A Short Introduction to BMO Space

Jiongji Guo, Chunkit Hung

HKUST

May 13, 2019

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Overview

1 Motivation for BMO space

2 BMO Space: An Introduction

3 BMO and L8: John-Nirenburg inequality

4 BMO and Lp

5 Applications of J-N’s inequality

6 BMO-H1 Duality

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Motivation for BMO

Fritz John’s work on (nonlinear) elasticity, based onAgmon-Douglis-Nirenberg’s previous work.

Deformation of an elastic body Ω ⊂ R3 is a function f : Ω ÞÑ R3.

Elasticity equation: Dfpxq “ RpxqEpxq, where Rpxq is a rotationand Epxq “ rpDfpxqqTDfpxqs12.

The nonlinear elastic energy controls the nonlinear strain|Epxq ´ I| but not the local rotation Rpxq.

Linear elasticity: Korn’s inequality gives L2 estimate of theinfinitesimal rotation in terms of L2 norm of the linear stain.

John’s result: extend it to nonlinear case, that is, if the nonlinearstrain is uniformly small on a cube then the BMO norm (as wecan call after John-Nirenberg define it) of Df is also small.

Epxq ´ IL8pQq ď ε implies that DfBMOpQq ď Cε for sufficientlysmall ε.

The estimate is actually an estimate on the oscillation of Rpxqknowing that E stays close to I by hypothesis.

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Motivation for BMO

Knowing Nirenbergs analytical power and his love of inequalities,John drew Nirenberg into exploring the implications of his resulton elasticity.

Take p “ 2, Rpxq stays close in L2 to its average on Q : Korn-styleinequality:1|Q|

ş

Q |Df ´ pDfqQ|2dx ď C sup

xPQ|Epxq ´ I|2.

Another motivation in Moser’s paper to study the Harnackinequality for the solution of a divergence-form elliptic equation:řni,j“1 BipaijqpxqBjuq “ 0, aijpxq

1s are L8 and uniformly elliptic.

Moser gave a third proof of the Holder regularity of such u afterDe Giorgi and John Nash. The proof of Harnack’s inequalityimplies Holder continuity by an elementary argument.

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Another feasible motivation for BMO

Another motivation to define BMO space can be found in the gap ofindex for Sobolev and Morrey embedding theorems:

Sobolev embedding: Continuous embeddings of Sobolev spacesinto appropriate Lq spaces:

k P p0, nq, 1 ă p ă nk , W

k,ppRnq ãÑ Lp˚

pRnq if 1p˚ “

1p ´

kn ;

If k P p0, nq is further an integer, the inclusion also exists whenp “ 1.

Special case k “ 1 : Gagliardo-Nirenberg-Sobolev inequality.

k “ n : the inclusion holds when p “ 8.

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Usefulness of L8

The Morrey embedding theorem describes continuous embeddings ofSobolev spaces into appropriate Holder spaces.

For k P p0, nq and nk ă p ă 8, the inclusion exists for

λ “ k ´ np : W k,ppRnq ãÑ ΛγpRnq.p “ n

k : Unknown for this case currently.

As usual, we may also expect to imbed W k,p into L8 for p “ nk.

Set k “ 1 and p “ n. The embedding into L8 only holds whenn “ 1.

Such claim is not universally right! The only thing we can say isabout the embedding into Lq, p ď q ă 8 when 1 ď p ă 8.

Counterexample: upxq “ log´

log´

11`|x|

¯¯

. It is in W 1,npB1q for

n ą 1, but not in L8pB1q.

Another failure of L8 can also be seen in the Calderon-Zygmundsingular integral operator, which does not preserve L8, we omit thedetails here.

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Definition of BMO Space

BMO Space

The bounded mean oscillation space is the vector space of all locallyintegrable function on Rn for which

supB

1

|B|

ż

B

|fpxq ´ fB |dx ă 8,

where the supremum is taken over all balls B ⊂ Rn, and fB “1|B|

ş

Bf

denotes the mean of f over B.

We note that the supremum is 0 for any constant function, so the definitionabove does not actually induce a proper norm.

To make the definition more make sense, we define BMO as the set of allequivalence functions (modulo equality almost everywhere) of locallyintegrable functions modulo additive constants. This set carries naturally thestructure of a vector space, the above supremum does then define a norm, andBMO is complete under this norm. Thus it is a Banach space.

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Basic Properties

Small exercise: For a function u whose BMO ”norm” is K, we canclaim that if u is bouneded, then |upxq ´ upyq| ď

?2K in any cube

C0.

A simple triangle inequality shows the definition is equivalent to ifwe replace fB as any constant aB depending on B, followingJohn-Nirenberg’s original setting.

The function in BMO space keeps scaling invariance andtranslation invariance.

As usual, we can find the equivalence to the ”norm” in which wereplace B by Q, arbitrary cube whose sides are parallel to thecoordinate axes.

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BMO and sharp maximal function

From the study of harmonic analysis, we may find that it isstrongly related to the sharp maximal function, which is a variantof Hardy-Littlewood maximal function learnt in real analysis:

f 7pxq “ supBQx

infbPC

1

|B|

ż

B|fpxq ´ b|dx,

where the maximum is taken over all ball balls containing x. (Theequivalence is verified later)

We thus know that a function f P L1locpRnq is in BMO if and only

if f 7 P L8.

If you have learnt something about the property of sharp maximaloperator, we will find another motivation to regard BMO as analternative limit of Lp as pÑ8 since f 7Lp is comparable tofLp for all 1 ă p ă 1, while fBMO is comparable to f 78.

We omit the details on sharp maximal function. We add this sectionfor those interests in harmonic analysis.

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BMO and L8

The BMO space is closely related to L8 in different ways:

It is easy to find that all bounded functions defined in Rn are alsoBMO functions. But the opposite is not right: log |x| is inBMOpRq, but sgnpxq log |x| is not.

Furthermore, we have the ”norm”-estimate: fBMO ď 2fL8 byobserving that |f ´ fQ|Q ď 2|f |Q ď 2fL8 .

Most transparent fact: the norm for BMO scales and translates inthe same way as for L8.

Similarity of value: John-Nirenburg inequality, which furtherasserts that any BMO function is nearly in L8. The result isnon-trivial, see Garnett-Jones’ paper for detailed discussion.

Similarity of function space: We will may the Sobolev embeddingtheorem for p “ nk, k P p0, nq by using Poincare’s inequality onlywhen k “ 1, while for other cases, we shall use Riesz potential orH1 ´ BMO duality. We omit the proof here.

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Proof of equivalence between BMO and sharp maximalfunction

For any b, we have

infbPC

1

|B|

ż

B

|fpxq ´ b|dx ď1

|B|

ż

B

|fpxq ´ fB |dx

ď1

|B|

ż

B

|fpxq ´ b|dx`1

|B|

ż

B

|b´ fB |dx

ď2

|B|

ż

B

|fpxq ´ b|dx;

The opposite side is given by taking inf over C.An optimal choice of b improves upon fB by at most a factor of two.

Note that b denotes an arbitrary complex number which is permitted todepend on B.

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One Motivation for J-N’s inequality

f : real-valued function on Rn such that fBMO “ 1.

We have |f ´ fB|B ď 1.

By Markov’s inequality, we have

|tx P B : |fpxq ´ fB| ě λu| ď |B|λ

for any λ ą 0.

Nontrivial for λ ą 1, f only exceeds its average by say 10 on atmost 110 of any ball/cube.

Iterate this fact to give far better estimates as λÑ8.

Important principle in harmonic analysis: bad behavior on a smallexceptional set can often be iterated away if we know that theexceptional sets are small at every scale.

f only exceeds its average by 20 on at most 110 of 110 of a givenball (i.e. on at most 1100 of the ball), f exceeds its average by 30on at most 0.1%, and so forth, leading to a bound which tails offexponentially in λ rather than polynomially.

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John-Nirenberg’s inequality

John-Nirenberg (1962)

For function f P BMOpRnq with fBMO ‰ 0, there exists constantsC1, C2 ą 0 depending only on dimension n, such that for any cube Q inRn, we have

| tx P Q : |f ´ fQ| ě λu | ď C1 exp

ˆ

´C2λ

fBMO

˙

|Q|

for any λ ą 0.

Here we note that in their original paper, they actually proved theversion for aQ, a number depending on Q instead of fQ, however theproof keeps almost the same expect some minor changing. We state thetheorem using fQ for more common applications.

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Sufficient condition for BMO

Before we come the proof of John-Nirenberg’s inequality, we first proveits converse which is much easier to verify inspired from simplecalculation. We can also use it to verify that log |x| is in BMOpp0, 1qnq :

Proposition

Let u : QÑ R be a measurable function such that for someb ą 0, B ě 0 and any cube C ⊂ Q, there exists a constant aC P R, suchthat

|tx P C : |u´ aC | ą σu| ď Be´bσ|C|,@σ ě 0,

then u P BMOpQq.

The proof of this proposition is a simple step of integration,

1

2

ż

C|u´ uC |dx ď

ż

C|u´ aC |dx

“

ż 8

0| tx P C : |upxq ´ aC | ě σu |dσ ď

B

b|C|.

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Some consequences

BMO implies locally Lp

If p P r1,8q and u P BMOpQq, then we haveˆ

1

|Q||u´ uQ|

p

˙1p

ď Cn,puBMO.

In particular, u P LplocpRnq, and the supremum of LHS can approximateuBMO.

This is a direct result from the integration:

1

|Q|

ż

Q

|u´ uQ|pdx “ p

ż 8

0

|tx P Q : |upxq ´ uQ| ą su| sp´1ds

ď p

ż 8

0

C1e´C2suBMOsp´1ds

and change of variables by setting λ :“ C2suBMO :

1

|Q|

ż

Q

|u´ uQ|pdx ď C1p

ˆ

uBMO

C2

˙p ż 8

0

λp´1e´λdλ

“ C1pC´p2 ΓppqupBMO. 15 / 36

Exponential Integrability of BMO functions

The exponential integrability of a function is defined over any compactsubset K ⊂ Rn in the sense that

ş

K exppc|gpxq|qdx ă 8 for any c ă n.For BMO function, the exponential integrability is guaranteed since

ż

Qexppc|u´ uQ|qdx “ c

ż 8

0ect |tx P Q : |upxq ´ uQ| ą tu| dt

ď cC1

ż 8

0expppc´ c2qtqdt “

cC1

C2 ´ c,

where we have assumed uBMO “ 1 and |Q| “ 1. We thus haveclaimed:

Exponential integrability

For any c ă C2 there exists Kpc, C1, C2q such that

1

|Q|

ż

Qexppc|u´ uQ|uBMOqdx ď Kpc, C1, C2q

for BMO function u.

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log |x| is a BMO function

Now we use the proposition to check the previously unverified result,that log |x| is in BMOpp0, 1qnq.

Fix a cube C, with h the length of the side of C.

Define ξ “ maxxPC

|x|, η “ minxPC

|x|, aC “ log ξ. Then aC ´ u ě 0.

Removing the trivial part, we assume that ξ ě ηeσ. Then we have

ξe´σ ě η ě ξ ´ diampCq ě ξ ´?nh, and

ξ ď

?nh

1´ e´σ.

1hn |tx P C : |u´ aC | ě σu| ď 1

hnBξe´σ ďp?nqnωn

p1´e´σqne´nσ.

For σ ą 1 and σ ď 1 we have the proposition holds with b “ n and

B “ max

en, p?nqnωnp1´ e

´1q´n(

.

From this, we also know that for ξ P L1, the function pξ ˚ logqpxq isalso in BMO space.

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Distance in BMO to L8

Garnett-Jones 1978, Ann.Math

Let A0 be the smallest constant for BMO function u that satisfy:

supQ

1

|Q||tx P Q : |upxq ´ uQ| ą λu| ď e´λA0 .

Then there are constants C1, C2 depending on n such that for A ą A0,we have u “ g ` f, where fBMO ď C1A0 and gL8 ď C2 max pA, λq .

The converse of this result is almost trivial knowing John-Nirenberg’sinequality.

Not knowing Garnett-Jones’s result, you can also get some intuitionthat BMO should not be much larger than L8. We can also prove thatfor BMO function f, for every ε ą 0 we have

ż

Rn

|fpxq|

p1` |x|qn`εdx ă 8.

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Idea Behind the Proof: Discrete Logarithm

We re-visit the idea for motivating J-N’s inequality:

Consider two cubes Q2 ⊂ Q1 ⊂ Rn with lpQ2q “ lpQ1q2, then we have

|fQ1´ fQ2

| ď|Q1|

|Q2|¨

1

|Q1|

ż

|fpyq ´ fQ1|dy

ď 2fBMO.

Iterate this operation to pn` 1q cubes Qn ⊂ Qn´1 ⊂ ¨ ¨ ¨ ⊂ Q0. We have|fQn ´ fQ0 | ď n ¨ 2nfBMO.

The difference of scaling cause an exponential decay of the ”boundedmean oscillation measure”:

Magnitudes of the corresponding supports: decrease geometrically inorder 2´n.Average: increase arithmetically in order n.

Consider discrete logarithm as a concrete example:gpxq “

ř8

n“0 anχr0,2´nspxq, x P p0, 1q with an ě 1 for n ě 0.

g is in BMO iff an “ Op1q and gpxq „ | logpxq| for 0 ă x ă 12.

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Key ingredient: Calderon-Zygmund decomposition

We use a very useful decomposition in harmonic analysis, invented byCalderon and Zygmund in their study of singular integral operators.Intuitively saying, it says all integrable functions can be decomposed into agood part, where the function is bounded by a small number, and a bad part,where the function can be large, but locally has average value zero; and wehave a guarantee that the bad part is supported on a relatively small set:

C-Z decomposition

Given a function f which is integrable and non-negative, and given a positivenumber λ, there exists a sequence tQju of disjoint dyadic cubes such that

1 (good part) fpxq ď λ for almost every x R ∪jQj ;2 (cube requirement) |∪jQj | ď 1

λf1;

3 (bad part) λ ă fQjď 2nλ.

The proof of this can be seen in any standard reference on harmonic analysis.We omit it here.

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Proof of J-N’s inequality

WLOG, we may assume that fBMO “ 1, so we have

1

|Q|

ż

Q|fpxq ´ fQ|dx ď 1.

Form the C-Z decomposition of pf ´ fQq at height 2, which gives afamily of disjoint cubes tQ1,ju such that

|f ´ fQ| ď 2

for x R ∪jQ1,j and

2 ă1

|Q1,j |

ż

Q1,j

|fpxq ´ fQ|dx ď 2n`1.

In particular, we haveÿ

j

|Q1,j | ď1

2

ż

Q|fpxq ´ fQ|dx ď

1

2|Q|,

and

|fQ1,j´fQ | “

ˇ

ˇ

ˇ

ˇ

ˇ

1

|Q1,j |

ż

Q1,j

pfpxq ´ fQqdx

ˇ

ˇ

ˇ

ˇ

ˇ

ď 2n`1.21 / 36

Proof of J-N’s inequality

Form the C-Z decomposition of pf ´ fQ1,j q at height 2 on eachcube Q1,j , (again we may assume that the pf ´ fQ1,j q ď 1)

We obtain a family of cubes tQ1,j,ku satisfies the following:1 |fpxq ´ fQ1,j | ď 2 for x P Q1,jzp∪kQ1,j,kq;2

ř

k |Q1,j,k| ď12 |Q1,j |;

3 |fQ1,j,k´ fQ1,j

| ď 2n`1.

Gather the cubes Q1,j,k corresponding to all Q1,j ’s and collectivelycalled tQ2,ju.

Now, we haveÿ

j

|Q2,j | ď1

4|Q|

and for x R ∪jQ2,j ,

|fpxq ´ fQ| ď |fpxq ´ fQ1,j | ` |fQ1,j ´ fQ| ď 2` 2n`1 ď 2 ¨ 2n`1.

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Proof of J-N’s inequality

Repeat the process infinitely:

For each N , we have a family of disjoint cubes tQN,ju, such that

|fpxq ´ fQ| ď N ¨ 2N`1

andÿ

j

|QN,j | ď1

2N|Q|

for x R ∪jQN,j .

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Proof of J-N’s inequality

Fix λ ě 2n`1, and choose N such that N2n`1 ď λ ă pN ` 1qn`1.

|tx P Q : |fpxq ´ fQ| ě λu| ďÿ

j

|QN,j | ď1

2N|Q|

“ e´N log 2|Q| ď e´C2λ|Q|.

(pick C2 “ log 22n`2)

If λ ă 2n`1, then C2λ ă log?

2. Hence,

|tx P Q : |fpxq ´ fQ| ě λu| ď |Q| ď expplog?

2´ C2λq|Q|

“?

2 expp´C2λq|Q|

so we can take C1 “?

2.

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Some Remarks

Here we should note that the constants in these estimates are notoptimal, for example, we can change the height for C-Zdecomposition to derive improvements for previous result. Theoptimal constant of C2 “ 2e is achieved by using rearrangementargument which we will not state here. For C1, the problemremains open while the main direction using rearrangementfunction keeps the same, where we can take C1 “ e4e2. See [Ana]for more discussion on such approach.

There are many alternative ways to prove John-Nirenberg’sinequality, for example, we can reduce this problem to a directconsequence of BMO´H1 duality by using the property for H1

function and Riesz’s representation theorem, see Stein’s harmonicanalysis for the proof.

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BMO and Lp

First we note that the John-Nirenburg’s inequality can be extendedconsidering the Lp norms, so that the case of BMO maps corresponds to thelimit as pÑ8 :

Corollary

For any p P r1,8q and u P LppQq we define

Kpp puq :“ sup

#

ÿ

i

|Qi|

ˆ

1

|Qi||upxq ´ uQi |dx

˙p

, Q “ ∪iQi

+

.

Then there exists a constant c “ cpp, nq such that

u´ uQLwpď cpp, nqKppuq,

where ¨ Lwp

refers to the weak Lp norm.

The proof is basically the same as John-Nirenberg’s inequality, the goal beingto prove the polynomial decay:

|tx P Q : |upxq ´ uQ| ą tu| ďcpp, nq

tpKppuq, t ą 0

instead of an exponential decay.26 / 36

BMO and Lp

We still invoke iteration on C-Z decomposition of u´ uQj at level

2jµ where µ is given according to µ ě 1|Q|1p.

After kth step we invoke the C-Z decomposition for the functionpu´ uQk´1

q1Qk´1at level 22pk´1qµ and we get a collection of open,

disjoint sub-cubes tQk,ju of Qk´1 so that |upxq| ď 22k´1µ a.e. inQk´1z ∪j Qk,j and

ÿ

all j’s

|Qk,j | ď

´

ş

Q |upxq|dx¯pk´1qp1

φpµ, kq(1)

where

φpu, kq “ p22pk´1qµqp22pk´2qµq1p1

p22pk´3qµq1p12¨ ¨ ¨µ1p

1k´1.

The proof reduces to the estimate for φ by property of series andtaking µ “ 2|Q|1p. We omit the details here.

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BMO and Lp

As a substitute for L8, we would like to be able to use it ininterpolation theory. This is indeed possible, as the followingtheorem shows:

Stanpacchia’s interpolation

Let T be a linear operator bounded on Lp0pRnq for some 1 ď p0 ă 8and bounded from L8 Ñ BMO. Then T is bounded on LppRnq for anyp0 ă p ă 8.

The proof also comes from Calderon-Zygmund decomposition andcomparison of Hardy-Littlewood maximal function and sharpmaximal function. We omit the proof here.

After knowing BMO´H1 duality, we can derive an analog for H1

space by using Marcinkiewicz’s interpolation theorem.

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An Unexpected Application

An important problem in classical analysis is to find smoothnesscondition for the existence of a continuous derivative. Weiss andZygmund proved a statement of giving some conditions:

Weiss-Zygmund smoothness condition

Let f be 2π-periodic and continuous. Suppose for some β ą 12, thatfpx` tq ` fpx´ tq ´ 2fpxq “ O

`

t| log t|β˘

uniformly in x as tÑ 0`.Then f is absolutely continuous and its derivative f 1 is in Lpr0, 2πs forevery p ą 1.

The original proof of this result is fairly short but utilizes a numberof deep results including a theorem of Littlewood and Paley.

The John-Nirenberg inequality approaches this result moredirectly, more general and as well shows that the derivative f 1 isnot merely in each Lp space but also of BMO.

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A New Proof

The proof relies on the following result:

Lemma

Let u P L1pC0q, where C0 is a finite cube in Rn. Assume that there is aconstant K and a constant β ą 12 such that if C1 and C2 are any twoequal subcubes having a full pn´ 1q-dimensional face in common, then|uC1 ´ uC2 | ď Kp1` | log h|βq, where h is the common side length.Then u is a BMO function.

The proof of using this lemma is direct:

WLOG, assume F is a smooth function, then we only need toestimate F 1Lp .

For n “ 1, we have the quantity as in the lemma for F 1. ” f.

The proof is complete by giving the John-Nirenberg inequality foru.

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Sketch of Proof

Sub-divide each edge of a sub-cube C into 2N equal parts to get2nN equal cubes tCru1ďrď2nN .

1

|C|

ż

C|u´ uC |dx “ lim

NÑ82´nN

ÿ

r

|ur ´ uC |.

The proof reduces to claim 2´nNř

r |ur ´ uC | ď κpβ,K, nq.

By Cauchy-Schwartz’s inequality,

2´nNř

r |ur ´ uC | ď“

2´nNř

r |ur ´ uC |2‰12

:“ a12N .

The proof reduces to show aN ’s are uniformly bounded so that

aN`1 ď aN `´

nK1`| log h|β

¯2, where h “ K2N`1.

Since a0 “ 0, aN`1 ď n2K2ř8j“1p1` |j log 2´ log k|βq´2 ď κ2 for

some constant χ independent of K. The convergence holds sinceβ12.

The remaining proof relies on the property of series andsubdivision of subcubes to derive iteration results.

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Some more applications of John-Nirenberg’s inequality

Before we end the presentation, let us introduce some other applications ofJohn-Nirenberg inequality not discussed in the presentation:

Back to the initial question on nonlinear elastic boundary value problem,John provides an attractive answer in 1972 by proving the uniqueness ofnonlinear elastic equilibria when the boundary displacement is fixed andonly deformations with uniformly small strain are considered. This isdone using John-Nirenberg’s inequality.

The John-Nirenberg’s inequality acts an important rule in the study forthe supersolution of elliptic PDE of divergence form, deriving generallyweak Harnack’s inequality by Moser, which resulting in the Holderregularity of the solution. The proof is greatly simplified by using J-Ninequality. Some generalization can further be considered. See Moser’spaper or [Han] for a systematic discussion.

If we know something about harmonic analysis, by T p1q theorem thereare some singular integral operator related to BMO which can beextended to L2. John-Nirenberg’s inequality acts an important rule inthe proof of T p1q theorem.

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Duality for BMO Space

Similar to the failure of preservance of L8 space for singularintegral operator, a similar problem still happens for L1. (Hilberttransform of an integrable function is not in general integrable)

We define a subspace of L1 whose image under any singularintegral is in L1 : H1 space:

Space Atomic H1

The H1 space is defined by the set of linear combination of specialatoms aj , with coefficients λj are complex and have finite sum ofcountable addition.The atom a1js are defined as complex-valued functions defined on Rnwhich is supported on a cube Q and is such that

ş

Q a “ 0 and

a8 ď 1|Q|.

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Duality for BMO Space

The norm of H1 is defined by

fH1 “ inf

#

ÿ

j

|λj | : f “ÿ

λjaj

+

.

So the H1 is a Banach space.

We can also check that singular integrals are bounded from H1 to L1.Just similar to the case for L8 to BMO.

A natural thinking is to connect H1 and BMO since they are thesubstitutes of L1 and L8 respectively and they have similar properties asthe dual to each other:

Fefferman-Stein, 1971

There exists constants 0 ă c ď C ă 8 such that

cfBMO ď supg:gH1ď1

ˇ

ˇ

ˇ

ˇ

ż

fpxqgpxqdx

ˇ

ˇ

ˇ

ˇ

ď CfBMO

and every bounded linear functional on H1 is of the above type.

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Useful References

Fritz John, Louis Nirenberg. On functions of bounded meanoscillation. Communications on pure and applied Mathematics14.3 (1961): 415-426

Anatolii Korenovskii. Mean Oscillations and EquimeasurableRearrangements of Functions.

Javier Duoandikoetxea. Fourier Analysis.

Loukas Grafakos. Modern Fourier Analysis. Third Edition,Graduate Texts in Mathematics 250, Springer.

Alberto Tochinsky. Real-Variable Methods in Harmonic Analysis.

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Useful References

Luigi Ambrosio. Lectures on Elliptic Partial DifferentialEquations. Publications of the Scuola Normale Superiore bookseries (PSNS, volume 18), Springer.

Qing Han and Fanghua Lin. Elliptic partial differential equations.Vol. 1. American Mathematical Soc., 2011.

Stein, E. M (2016). Harmonic Analysis (PMS-43), Volume 43:Real-Variable Methods, Orthogonality, and OscillatoryIntegrals.(PMS-43). Princeton University Press.

John B. Garnett and Peter W. Jones. The Distance in BMO toL8. Annals of Mathematics, Vol.108, No.2 (Sep 1978), pp. 373-393

Helge Holden and Ragni Piene. The Abel Prize 2013-2017,Springer.

Brian Thomson. Symmetric Properties of Real Functions.

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