-Zygmund decomposition (1952)). Assume that f∈L¹(Rⁿ). Then for everyt>0 there are countably or finitely many dyadic cubes Qi,i=1, 2, . . . , inRⁿsuch that

(1) cubesQ_i,i=1, 2, . . . , are pairwise disjoint, (2) t<

Z

Q_i|f(y)|d yÉ2ⁿtfor everyi=1, 2, . . . and (3) |f(x)| Étfor almost everyx∈Rⁿ\S_∞

i=1Q_i.

The collection of cubesQ_i,i=1, 2, . . . , is called the Calderón-Zygmund cubes inRⁿ at levelt.

TH E M O R A L: The difference to the Calderón-Zygmund in a cube is that we assume global integrability instead of local inregrability. With this assumption, we obtain the Calderón-Zygmund decomposition at every levelt>0. Note that, if the function is bounded, the Calderón-Zygmund decomposition may be empty for some values oft>0.

Proof. As in the proof of Theorem1.3, consider the collection (possible empty) of dyadic cubesQ⁰∈DinRⁿthat satisfyR

Q⁰|f(y)|d y>t. Note that l(Q⁰)ⁿ= |Q⁰| <1

t Z

Q⁰|f(y)|d yÉ1 t Z

Rⁿ|f(y)|d y for every cubeQ⁰∈Dthat satisfiesR

Q⁰|f(y)|d y>t. Thus for every cubeQ⁰∈D⁰ there exists a maximal cubeQ_i∈D⁰. Otherwise, the proof is similar as the proof

of Theorem1.3. ä

Example 1.9. Consider the Calderón-Zygmund decomposition forf:R→R,f(x)= χ[0,1](x) at levelt>0. We may assume that 0<t<1, sinceR

Q|f(x)|dxÉ1 for every intervalQ⊂R. In other words, iftÊ1, there are no intervalsQ inRfor whichR

Q|f(x)|dx>t.

For 0<t<1, choosek∈{0,−1,−2, . . . } such that 2^k−1Ét<2^k. We claim that the Calderón-Zygmund decomposition at leveltconsists only of one interval [0, 2^−k).

To see this, we observe that 1

|Q| Z

Q|f(x)|dx=2^k Z 2^−k

0 χ[0,1](x)dx=2^k>t.

On the other hand, if Q⊂Q⁰, Q6=Q⁰, where Q⁰∈D, then Q⁰=[0, 2^−k+l), l∈ {1, 2, . . . }, and thus

1

|Q⁰| Z

Q⁰|f(x)|dx=2^k−l<2^k−1Ét so thatQis the maximal cube with the propertyR

Q|f(x)|dx>t.

1.3 Calderón-Zygmund decomposition of a function

For a function f∈L¹(Rⁿ), and any levelt>0, we have the decomposition

f=fχ{|f|Ét}+fχ{|f|>t} (1.3) into good partg=fχ_{|f|Ét}, which is bounded, and the bad partb=fχ_{|f|>t}. These parts can be analyzed separately using real variable techniques. For the good part we have the bounds

kgk1É kfk1 and kgk∞Ét and for the bad part

kbk1É kfk1 and |{x∈Rⁿ:b(x)6=0}| É |{x∈Rⁿ:|f(x)| >t}| É1 tkfk1. The last bound follows from Chebyshev’s inequality and tells that the measure of the support of the bad part is small. This truncation method is will be useful in later in connection with interpolation, see Lemma2.1, but here we consider a more refined way to decompose an arbitrary integrable function into its good and large bad parts so that not only the absolute value but also the local oscillation is in control.

Theorem 1.10 (Calderón-Zygmund decomposition of a function (1952)). As- sume thatf∈L¹(Rⁿ) and lett>0. Then there are functionsgandb, and countably or finitely many pairwise disjoint dyadic cubesQ_i,i=1, 2, . . . , such that

(1) f=g+b,

(2) kgk1É kfk1,

(3) kgk∞É2ⁿt,

(4) b= X∞ i=1

b_i, whereb_i=0 inRⁿ\Q_i,i=1, 2, . . .,

(5) Z

Q_i

b_i(x)dx=0,i=1, 2, . . .,

(6) Z

Q_i|b_i(x)|dxÉ2ⁿ⁺¹tand (7)

¯

¯

¯

¯

¯ [∞ i=1

Qi

¯

¯

¯

¯

¯ É1

tkfk1.

TH E M O R A L: Any function f ∈L¹(Rⁿ) can be represented as a sum of a good and a bad functionf=g+b, wheregis bounded andb=P_∞

i=1bi, wherebi, i=1, 2, . . ., are highly oscillating localized function with zero integral averages.

Note that the bad functionbcontains the unbounded part of functionf. Remarks 1.11:

(1) It follows from (1) and (2) that

kbk1É kf−gk1É kfk1+ kgk1É2kfk1

and thusb∈L¹(Rⁿ).This shows that the bad functiongis integrable.

(2)

kgkp= µZ

Rⁿ|g(x)|^pdx

¶¹_p

= µZ

Rⁿ|g(x)|^p−1|g(x)|dx

¶¹_p

É µZ

Rⁿkgk∞^p−1|g(x)|dx

¶¹_p É kgk

1 p 1kgk¹⁻

1 p

∞

É kgk

1 p

1(2ⁿt)^1−1pÉ kfk

1 p

1(2ⁿt)^1−1p

and thus g∈L^p(Rⁿ) whenever 1É pÉ ∞. This shows that the good functiongis essentially bounded and belongs to allL^p-spaces.

Proof. LetQ_i,i=1, 2, . . ., be the Calderón-Zygmund cubes forf at levelt>0, see Theorem1.8. Define

g(x)=











f(x), x∈Rⁿ\ [∞ i=1

Q_i, Z

Q_i

f(y)d y, x∈Qi, i=1, 2, . . . . and

bi(x)=(f(x)−f_Qi)χQi(x), i=1, 2, . . . .

TH E M O R A L: The functiongis defined so that it is equal to f outside the Calderón-Zygmund cubes and in a Calderón-Zygmund cube the it is the average of the function in that cube.

(1)

g(x)=f(x)− X∞ i=1

(f(x)−f_Qi)χQi(x)=f(x)− X∞ i=1

b_i(x)=f(x)−b(x).

(2) Z

Rⁿ|g(x)|dx= Z

Rⁿ\S_∞

i=1Q_i|g(x)|dx+ Z

S_∞

i=1Q_i|g(x)|dx

= Z

Rⁿ\S_∞

i=1Q_i|f(x)|dx+ X∞ i=1

Z

Q_i|g(x)|dx

= Z

Rⁿ\S_∞ i=1Qi

|f(x)|dx+ X∞ i=1

Z

Qi

|f_Qi|dx É

Z

Rⁿ\S_∞

i=1Q_i|f(x)|dx+ X∞ i=1

Z

Q_i|f|Q_idx

= Z

Rⁿ\S_∞

i=1Q_i|f(x)|dx+ X∞ i=1

Z

Q_i|f(x)|dx|Qi|

= Z

Rⁿ\S_∞ i=1Qi

|f(x)|dx+ Z

S_∞ i=1Qi

|f(x)|dx= Z

Rⁿ|f(x)|dx.

(3) By Theorem1.8, we have|f(x)| Étfor almost everyx∈Rⁿ\S_∞

i=1Q_iand

¯

¯

¯

¯ Z

Q_i

f(y)d y

¯

¯

¯

¯É Z

Q_i|f(y)|d yÉ2ⁿt, i=1, 2, . . . . This implies that|g(x)| É2ⁿtfor almost everyx∈Rⁿ.

(4) See the proof of (1).

(5)

Z

Qi

b_i(x)dx= Z

Qi

(f(x)−f_Qi)χQ_i(x)dx

= Z

Qi

f(x)dx−f_Qi=0, i=1, 2, . . . . (6) By Theorem1.8, we have

Z

Qi

|b_i(x)|dxÉ Z

Qi

(|f(x)| + |f|Q_i)dx É2

Z

Q_i|f(x)|dxÉ2ⁿ⁺¹t|Q_i|, i=1, 2, . . . . (7)

¯

¯

¯

¯

¯ [∞ i=1

Q_i

¯

¯

¯

¯

¯

= X∞ i=1

|Q_i| É1 t

X∞ i=1

Z

Q_i|f(y)|d y=1 tkfk1.

ä

1.4 Dyadic maximal function on _R ⁿ

There is an interpretation of the Calderón-Zygmund decomposition in terms of maximal functions. The dyadic maximal function off∈L¹_loc(Rⁿ) is

M_df(x)=sup Z

Q|f(y)|d y, (1.4)

where the supremum is taken over all dyadic cubesQcontainingx. By Lemma 1.2, for almost everyx∈Rⁿ, we have

|f(x)| = lim

k→∞

Z

Q_k|f(y)|d yÉM_df(x),

whereQ_k∈Dk. Thus the dyadic maximal function is bigger than the absolute value of the function almost everywhere. This explains the name maximal func- tion.

WA R N I N G: The dyadic maximal function is not comparable to the standard Hardy-Littlewood maximal function

M f(x)=sup Z

Q|f(y)|d y, (1.5)

where the supremum is taken over all cubesQinRⁿcontainingx. It is clear that M_df(x)ÉM f(x) for everyx∈Rⁿ, but the inequality in the reverse direction does not hold.

For example, considerf:Rⁿ→R,

f(x)=







1, x_nÊ0, 0, x_n<0.

ThenM_df(x)=f(x) for everyx∈Rⁿ, but the standard Hardy-Littlewood maximal function is strictly positive everywhere.

Lemma 1.12. Assume thatf∈L¹(Rⁿ) and lett>0 such that the set E_t={x∈Rⁿ:M_df(x)>t}

has finite measure. ThenEt is the union of pairwise disjoint dyadic Calderón- Zygmund cubesQ_i, i=1, 2, . . . , given by Theorem1.8. In particular, cubesQ_i, i=1, 2, . . . , satisfy properties (1)-(3) in Theorem1.8.

Proof. We show thatE_t=S_∞

i=1Q_i. Ifx∈E_t, thenM_df(x)>tand thus there exists a dyadic cubeQ∈Dsuch thatx∈Qand

Z

Q|f(y)|d y>t.

The Calderón-Zygmund cubesQ_i,i=1, 2, . . . , given by Theorem1.8is a collection of maximal dyadic cubes with this property. This implies thatx∈S_∞

i=1Q_iand thus E_t⊂S_∞

i=1Q_i.

On the other hand, ifx∈S_∞

i=1Q_i, thenx∈Q_ifor somei=1, 2, . . . and by the Calderón-Zygmund decomposition

M_df(x)Ê Z

Qi

|f(y)|d y>t.

This shows thatx∈E_tand thusS_∞

i=1Q_i⊂E_t. This completes the proof. ä

Figure 1.9:The distribution set of the dyadic maximal function.

TH E M O R A L: The union of the Calderón-Zygmund cubes is the distribution set of the dyadic maximal function. This means that the Calderón-Zygmund decomposition is more closely related tof=fχ{MdfÉt}+fχ{Mdf>t}instead of f= fχ{|f|Ét}+fχ{|f|>t}in (1.3). Note carefully, that this is not the Calderón-Zygmund decomposition of a function constructed in the proof of Theorem1.8, but Lemma 1.12shows that {M_df >t}is the union of the Calderón-Zygmund cubes. This suggest another point of view to the Calderón-Zygmund decomposition, in which we analyse the distribution set of the dyadic maximal function, for example, using the Whitney covering theorem.

Remarks 1.13:

(1) By summing up over all Calderón-Zygmund cubes, we have

|E_t| = X∞ i=1

|Q_i| É1 t

X∞ i=1

Z

Q_i|f(y)|d y=1 t Z

E_t|f(y)|d y.

This is a weak type estimate for the dyadic maximal function. Observe that in contrast with the standard Hardy-Littlewood maximal function, there is no dimensional constant in the estimate. This estimate holds true also for other measures than Lebesgue measure.

(2) We also have an inequality to the reverse direction, since Z

Et

|f(y)|d y= X∞ i=1

Z

Qi

|f(y)|d yÉ2ⁿt X∞ i=1

|Q_i| =2ⁿt|E_t|. (1.6) This is a reverse weak type inequality for the dyadic maximal function.

(3) Ift>s, thenE_t⊂E_sand, by maximality of the Calderón-Zygmund cubes, each cube in the decomposition at leveltis contained in a cube in the de- composition at levels. In this sense, the Calderón-Zygmund decompostions are nested.

(4) Observe that the Calderón-Zygmund decomposition in Lemma1.12can be done under the assumption that|{x∈Rⁿ:M_df(x)>t}| < ∞. By the weak type estimate in (1), this is weaker than assuming f∈L¹(Rⁿ).

Next we show how we can use the Calderón-Zygmund decomposition to obtain estimates for the standard maximal function defined by (1.5).

Lemma 1.14. Assume that f ∈L¹(Rⁿ) and letQ_i,i=1, 2, . . . , be the Calderón- Zygmund cubes off at levelt>0 given by Theorem1.8. Then

(1) [∞ i=1

Q_i⊂{x∈Rⁿ:M f(x)>t}and (2) {x∈Rⁿ:M f(x)>4ⁿt}⊂

[∞ i=1

3Qi.

TH E M O R A L: The first claim is essentially a restatement of the fact that M_df(x)ÉM f(x) for everyx∈Rⁿand thus

{x∈Rⁿ:M_df(x)>t}⊂{x∈Rⁿ:M f(x)>t}.

The second claim implies the following inequality in the reverse direction

|{x∈Rⁿ:M f(x)>4ⁿt}| É X∞ i=1

|3Q_i| =3ⁿ X∞ i=1

|Q_i| =3ⁿ|{x∈Rⁿ:M_df(x)>t}| (1.7) for everyt>0. In particular, this gives the weak type estimate for the standard Hardy-Littlewood maximal function as well, since

|{x∈Rⁿ:M f(x)>t}| É3ⁿ

¯

¯

¯

¯

½

x∈Rⁿ:M_df(x)> t 4ⁿ

¾¯

¯

¯

¯É3ⁿ4ⁿ t

Z

Rⁿ|f(y)|d y for everyt>0. Thus the weak type estimate for the standard Hardy-Littlewood maximal function follows from the corresponding estimate for the dyadic maximal function. This shows that information on dyadic cubes can be used to obtain information over all cubes, see also Example2.6(2).

Proof. (1) Ifx∈S_∞

i=1Q_i, thenx∈Q_ifor somei=1, 2, . . . . By Theorem1.8 M f(x)Ê

Z

Q_i|f(y)|d y>t and thusS_∞

i=1Q_i⊂{x∈Rⁿ:M f(x)>t}.

(2) Assume thatx∈Rⁿ\S_∞

i=13Q_iand letQany closed cube inRⁿcontaining x. Choosek∈Zsuch that 2^−k−1<l(Q)É2^−k. Then there exists at most 2ⁿ such dyadic cubes R₁, . . . ,R_m∈Dk, which intersect the interior of Q. We note that

Figure 1.10:At most 2ⁿdyadic cubesR₁, . . . ,R_m∈Dkintersect the interior ofQ.

Q⊂3R_jfor everyj=1, . . . ,m. Each cubeR_j, j=1, . . . ,m, cannot be a subset of any of the cubesQ_i,i=1, 2, . . . , since otherwisex∈Q⊂3Q_ifor somei=1, 2, . . . , which is not possible, sincex∈Rⁿ\S_∞

i=13Q_i. SinceR_jis not contained in the union of the Calderón-Zygmund cubes, by the proof of Theorem1.8and Theorem1.3, we

have 1

|Rj| Z

Rj

|f(y)|d yÉt, i=1, . . . ,m.

On the other hand,|Rj| =2^−kn=2ⁿ2^−kn−nÉ2ⁿl(Q)ⁿÉ2ⁿ|Q|andmÉ2ⁿ, thus 1

|Q| Z

Q|f(y)|d y= 1

|Q|

m

X

j=1

Z

Q∩Rj

|f(y)|d y É

m

X

j=1

|Rj|

|Q| 1

|R_j| Z

R_j|f(y)|d yÉm2ⁿtÉ4ⁿt.

Since this holds true for every cubeQcontainingx, we haveM f(x)É4ⁿtfor every x∈Rⁿ\S_∞

i=13Q_i. In other words, Rⁿ\

[∞ i=1

3Qi⊂{x∈Rⁿ:M f(x)É4ⁿt},

from which the claim follows. ä

1.5 Dyadic maximal function on a cube

Next we discuss briefly the dyadic maximal function with respect to the dyadic subcubes of a cube. LetQ0⊂Rⁿbe a cube and assume thatf∈L¹(Q0). The dyadic maximal functionM_d;Q0f atx∈Q₀is

M_d,Q0f(x)=sup

Q3x

Z

Q|f(y)|d y, (1.8)

where the supremum is taken over all dyadic cubesQ∈D(Q₀) withx∈Q.

Letf,g∈L¹(Q₀) and x∈Q₀. It follows immediately from the definition that M_d,Q0f(x)Ê0,

M_d,Q0(f+g)(x)ÉM_d,Q0f(x)+M_d,Q0g(x), and

M_d,Q0(a f)(x)= |a|M_d,Q0f(x) for everya∈R.

LetE_t={x∈Q₀:M_d,Q0f(x)>t}, t>0. For everytÊ |f|Q0 the set E_t is the union of pairwise disjoint dyadic Calderón-Zygmund cubesQi,i=1, 2, . . . , given by Theorem1.3. In particular, cubesQ_i,i=1, 2, . . . , satisfy properties (1)-(3) in Theorem1.3. For 0<t< |f|Q₀, we haveEt=Q0.

Theorem1.3gives simple proofs for norm estimates for the dyadic maximal function. The next result is a weak type estimate.

Lemma 1.15. LetQ₀⊂Rⁿ be a cube. Assume that f∈L¹(Q₀) and let E_t={x∈ Q₀:M_d,Q0f(x)>t}. Then

|E_t| É1 t Z

Et

|f(x)|dx for everyt>0.

Proof. Lett>0. Ift< |f|Q0, thenE_t=Q₀and thus

|Et| = |Q0| É1 t Z

Q0

|f(x)|dx=1 t Z

Et

|f(x)|dx.

Then assume thattÊ |f|Q₀. LetQ_i,i=1, 2, . . . be the collection of dyadic subcubes ofQ₀given by Theorem1.3. By using the properties of the Calderón-Zygmund cubes, we obtain

|E_t| =

¯

¯

¯

¯

¯ [∞ i=1

Q_i

¯

¯

¯

¯

¯

= X∞ i=1

|Q_i| É1 t

X∞ i=1

Z

Qi

|f(x)|dx=1 t Z

Et

|f(x)|dx.

ä

There is also a reverse weak type estimate for the dyadic maximal function.

Lemma 1.16. LetQ₀⊂Rⁿ be a cube. Assume thatf ∈L¹(Q₀) is a nonnegative function and letE_t={x∈Q₀:M_d,Q0f(x)>t}. Then

Z

E_t|f(x)|dxÉ2ⁿt|E_t| for everytÊ |f|Q₀.

Proof. Let Q_i, i=1, 2, . . . be the collection of dyadic subcubes ofQ₀ given by Theorem1.3. By using the properties of the Calderón-Zygmund cubes, we obtain

Z

E_t|f(x)|dx= X∞ i=1

Z

Q_i

f(x)dxÉ2ⁿt X∞ i=1

|Q_i| =2ⁿt|E_t|.

ä

Marcinkiewicz interpolation 2

theorem

Interpolation of operator is an important tool in harmonic analysis. Consider an operator, which maps Lebesgue measurable functions to functions. A typical example is the maximal operator. The rough idea of interpolation is that if we know that the operator is a bounded in two different function spaces, then it is bounded in the intermediate function spaces.

We are mainly interested inL^p(Rⁿ) spaces with 1ÉpÉ ∞and we begin with a useful decomposition of anL^p(Rⁿ) function into two parts. To this end, we define L^p1(Rⁿ)+L^p2(Rⁿ), 1Ép1<p2É ∞, to be the space of all functions of the form f=f₁+f₂, wheref₁∈L^p1(Rⁿ) andf₂∈L^p2(Rⁿ).

Lemma 2.1. Let 1Ép₁<p₂É ∞and p₁ÉpÉp₂. Then L^p(Rⁿ)⊂L^p1(Rⁿ)+ L^p2(Rⁿ).

TH E M O R A L: EveryL^p(Rⁿ) function can be written as a sum of anL^p1(Rⁿ) function and anL^p2(Rⁿ) function wheneverp₁ÉpÉp₂.

Proof. If p=p₁ or p=p₂, there is nothing to prove, since f =f+0. Thus we assume that p₁<p<p₂. Assume that f∈L^p(Rⁿ) and lett>0. Define

f₁(x)=f(x)χ_{|f|>t}(x)=







f(x), if |f(x)| >t, 0, if |f(x)| Ét, and

f2(x)=f(x)χ_{|f|Ét}(x)=







f(x), if |f(x)| Ét, 0, if |f(x)| >t.

Clearly

f(x)=f(x)χ_{|f|>t}(x)+f(x)χ_{|f|Ét}(x)=f1(x)+f2(x), 21

see (1.3).

First we show thatf₁∈L^p1(Rⁿ). Sincep₁<p, we obtain Z

Rⁿ|f₁(x)|^p1dx= Z

{|f|>t}|f(x)|^p1dx= Z

{|f|>t}|f(x)|^p1−p|f(x)|^pdx Ét^p1−p

Z

Rⁿ|f(x)|^pdxÉt^p1−p||f||^pp< ∞. Then we show thatf₂∈L^p2(Rⁿ). Sincep₂>p, we have

Z

Rⁿ|f₂(x)|^p2dx= Z

{|f|Ét}|f(x)|^p2dx= Z

{|f|Ét}|f(x)|^p2−p|f(x)|^pdx Ét^p2−p

Z

Rⁿ|f(x)|^pdxÉt^p2−p||f||^pp< ∞.

Thusf=f₁+f₂with f₁∈L^p1(Rⁿ) andf₂∈L^p2(Rⁿ), as required. ä Definition 2.2. LetTbe an operator fromL^p(Rⁿ) to Lebesgue measurable func- tions onRⁿ.

(1) Tis sublinear, if for everyf,g∈L^p(Rⁿ),

|T(f+g)(x)| É |T f(x)| + |T g(x)| and

|T(a f)(x)| = |a||T f(x)|, a∈R, for almost everyx∈Rⁿ.

(2) Tis of strong type (p,p), 1ÉpÉ ∞, if there exists a constantc, indepen- dent of the functionf, such that

||T f||pÉc||f||p

for everyf∈L^p(Rⁿ).

(3) Tis of weak type (p,p), 1Ép< ∞, if there exists a constantc, independent of the function f, such that

|{x∈Rⁿ:|T f(x)| >t}| É

³c t||f||p

´p

for everyt>0 andf∈L^p(Rⁿ).

TH E M O R A L: Operator is of strong type (p,p) if and only if it is a bounded operator from L^p(Rⁿ) toL^p(Rⁿ). The corresponding weak type condition is a substitute for this for several operators in harmonic analysis which fail to be bounded in certainL^p(Rⁿ) spaces. For example, the Hardy-Littlewood maximal operator is a sublinear operator which is not of strong type (1, 1) but it is of weak type (1, 1).

Remarks 2.3:

(1) Every linear operatorTis sublinear, since

|T(f+g)(x)| = |T f(x)+T g(x)| É |T f(x)| + |T g(x)| and

|T(a f)(x)| = |aT f(x)| = |a||T f(x)|.

(2) The notion of strong type (p,p) is stronger than weak type (p,p). If kT fkpÉckfkpfor everyf∈L^p(Rⁿ), by Chebyshev’s inequality

|{x∈Rⁿ:|T f(x)| >t}| É 1 t^p

Z

Rⁿ|T f(x)|^pdx= 1

t^pkT fk^ppɳc tkfkp

´p

. Theorem 2.4 (Marcinkiewicz interpolation theorem (1939)). Let 1Ép₁<

p₂É ∞ and assume that T is a sublinear operator from L^p1(Rⁿ)+L^p2(Rⁿ) to Lebesgue measurable functions on Rⁿ, which is simultaneously of weak type (p₁,p₁) and (p₂,p₂). ThenTis of strong type (p,p) wheneverp₁<p<p₂. TH E M O R A L: Weak type estimates at the endpoint spaces imply strong type estimates spaces between.

Proof. p2< ∞ Assume that if there exist constantc1andc2, independent of the function f, such that

|{x∈Rⁿ:|T f(x)| >t}| ɳc1

t ||f||p₁

´p1

, t>0, for everyf∈L^p1(Rⁿ) and

|{x∈Rⁿ:|T f(x)| >t}| ɳc2

t ||f||p₂

´p2

, t>0.

for everyf∈L^p2(Rⁿ) . Consider the decomposition f=f1+f2=fχ_{|f|>t}+fχ_{|f|Ét},

wheref₁∈L^p1(Rⁿ) andf₂∈L^p2(Rⁿ), given by Lemma2.1. Sublinearity|T f(x)| É

|T f₁(x)| + |T f₂(x)|implies that for almost every x for which|T f(x)| >t, either

|T f₁(x)| >₂^t or|T f₂(x)| >₂^t. Thus

|{x∈Rⁿ:|T f(x)| >t}| É

¯

¯

¯

¯

½

x∈Rⁿ:|T f₁(x)| > t 2

¾

∪

½

x∈Rⁿ:|T f₂(x)| > t 2

¾¯

¯

¯

¯ É

¯

¯

¯

¯

½

x∈Rⁿ:|T f₁(x)| > t 2

¾¯

¯

¯

¯+

¯

¯

¯

¯

½

x∈Rⁿ:|T f₂(x)| > t 2

¾¯

¯

¯

¯ É

Ãc1 t 2

||f1||p₁

!p1

+ Ãc2

t 2

||f2||p₂

!p2

É µ2c₁

t

¶p1Z

{x∈Rⁿ:|f(x)|>t}|f(x)|^p1dx +

µ2c₂ t

¶p2Z

{x∈Rⁿ:|f(x)|Ét}|f(x)|^p2dx.

By Cavalieri’s principle Z

Rⁿ|T f(x)|^pdx=p Z _∞

0

t^p−1|{x∈Rⁿ:|T f(x)| >t}|dt É(2c₁)^p1p

Z_∞

0

t^p−p1−1 Z

{x∈Rⁿ:|f(x)|>t}|f(x)|^p1dx dt +(2c2)^p2p

Z _∞

0

t^p−p2−1 Z

{x∈Rⁿ:|f(x)|Ét}|f(x)|^p2dx dt, where the integrals on the right-hand side are computed by Fubini’s theorem as

Z _∞

0

t^p−p1−1 Z

{x∈Rⁿ:|f(x)|>t}|f(x)|^p1dx dt= Z

Rⁿ|f(x)|^p1 Z _|f(x)|

0

t^p−p1−1dt dx

= 1

p−p₁ Z

Rⁿ|f(x)|^p−p1|f(x)|^p1dx

= 1

p−p₁ Z

Rⁿ|f(x)|^pdx and

Z _∞

0

t^p−p2−1 Z

{x∈Rⁿ:|f(x)|Ét}|f(x)|^p2dx dt= Z

Rⁿ|f(x)|^p2 Z _∞

|f(x)|

t^p−p2−1dt dx

= 1

p₂−p Z

Rⁿ|f(x)|^p2|f(x)|^p−p2dx

= 1

p2−p Z

Rⁿ|f(x)|^pdx.

Thus we arrive at

||T f||^pp= Z

Rⁿ|T f(x)|^pdx É(2c1)^p1 p

p−p1

Z

Rⁿ|f(x)|^pdx+(2c2)^p2 p p2−p

Z

Rⁿ|f(x)|^pdx

=p

µ(2c₁)^p1

p−p₁ +(2c₂)^p2 p₂−p

¶

||f||^pp.

p₂= ∞ Assume that||T f||∞Éc₂||f||∞for everyf∈L^∞(Rⁿ) and write f=f₁+f₂=fχ_{|f|> ^t

2c2}+fχ_{|f|É ^t

2c2}.

Thenf1∈L^p1(Rⁿ) as in Lemma2.1andf2∈L^∞(Rⁿ), sincekf2k∞É_2c^t₂. We apply strong (∞,∞) estimate forf₂and obtain

|T f₂(x)| É ||T f₂||∞Éc₂||f₂||∞Éc₂ t 2c₂=t

2 for almost everyx∈Rⁿand, consequently,

¯

¯

¯

¯

½

x∈Rⁿ:|T f₂(x)| > t 2

¾¯

¯

¯

¯=0.