Carleson measures, trees, extrapolation, and T(b) theorems

Publ. Ma . 46 (2002), 257–325
CARLESON MEASURES, TREES, EXTRAPOLATION,
AND T(b) THEOREMS
P. Ausche , S. Ho mann, C. Muscalu, T. Tao and C. Thiele
Abs ac
The heo y o Ca leson measu es, s opping ime a gumen s, and
a omic decomposi ions has been well-es ablished in ha monic
analysis. Mo e ecen is he heo y o phase space analysis om
he poin o iew o wa e packe s on iles, ee selec ion algo-
i hms, and ee size es ima es. The pu pose o his pape is o
demons a e ha he wo heo ies a e in ac closely ela ed, by
aking exis ing esul s and ep o ing hem in a unified se ing. In
pa icula we gi e a dyadic e sion o ex apola ion o Ca leson
measu es, as well as a wo-sided local dyadic T(b) heo em which
gene alizes ea lie T(b) heo ems o Da id, Jou n´e, Semmes, and
Ch is .
Con en s
1. In oduc ion 258
2. No a ion 261
2.1. Tiles and ees 261
2.2. Size and Ca leson measu es 266
2.3. Wa ele s, phase space p ojec ions, and BMO 266
2.4. Mean 268
3. The “L∞ heo y” 269
3.1. Chopping big ees in o li le ees, and ex apola ion o
Ca leson measu es 273
3.2. An al e na e a gumen 281
4. The “Lp heo y” 284
4.1. Example: A omic decomposi ion o dyadic Hp287
5. The Ca leson embedding heo em and pa ap oduc s 290
5.1. Weak- ype es ima es 294
6. Calde ´on-Zygmund ope a o s, and T(b) heo ems 298
6.1. Acc e i i y and one-sided T(b) heo ems 301
6.2. Adap ed Haa bases, and wo-sided T(b) heo ems 304
Re e ences 320
2000 Ma hema ics Subjec Classifica ion. 42B20, 42B25.
Key wo ds. Haa wa ele s, BMO, acc e i i y, T(b) heo ems, ex apola ion, Cal-
de ´on-Zygmund heo y.
258 Ausche , Ho mann, Muscalu, Tao, Thiele
1. In oduc ion
The pu pose o his a icle is o demons a e he close connec ion
be ween wo se s o echniques in ha monic analysis: he heo y o Ca -
leson measu es and ela ed objec s, and he heo y o ees and ela ed
objec s.
A Ca leson measu e is a posi i e measu e µon he uppe hal
space such ha µ(I×(0,(I))) |I| o e e y cube I⊆Rnwi h side
leng h (I). The e is also an analogous no ion o domains mo e gene al
han he hal -space, as well as a disc e e e sion: i µis a mapping om
dyadic cubes in o he non-nega i e eals, hen µsa isfies a (disc e e)
Ca leson measu e condi ion I⊆JµI|J| o e e y dyadic cube J,
whe e he sum uns o e all dyadic sub-cubes o J. Ca leson measu es
a e in ima ely connec ed wi h many aspec s o ha monic analysis, in-
cluding non- angen ial beha io o unc ions in he hal -space (o in a
domain) (see e.g. [51]), Hp heo y and BMO [27], boundedness o sin-
gula in eg als, squa e unc ions and maximal unc ions (e.g. [12], [22],
[35], [50], [37], [38]), geome ic measu e heo y (e.g. [24], [35], [36]),
and PDE (e.g. [3], [28], [32], [42]). Mo eo e , ia hei connec ion wi h
he heo y o ees, Ca leson measu es ha e played a significan ole in
ecen wo k on Bilinea Singula In eg als [39], [40], [45], [46], [53],
and ( a he app op ia ely!) Ca leson’s heo em on a.e. con e gence o
Fou ie Se ies [25], [41]. In hese la e connec ions i is mo e con enien
o wo k in he phase plane han in he Ca leson hal -space, and we ha e
delibe a ely chosen ou no a ion o eflec his ac .
This a icle is mainly exposi o y. Apa om one main new esul
(a local T(b) heo em), we shall mos ly ake exis ing esul s (a omic
decomposi ions, pa ap oduc es ima es, Ca leson embedding) and e-
p o e hem in a amewo k which unifies bo h he Ca leson measu e
heo y and he heo y o ees and iles. (As such he e is some o e lap
wi h he ecen lec u e no es in [49].)
Since his is an exposi o y a icle, we shall simpli y ma e s and only
wo k in one dimension R. Also, we shall mos ly wo k in he dyadic
se ing ins ead o he con inuous one, o a oid issues such as apidly
dec easing ails o use o he Vi ali co e ing lemma. Thus, ou esul s will
be ph ased using dyadic in e als and he Haa basis ins ead o a bi a y
in e als and Gaussians (o simila smoo h ke nels). Howe e mos o ou
esul s ha e con inuous analogues (see e.g. [29] o a compa ison be ween
dyadic and con inuous ha monic analysis). We also will unca e all ou
spaces o be fini e-dimensional o a oid echnicali ies.
T ees, Ex apola ion, T(b)259
The pape is o ganized as ollows. A e se ing up he no a ion
o dyadic Ca leson measu es and BMO, Haa wa ele s, and iles and
ees, we will gi e a quick e iew o he s anda d “L∞” heo y o BMO
(i.e. measu ing he ways in which BMO is close o L∞), bu om he
pe spec i e o ees and iles. As pa o his L∞ heo y, we gi e a
ees-based p oo o he (dyadic analogue o he) ex apola ion lemma
o Ca leson measu es de eloped ecen ly in [3], [32], [42]. We also gi e
an al e na e p oo o he ex apola ion lemma due o John B. Ga ne .
We hen show how BMO is also use ul in “Lp” con ex s, mainly
h ough a BMO e sion o he Calde ´on-Zygmund decomposi ion. This
ype o lemma is used o en in he ecen wo k on Ca leson’s heo em
and he bilinea Hilbe ans o m, and is implici in ea lie wo k on
Ca leson measu es and simila objec s; we illus a e his by using he
BMO Chebyshe inequali y o e-p o e he s anda d a omic decomposi-
ion o Hp.
Nex , we p o e he Ca leson embedding heo em and gi e i s usual
applica ions o pa ap oduc es ima es and he T(1) heo em. We also
gi e a sho p oo o he boundedness o pa ap oduc s below L1; he
p oo is mo e di ec han ea lie p oo s in ha one does no go explici ly
h ough he T(1) heo em.
Finally, we conside Calde ´on-Zygmund ope a o s. We p o e a wo-
sided local T(b) heo em which gene alizes he exis ing local and global
T(b) heo ems [23], [4], [11], [50]; o ins ance, we can p o e he
s anda d global T(b) heo em assuming ha bis only in BMO a he
han L∞.
The T(b) Theo em, in i s a ious guises, has i s oo s in a ques ion
posed by Y es Meye , who asked whe he he T(1) Theo em o Da id
and Jou n´e[22] (see also Sec ion 6 below) emains ue i he cons an
unc ion 1 is eplaced by some unc ion b∈L∞wi h Re b≥δ(such ba e
said o be “acc e i e”). The ques ion was mo i a ed by i s applicabili y
o he L2boundedness o he Cauchy in eg al ope a o on a Lipschi z
g aph. Indeed, i Γ deno es he g aph, in he plane, o a eal- alued
Lipschi z unc ion A, hen by Cauchy’s heo em, we ha e ha in he
sense o BMO ( ha is, modulo cons an s),
0 = p. . Γ
1
z−wdw,
o z∈Γ. Bu in g aph co-o dina es, his amoun s o saying ha (again
in he sense o BMO),
0 = p. . ∞
−∞
1+iA(y)
x−y+i(A(x)−A(y)) dy ≡T(b)(x),
260 Ausche , Ho mann, Muscalu, Tao, Thiele
whe e bis he acc e i e unc ion 1 + iA, and T is he singula in eg al
ope a o na u ally associa ed o he an isymme ic Calde ´on-Zygmund
ke nel K(x, y)=(x−y+i(A(x)−A(y)))−1.
The L2boundedness o T, and hence also ha o he Cauchy in eg al
ope a o
CΓ (x)≡p. . Γ
(w)
z−wdw,
hus ollows om an analogue o he T(1) Theo em in which he condi ion
T(1), T∗(1) ∈BMO is eplaced by he condi ion T(b)=0=T
∗(b), o
some acc e i e unc ion b. Jus such a esul was p o ed by McIn osh
and Meye [43], who consequen ly ob ained an al e na i e p oo o hei
ea lie join esul wi h Coi man [13] conce ning he L2boundedness o
he ope a o CΓ.
The “T(b) Theo em” o [43] was gene alized by Da id, Jou n´e and
Semmes [23] o allow T(b),T∗(b)∈BMO (indeed, hey allowed o he
gene aliza ions as well, o example ha he e could be wo diffe en
acc e i e unc ions b1,b2such ha T(b1),T∗(b2)∈BMO, and mo eo e
ha he poin wise acc e i i y condi ion could be elaxed o a condi ion
holding on a ious so s o a e ages —see, e.g., he no ion o “pseudo-
acc e i i y” defined in Sec ion 6.1 below).
This led o a p oo o he T(b) Theo em by cons uc ing Haa wa ele s
adap ed o he unc ion b[20] (we shall base ou p oo on a a ia ion o
hese adap ed Haa wa ele s).
A e y simple p oo o a “one-sided e sion” o he T(b) Theo em was
ob ained by Semmes [50], who obse ed ha in he special case T(b)∈
BMO, T∗(1) = 0, one can eadily show ha T(1) ∈BMO, hus educing
ma e s o he T(1) Theo em. I is wo h no ing ha a sui able adap a-
ion o Semmes’s a gumen is applicable o he solu ion o he squa e oo
p oblem o Ka o. Indeed, one o he p esen au ho s (Ausche ), along
wi h Tchami chian [5], o mula ed a e sion o he T(b) Theo em whose
p oo was based upon he a gumen o [50], and which was subsequen ly
used o sol e he Ka o p oblem in highe dimensions [1], [31], [2]. We
u he no e ha he e a e local e sions o he T(b) Theo em, due o
M. Ch is [11] (c . Theo em 6.8 below), which also ha e in e es ing ap-
plica ions, namely o ques ions o analy ic capaci y; see o ins ance [54]
o u he discussion.
This wo k was conduc ed a Uni e si y o Missou i, Uni e si y o Cal-
i o nia a Los Angeles (UCLA), and he Cen e o Ma hema ics and i s
Applica ions (CMA) a he Aus alian Na ional Uni e si y (ANU). The
T ees, Ex apola ion, T(b)261
au ho s a e pa icula ly g a e ul o CMA o hei wa m hospi ali y du -
ing he isi o h ee o us (P. Ausche , C. Thiele, T. Tao). S. Ho mann
is suppo ed by NSF g an DMS 0088920. C. Muscalu is suppo ed by
NSF g an DMS 0100796. T. Tao is a Clay P ize Fellow and is sup-
po ed by a g an om he Packa d Founda ion. C. Thiele is suppo ed
by a Sloan ellowship and NSF g an s DMS 9985572 and DMS 9970469.
The au ho s a e indeb ed o S ephanie Molna , John B. Ga ne , Joan
Ve de a and he e e ee o many help ul co ec ions and commen s.
2. No a ion
We use AB o deno e he es ima e A≤CB o some absolu e
cons an Cwhich may a y om line o line.
I Eis a se , we use |E| o deno e he Lebesgue measu e o E.We
will always be igno ing se s o measu e ze o, hus we only conside wo
se s E,F o be in e sec ing i |E∩F|>0.
Al hough ou unc ions may be complex alued, we shall use he eal
inne p oduc
 ,g:=  (x)g(x)dx
h oughou .
2.1. Tiles and ees. We shall be wo king wi h dyadic in e als
h oughou he pape . The numbe o dyadic in e als is infini e, bu
o simpli y he a gumen s we shall es ic ou sel es o a fini e se on
he hal -line; in applica ions, his es ic ion can always be emo ed by a
s anda d ansla ion and limi ing a gumen . Specifically, we fix a la ge
in ege M>0; none o ou es ima es will depend on M. We define
dyadic in e al o be an in e al1o he o m I=[j2k,(j+ 1)2k], whe e
j,ka e in ege s such ha −M≤k≤Mand I⊆[0,2M]. Le Ideno e
he se o all dyadic in e als; obse e ha Iis fini e. All sums and
unions in ol ing Io Jwill be assumed o be o e Iunless o he wise
specified. I is a unc ion on R, we define [ ]I:= 1
|I|I o deno e
he mean o on I. We use 2I o deno e he pa en 2o I, and Il,I o
deno e he le and igh child en o I( hese a e undefined i |I|=2
Mo
|I|=2
−M espec i ely). We e e o he in e als Iland I as siblings.
1We will be ca eless abou whe he ou in e als a e closed, hal -open, o open because
o ou con en ion o igno ing se s o measu e ze o.
2On he non-dyadic heo y 2Iis o en used o deno e he in e al wi h he same
cen e as Ibu wice he leng h; his can be hough o as a non-dyadic e sion o
he pa en o I. Howe e , in his pape we use 2I o exclusi ely e e o he dyadic
pa en o I, i.e. he unique dyadic in e al o wice he leng h which con ains I.

262 Ausche , Ho mann, Muscalu, Tao, Thiele
Since ou dyadic in e als ha e been es ic ed o a fini e se , all no ms
will au oma ically be fini e and all s opping ime p ocesses will au oma -
ically e mina e. This allows us o a oid some mino echnicali ies in ou
a gumen s, al hough i also means ha we occasionally ha e o ea he
smalles scale |I|=2
−Mo he la ges scale |I|=2
Ma li le diffe en ly
om all he o he scales.
A majo ad an age o he dyadic se ing is he nes ing p ope y:i
I,Ja e dyadic in e als which in e sec each o he , hen ei he I⊆Jo
J⊆I. In pa icula , o any collec ion o dyadic in e als, he maximal
in e als in his collec ion will always be disjoin .
The heo y o Ca leson measu es is usually se in he uppe hal -space
R2
+:= {(x, ):x∈R, ∈R+}.
Ac ually, because o ou unca ion pa ame e Mwe will wo k in he
compac subse
(R2
+)M:= {(x, ):x∈[0,2M], ∈[2−M,2M−1]}.
The a iable x ep esen s spa ial posi ion, while he a iable ep esen s
ime, wa eleng h, o spa ial scale. Fo e e y dyadic in e al I∈I,
we le l(I)=|I|deno e he side-leng h o I, and define he Ca leson
box Q(I)⊂(R2
+)Mby
Q(I):=I×[2−M,l(I)]
and he Whi ney box Q+(I)⊂Q(I)by
Q+(I):=I×l(I)
2,l(I).
We ema k ha we ha e he pa i ion Q(I)=J:J⊆IQ+(J).
Meanwhile, he heo y o ees and iles is usually se in phase space
R2:= {(x, ξ):x∈R,ξ∈R}.
Because o ou unca ion, and because we a e in he dyadic se ing, we
will ins ead wo k in he egion3
(R2)M:= {(x, ξ):x∈[0,2M],ξ∈[0,2M]}.
The a iable x ep esen s spa ial posi ion, while ξ ep esen s equency.
AHeisenbe g ile o simply ile is a ec angle in R2o he o m P:= IP×
ωP, whe e IPand ωPa e dyadic in e als such ha |P|=|IP||ωP|=1.
I Pand Qa e iles, we say ha P≤Qi Pin e sec s Qand IP⊆IQ.
This is a pa ial o de on iles.
3In u h, we a e wo king no wi h he Euclidean field R, bu wi h he Walsh
field R+≡(Z2)Z. See e.g. [52].
T ees, Ex apola ion, T(b)263
10 x
ξ
x
10
0
2
0
1
1
Figu e 1. The geome y o he Ca leson hal -plane
(pa i ioned in o Whi ney boxes) and phase space (pa -
i ioned in o non-lacuna y iles). The heu is ic =1/ξ
p o ides a one- o-one co espondence be ween he wo
pa i ions.
I Iis a dyadic in e al, we define he lacuna y ile P+(I)by
P+(I):=I×1
l(I),2
l(I)
and he non-lacuna y ile P0(I)by
P0(I):=I×0,1
l(I).
Le P+deno e he se o all lacuna y iles, and P0 he se o all non-
lacuna y iles. We define P+(I)≤P+(J) i and only i P0(I)≤P0(J),
o equi alen ly i I⊆J.Thus≤is a pa ial o de ing on P+. O cou se,
he e a e many iles which a e no in ei he o hese wo se s, and many
esul s in his pape can be ex ended o gene al iles. Howe e , o
simplici y we shall mos ly es ic ou sel es o he lacuna y and non-
lacuna y iles. We w i e [ ]Pas sho hand o he a e ages [ ]IP.
I P+(I) is a lacuna y ile, we define he pa en 2P+(I)o P+(I)by
2P+(I):=P+(2I). Simila ly define 2P0(I):=P0(2I).
264 Ausche , Ho mann, Muscalu, Tao, Thiele
Alacuna y4 ee (hence o h abb e ia ed as ee) is a collec ion T⊆
P+o lacuna y iles wi h a op ile PT∈T, such ha P≤PT o all
P∈T. We use ITas sho -hand o IPT.I P∈P+, we define he
comple e ee T ee(P) o be he ee
T ee(P):={Q∈P+:Q≤P}
wi h op P. We some imes w i e T ee(I) o T ee(P+(I)). No e ha
e e y ee Tlies inside a comple e ee T ee(PT). I Tis a ee inside
a collec ion Po iles, we say ha Tis comple e wi h espec o Pi
T= T ee(PT)∩P.
Le α>0 and Tbe a ee. We define an α-packing o T o be a
se P⊂To iles such ha

P∈P
|IP|≤α|IT|.
We say ha Pis a uni o m α-packing5o Ti

P∈P:IP⊂J
|IP|≤α|J|
o all dyadic in e als J.
I α<1/2 and Pis an α-packing o T, obse e ha he pa en
iles 2P:= {2P:P∈P} o m a 2α-packing o T. Simila ly i Pis a
uni o m α-packing o T, hen 2Pis a uni o m 2α-packing o T.
We say ha a collec ion o lacuna y iles Pis con ex i o e e y pai
o iles P1≤P2in P, he se {P∈P+:P1≤P≤P2}is also con ained
in P. We will usually be dealing wi h con ex ees in his pape .
The co espondence be ween he uppe hal -space and phase space is
gi en by he heu is ic o mula6
=1/ξ;(1)
in o he wo ds, equency is he ecip ocal o wa eleng h. This co espon-
dence iden ifies Whi ney boxes Q+(I) wi h lacuna y iles P+(I), and
iden ifies a Ca leson box Q(I) wi h he comple e ee T ee(I).
4Non-lacuna y ees T⊆P0a e also use ul in he s udy o he bilinea Hilbe
ans o m and Ca leson’s ope a o ; mo e p ecisely, when ea ing he bilinea Hilbe
ans o m B( ,g),hone uses a iple o ees associa ed o ,g,h espec i ely, wi h
wo o he ees lacuna y and he hi d non-lacuna y (bu possibly wi h a non-ze o e-
quency o igin). Simila ly when ea ing he Ca leson ope a o CN(x) ,χEone uses
a pai o ees associa ed o and χE espec i ely, wi h one lacuna y and one non-
lacuna y. See [40], [41]. Howe e we will no use non-lacuna y ees explici ly in his
pape , al hough hey appea implici ly in Lemma 4.2 and in he pa ap oduc heo y.
5This is oughly equi alen o P∈PχIPha ing a BMO no m bounded by α.
6To be comple ely p ecise, one would ha e o adjus his o mula when ξ≤21−M,
bu as his is only a heu is ic anyway we will no bo he o do his.
T ees, Ex apola ion, T(b)265
(Incomple e ees Ta e iden ified wi h he po ion o a Ca leson box
abo e a “dyadic Lipschi z g aph”, c . [3].) No e how his co espondence
clea ly gi es a p i ileged posi ion o he equency o igin ξ=0.
The hesis o his pape is ha he heo y o Ca leson measu es can
be equa ed wi h he heo y o lacuna y iles. The heo y o gene al iles
—which is needed o applica ions such as Ca leson’s heo em and he
bilinea Hilbe ans o m, in which he equency o igin plays no dis-
inguished ole— can hen be hough o as a gene aliza ion o Ca leson
measu e heo y7.
Tiles and ees Ca leson measu es
Phase space R2Uppe hal -plane R2
+
Lacuna y ile P+(I) Whi ney box Q+(I)
Non-lacuna y ile P0(I) “Towe ” I×[l(I),∞)
Comple e ee T ee(I) Ca leson box Q(I)
Con ex ee Ca leson box abo e a Lipschi z g aph
Size µsize(T)No malized mass µ(Q(I))/|I|
Bounded maximal size Ca leson measu e (o BMO unc ion)
ξ1/
Table 1. A pa ial dic iona y be ween ee e minol-
ogy, and Ca leson measu e e minology. In ou pape
he wo iewpoin s a e essen ially equi alen , howe e
he phase space iewpoin is be e adap ed o handle
mo e gene al si ua ions whe e one needs o modula e
in equency. Con e sely, Ca leson measu es a e be e
adap ed o complex analysis applica ions.
7In ou pape , we will only need iles which a e cen e ed a o nea he equency
o igin, in which case i does no pa icula ly ma e whe he we use he Ca leson
hal -plane o he phase plane. Howe e , we ha e chosen o use phase space no a ion
(using equency ξins ead o wa eleng h ) as his is mo e compa ible wi h he mo e
gene al heo y o mul ilinea ope a o s such as he bilinea Hilbe ans o m (o he
Ca leson maximal ope a o ), which a e in a ian unde ansla ions o he equency
a iable. No e ha he modula ion ope a ion → e2πiξ0x can be ep esen ed easily
in he phase plane as a ansla ion by ξ0in he ξ a iable, bu is no so elegan ly
ep esen able in he Ca leson hal -plane. Ne e heless, we will no need o modula e
in equency in his pape , so he Ca leson iewpoin and he phase space iewpoin
a e essen ially equi alen he e.
272 Ausche , Ho mann, Muscalu, Tao, Thiele
i suffices o find a collec ion To ees in Twhose ops a e a 1
2-packing
o Tsuch ha
|W |2size(T T∈TT)1.(8)
Fi s obse e om (4) ha i J⊂Iand x∈J hen
ΠT ee(I) (x)−[ΠT ee(I) ]J=Π
T ee(J) (x)
so om hypo hesis we ha e
J
|ΠT ee(I) −[ΠT ee(I) ]J|p|J|.(9)
Now le Cpbe a la ge cons an o be chosen la e . Le Qdeno e he
iles Q∈T ee(I) such ha |[ΠT ee(I) ]Q|≥Cpand ha Qis maximal
wi h espec o ≤.I Cpis sufficien ly la ge, we see om (9)
IQ
|ΠT ee(I) |pCp
p|IQ|
and hence ha Qis a 1
4-packing o T. In pa icula he collec ion 2Q=
{2Q:Q∈Q}o pa en s o iles in Qis a 1
2-packing o T.
We se T:= {T ee(2Q):2Q∈2Q}, and define
F:= ΠT T∈TT =Π
T ee(I) −
Q∈Q
ΠT ee(2Q) .
Then we can ew i e he le -hand side o (8) as 1
|I|F2
2.
By cons uc ion we see ha Fis suppo ed on I. Since [ΠT ee(2Q) ]2Q=
0, we see ha Fis cons an on each I2Qand ha
FL∞(2Q)=|[F]2Q|=|[ΠT ee(I) ]2Q|Cp.
I x∈Iis no in any o he I2Q, hen
|F(x)|=|ΠT ee(I) (x)|Cp.
Thus we ha e F∞Cp. Combining his wi h he p e ious we ob-
ain (8) as desi ed.
We now gi e he well-known con e se o he abo e lemma:
Lemma 3.4 (John-Ni enbe g inequali y).Le Ibe a dyadic in e al,
and le ∈S0(I)be eal- alued. Then we ha e
 p(1 + p)|I|1/p BMO
o all 0<p<∞and
|{x∈I: (x)>2n BMO}| ≤ 2−n+1|I| o all n∈Z+.
P oo : I suffices o p o e he la e inequali y, as he o me easily ol-
lows.
We p o e he claim by induc ion on n. The claim is clea o n=1.
Now suppose ha n>1 and he claim has al eady been p o en o n−1.

T ees, Ex apola ion, T(b)273
Fix I, . Le Pdeno e hose iles Pin T ee(I) such ha [ ]P>
2 BMO, and such ha Pis maximal wi h espec o ≤. Fo each P
we ha e IP
| |2≥|IP||[ ]P|2≥4|IP| 2
BMO.
On he o he hand, om (6), (7) we ha e
I
| |2≤|I||W |2size(T ee(I)) ≤|I| 2
BMO.
Thus Pis a 1
4-packing o T ee(I), so ha he collec ion 2P={2P:P∈
P}o pa en s o iles in P o m a 1
2-packing o T ee(I).
By cons uc ion we ha e [ ]2P≤2 BMO o all P∈P, and (x)≤
2 BMO o all x∈ P∈PI2P.Thus
{x∈I: (x)>2n BMO}⊆ 
2P∈2P
{x∈I2P: −[ ]2P(x)>2(n−1) BMO}.
The claim hen ollows om he induc i e hypo hesis.
3.1. Chopping big ees in o li le ees, and ex apola ion o
Ca leson measu es. Le a:P+→R+be a unc ion. Suppose we ha e
a con ex lacuna y ee T0wi h a la ge size, le ’s say
asize∗(T0)≤C0.(10)
Le 0 <δ≤C0be a small numbe . An ob ious ques ion o ask is
whe he one can decompose he la ge ee T0in o small ees, each o
which has size less han o equal o δ. This is clea ly impossible, as he
example o a single on ee T0wi h la ge size demons a es. Howe e ,
one can do he nex bes hing:
Theo em 3.5. Wi h he abo e assump ions, we ha e he disjoin pa i-
ion
T0=
T∈T
T∪P(11)
whe e he ees Tin Ta e con ex and sa is y
asize∗(T)≤δ(12)
while he iles P∈Pobey he es ima e
a(P)≤C0|IP|.(13)
Fu he mo e, he iles Pand he ee ops {PT:T∈T}a e bo h uni o m
C(C0,δ)-packings o T0.
274 Ausche , Ho mann, Muscalu, Tao, Thiele
No e ha Lemma 2.1 gi es an easy con e se o he abo e heo em:
i T0can be pa i ioned by (11) wi h he abo e p ope ies hen asize∗(T0)
is bounded (bu by a much la ge cons an han C0). Thus, i one is will-
ing o igno e losses in cons an s, he abo e heo em gi es a comple e
cha ac e iza ion o ees o la ge size in e ms o ees o small size. As
we shall see in his sec ion, his heo em can be applied o gi e ex ap-
ola ion lemma o Ca leson measu es, and seems likely o be use ul in
o he con ex s also.
A con inuous pa ame e e sion o Theo em 3.5 is a leas implici
in [3], whe e, as he e, i is used o p o e he “Ex apola ion Lemma
o Ca leson Measu es” (see Co olla y 3.9 below). The la e , in i s
con inuous pa ame e o m, was hen used o es ablish he “ es ic ed
e sion” o he Ka o squa e oo conjec u e, o L∞pe u ba ions o
eal, symme ic, ellip ic coefficien ma ices. The essen ial idea o he
ex apola ion me hod had p e iously been in oduced by J. L. Lewis
in his wo k wi h M. A. M. Mu ay [42] on he hea equa ion in non-
cylind ical domains, and efined u he by Lewis and one o he p esen
au ho s [32] in hei wo k on pa abolic and ellip ic equa ions. Simila
ideas had also appea ed p e iously in he wo k o Da id and Semmes on
uni o m ec ifiabili y: indeed Theo em 4.5 is e y closely ela ed o he
“Co ona Decomposi ion” o [24].
Roughly speaking, in applica ions o he ex apola ion me hod, he
idea is fi s o show ha some “scale-in a ian es ima e on cubes” (like a
Ca leson measu e es ima e, a BMO es ima e, o a e e se Holde o A∞
es ima e o a weigh ) holds when some con olling Ca leson measu e is
sui ably small in a ce ain sense, which will be made p ecise in he se-
quel. The e m “ex apola ion” e e s o he emo al o he smallness
es ic ion. In ha sense i is analogous o G. Da id’s echnique o boo -
s apping he Lipschi z cons an (see, e.g., [21]), al hough i is no clea
whe he he e exis s an explici connec ion be ween he wo me hods.
In [42], [32], o example, he con olling Ca leson measu e was a con-
di ion on ei he he bounda y o he domain, o on he coefficien s o he
ellip ic o pa abolic ope a o , and one p o ed e e se Holde iequali ies
o he associa ed ellip ic-ha monic o pa abolic measu es. In pa icula ,
in [32], he au ho s gi e an al e na i e p oo , ia ex apola ion, o he
main heo em o R. A. Feffe man, C. E. Kenig and J. Piphe [28], in
which he con olling Ca leson measu e is a condi ion on he disag ee-
men be ween he coefficien s o wo ellip ic (o pa abolic) ope a o s, in
he case ha e e se Holde es ima es a e known o hold o he ellip ic-
ha monic measu es associa ed o he fi s ope a o , and one wishes o
p o e such es ima es o he second.
T ees, Ex apola ion, T(b)275
In [3], he au ho s exploi he ac ha p o ing Ka o’s squa e oo es-
ima e is equi alen (by “T(1)” ype easoning) o p o ing ha a ce ain
posi i e measu e in he uppe hal space is Ca leson. He e, he con ol-
ling Ca leson measu e was he one associa ed o he o iginal, sel -adjoin
ope a o , and he ex apola ion echnique was used o p o e ha he
analogous measu e, ela ed o he squa e oo es ima e o he pe u bed
ope a o , was also Ca leson. I is in his se ing ha Co olla y 3.9, o
a he i s con inuous pa ame e analogue, is di ec ly applicable.
The p oo we gi e he e ollows he app oach in [3]. A he end o his
sec ion we gi e an al e na e p oo , due o John B. Ga ne , which gi es
be e dependence on cons an s.
Be o e we gi e he igo ous p oo , we fi s in o mally desc ibe he idea
o he a gumen . Suppose he o iginal ee T0has size asize(T0)=c.
Then 0 ≤c≤C0by (10). To c ea e a ee o maximal size less han δ,
we s a wi h T0and emo e om i some sub- ees o size be ween
c+δ/2 and c−δ/2, which we selec by a s aigh o wa d s opping ime
a gumen . I hen emains o con ol he sub- ees ha we e emo ed.
By sh inking he ees sligh ly (pu ing he e o in o P) we can assume
ha he ees ha e size ei he g ea e han c+δ/2 o less han c−δ/2 (so
ha he ee ha emains mus ha e size a mos δ). We call he fi s
ype o ee “hea y” and he second ype “ligh ”. Because he o iginal
ee had size c, i canno be he case ha IT0is co e ed by hea y sub-
ees, and so a posi i e p opo ion o IT0mus be co e ed by ligh ees o
by no hing. We hen pass o he ligh sub- ees and i e a e his p ocess,
finding a posi i e p opo ion IT0occupied by inc easingly ligh e sub-
ees. A e abou O(C0/δ) s eps, we mus e mina e, finding a posi i e
p opo ion o IT0which a e no co e ed by any u he sub- ees. We
hen pass o he emaining po ion o IT0and all he hea y ees which
ha e un il now been neglec ed, and i e a e once again; since we ha e
eplaced IT0wi h a s ic ly smalle ac ion o IT0, his p ocedu e will
con e ge geome ically o ob ain he desi ed es ima es.
We now p o e Theo em 3.5. We shall d op (13) since i ollows
om (10). In he spi i o Lemma 3.1, i will suffice o p o e he appa -
en ly weake
Theo em 3.6. Wi h he abo e assump ions, we can find a (possibly
emp y) collec ion Ti e a e o disjoin con ex ees in T0whose ops ha e
disjoin spa ial in e als and o m a (1 −η)-packing o T0 o some
η=η(C0,δ)>0, such ha we ha e he disjoin pa i ion
T0=
T∈Ti e a e
T∪
T∈T
T∪P(14)
276 Ausche , Ho mann, Muscalu, Tao, Thiele
whe e he ees T∈Tobey (12), and Pand he ee ops o Ta e bo h
uni o m C(C0,δ)-packings o T0.
Indeed, i Theo em 3.6 holds, hen we can cons uc he collec ion in
Theo em 3.5 by s a ing wi h he pa i ion (14), and hen aking each
o he ees in Ti e a e and b eaking hem up by a u he applica ion
o Theo em 3.6. We con inue on in his way un il he o iginal ee T0
is comple ely b oken up in o ees Tobeying (12) and iles Pobey-
ing (13). The ac ha Pand he ee ops o Ta e C(C0,δ)-packings
o T0 hen ollows om Theo em 3.6 and he ac ha he geome ic
se ies n(1 −η)ncon e ges. A simila a gumen can hen be used o
imp o e “C(C0,δ)-packing” o “uni o m C(C0,δ)-packing”. We omi
he de ails.
P oo o Theo em 3.6: Define he quan i y cby
c:= asize(T0),(15)
hus 0 ≤c≤C0. We shall p o e he heo em by induc ion on c. Speci -
ically, we fix 0 ≤c≤C0and assume ha he heo em has al eady been
p o en in he case asize(T0)≤c−δ/2. No e ha we only ha e o apply
his induc ion a fini e numbe o imes (abou O(C0/δ)) so we will be
allowed o le he cons an s ge wo se wi h each induc ion s ep.
The main lemma used in he p oo o he heo em will be
Lemma 3.7. We can pa i ion
T0=
T∈Tsmall
T∪Pbuffe ∪
T∈Thea y
T∪
T∈Tligh
T(16)
whe e Tsmall is a collec ion o con ex ees which all obey (12) and whose
ee ops a e a uni o m 4-packing o T0,Pbuffe is uni o m 3-packing o
T0, and Thea y,Tligh a e collec ions o disjoin con ex sub- ees o T0
which a e comple e wi h espec o T0, and a e such ha we ha e he ee
coun ing es ima es

T∈Thea y
|IT|+
T∈Tligh
|IT|≤|IT0|(17)

T∈Thea y
|IT|≤ c
c+δ/2|IT0|(18)
and he size bounds
asize(T)≤c−δ/2 o all T∈Tligh .(19)
T ees, Ex apola ion, T(b)277
b
l
h
s
2M
s
bbs
sshhbh h
bbb bb
l hhhhhh hhll
Figu e 3. A con ex ee T0and i s decomposi ion om
Lemma 3.7. The ci cled hand l iles a e he ops o
maximal sub- ees o T0 o which he size o afluc ua es
by a leas δ/2 om c; he unci cled hand l iles a e
he emaining iles in hose maximal sub- ees. Thea y
hus consis s o he ( h ee) h ees while Tligh consis s
o he ( wo) l ees. Pbuffe consis s o hose emaining
iles (labeled b) which lie jus below a hea y o ligh ile,
o a e a he e y op o he phase plane. The emaining
iles (labeled s) o m he ( h ee) small ees Tsmall.
P oo : Define Tfluc ua e o be hose sub- ees To T0which a e comple e
wi h espec o T0, such ha |asize(T)−c|≥δ/2, and such ha Tis
maximal wi h espec o se inclusion and he abo e wo p ope ies. No e
ha such ees a e au oma ically con ex.
By cons uc ion, none o he ees in Tfluc ua e con ain he op ile PT0.
We may subdi ide11
Tfluc ua e =Thea y ∪Tligh
11Wi h e e ence o Figu e 3, Tfluc ua e consis s o he hand l ees, T1consis s o
he sand b iles, and T2consis s o jus he s iles.

278 Ausche , Ho mann, Muscalu, Tao, Thiele
whe e Thea y consis s o hose ees T∈Tfluc ua e wi h
asize(T)≥c+δ/2(20)
and Tligh consis s o hose ees T∈Tfluc ua e wi h
asize(T)≤c−δ/2.
The ees in Tfluc ua e a e disjoin , con ex, and ha e disjoin spa ial
suppo s, so (17) holds. On he o he hand i one mul iplies (20) by |IT|
and sums o e all T∈Thea y one ob ains
|IT0|c=
P∈T0
a(P)≥
P∈T∈Thea y
T
a(P)≥(c+δ/2) 
T∈Thea y
|IT|.
Di iding by c+δ/2 we ob ain (18).
Le T1deno e he con ex ee T1:= T0 T∈T luc ua e Twi h op PT0.
In o mally, T1 ep esen s he po ion o T0below he fluc ua ing iles.
The ee T1con ains PT0and is hence non-emp y. Le Pbuffe deno e he
iles12
Pbuffe := {P∈T1:P=2Q o some Q∈ T1}∪{P∈T1:|IP|=2
−M}.
In o he wo ds, Pbuffe consis s o hose iles in T1which ouch he uppe
bounda y o T1(which in pa icula may include he iles o minimal
wid h |IP|=2
−M). Since he iles Qin he defini ion o Pbuffe ha e
disjoin spa ial suppo s and |IP|=2|IQ|we see ha {P∈T1:P=
2Q o some Q∈ T1}is a uni o m 2-packing o T1. Since {P∈T1:
|IP|=2
−M}is clea ly a uni o m 1-packing o T1, we hus see ha
Pbuffe is a uni o m 3-packing o T1.
Le T2deno e he (possibly emp y) ee T2:= T1 Pbuffe wi h op PT0.
This ee is no necessa ily con ex, howe e we shall in oke he ollowing
lemma o spli i in o con ex ees.
Lemma 3.8. Le Tbe a con ex ee, and le P⊂Tbe a uni o m α-pack-
ing o T o some α>0. Then T Pcan be pa i ioned in o
T P=
T∈T
T
whe e Tis a collec ion o con ex ees Twhose ops {PT:T∈T} o m
a uni o m (α+1)-packing o T.
12He e we a e aking ad an age o ou decision o wo k in a fini e model, whe e he
iles ha e a minimal wid h 2−M. One can eplica e his a gumen in he infini e
se ing bu one has o ea he po ion o T1which “goes all he way o infini y”
sepa a ely. See [3].
T ees, Ex apola ion, T(b)279
P oo : Le Qdeno e hose dyadic in e als Q⊂ITsuch ha Q∈T P
and 2Q∈ T P. Fo any Q∈Q, we see ha ei he Q=PTo 2Q∈P.
Since Pis a uni o m α-packing, his implies ha Qis a uni o m (α+1)-
packing.
Fo each Q∈Q, define he con ex ee TQwi h op Qby
TQ:= {P∈T P:P≤Q,
and he e does no exis Q∈Qsuch ha P<
Q<Q}.
I we hen se T:= {TQ:Q∈Q}we see ha he lemma ollows.
By Lemma 3.8 we may w i e T2=T∈Tsmall Twhe e he ees in Tsmall
a e dis inc and he ee ops o Tsmall a e a uni o m 4-packing o T.
We now e i y ha each ee T∈Tsmall obeys (12). I suffices o
show ha

P∈T ee(I)∩T
a(P)≤δ|I|(21)
o all I⊆IT0.
Fix I. The idea is o w i e T ee(I)∩Tas he diffe ence o ees, each
o which has size c+O(δ).
We may assume ha P+(I)∈Tsince he claim is i ial o he wise.
We obse e ha
T ee(I)∩T= (T ee(I)∩T0) 
J∈J
(T ee(J)∩T0)
whe e Jconsis s o hose in e als J⊆Isuch ha P+(J)∈ T, and
which a e maximal wi h espec o his p ope y.
The ile P+(I)isinTand hence in T1. By cons uc ion o T1,we
hus ha e

P∈T ee(I)∩T0
a(P)=|I|asize(T ee(I)∩T0)≤|I|(c+δ/2)
(since o he wise T ee(I)∩T0would belong o Thea y, a con adic ion).
Simila ly, o e e y J∈J, he ile P+(J) is con ained in T1(o he wise
P+(2J) would be bo h in Tand in Pbuffe , a con adic ion), so

P∈T ee(J)∩T0
a(P)≥|J|(c−δ/2).
By he cons uc ion o J, he in e als Jin Jpa i ion I, hus

J∈J
P∈T ee(J)∩T0
a(P)≥|I|(c−δ/2).
Sub ac ing his om he p e ious we ob ain (21) as desi ed.
280 Ausche , Ho mann, Muscalu, Tao, Thiele
We apply he abo e lemma and place Pbuffe in o P,Tsmall in o T, and
Thea y in o Ti e a e. Fo he emaining ees Tligh we use he induc ion
hypo hesis, which spli s each o he ees in Tligh in o Ti e a e,T, and P.
All he desi ed conclusions o Theo em 3.6 a e easily e ified excep
pe haps o he claim ha he ops o Ti e a e o m a (1 −η)-packing
o T, o in o he wo ds

T∈Ti e a e
|IT|≤(1 −η)|IT0|.
To p o e his inequali y, no e ha he ees in Thea y con ibu e
T∈Thea y |IT| o he le -hand side, while om he induc ion hypo hesis
he ees in Tligh con ibu e a mos (1−η)T∈Tligh |IT| o some η>0.
The e a e no o he con ibu ions. The claim hen ollows om (17), (18)
( educing he alue o ηas necessa y).
The cons an s C(C0,δ) gi en by his a gumen a e abou (C0/δ)CC0/δ.
This bound can be imp o ed subs an ially; see below.
The ollowing co olla y allows one o use one Ca leson measu e µ o
p o e he Ca leson measu e p ope y o a ela ed measu e µ.I is he
dyadic e sion o an ex apola ion lemma in [3], which in u n is based
on ideas in [42], [32].
Co olla y 3.9 (Ex apola ion o Ca leson measu es).Le µ:P+→R+
ha e bounded maximal size and le δ>0. Le µbe a non-nega i e
measu e on R2
+obeying he “weak Ca leson condi ion”
µ(P)≤C1|IP| o all P∈P+
and such ha µsize(T)≤C2 o all con ex ees Tsuch ha µsize∗(T)≤
δ. Then µalso has bounded maximal size:
µsize∗(P+)≤C(µsize∗(P+),δ)(C1+C2).
P oo : Le T0be any con ex ee. We need o show ha
µsize(T0)≤C(µsize∗(P+),δ)(C1+C2).
By Theo em 3.5, we can pa i ion T0=T∈TT∪Pwhe e µsize∗(T)≤δ
o all T∈T, and

T∈T
|IT|+
P∈P
|IP|≤C(µsize∗(P+),δ)|IT0|.
T ees, Ex apola ion, T(b)281
F om his and assump ions on µwe see ha

P∈T0
µ(P)= 
T∈T
P∈T
µ(P)+ 
P∈P
µ(P)
≤C(µsize∗(P+),δ)C2|IT0|+C(µsize∗(P+),δ)C1|IT0|
and he claim ollows.
As men ioned ea lie , his lemma has applica ions o he Ka o p ob-
lem. In [3], his lemma was used o es ablish a es ic ed e sion o
Ka o’s conjec u e, o pe u ba ions o eal, symme ic coefficien ma i-
ces. In ha case, µwas a Ca leson measu e which con olled he o iginal
ope a o , and µwas he analogous measu e con olling he pe u bed
ope a o . The poin was o es ablish ha µwas also a Ca leson mea-
su e. We ema k ha he ac ha he final bound on µwas linea
in C2was c ucial o his applica ion.
I is possible o elimina e he weak Ca leson condi ion by allowing he
ee measu ed by µ o be a li le la ge han he ee measu ed by µ,
bu we will no pu sue his ype o gene aliza ion he e.
3.2. An al e na e a gumen . In his sec ion we gi e an al e na e
p oo o Theo em 3.5, due o John B. Ga ne (pe sonal communica ion).
The idea o his a gumen is simila o some a gumen s in [8].
Fix T0,a. We fi s obse e ha i suffices o p o e he heo em unde
he addi ional “weak Ca leson” assump ion
a(P)≤δ
2|IP| o all P∈T0.(22)
To see his, suppose ha we a e in he gene al case when (22) need no
hold. We se P o be he se o iles whe e (22) ails:
P:= P∈T0:a(P)≥δ
2|IP|.(23)
F om (10) we see ha Pis a uni o m 2C0/δ-packing o T0(c . Lem-
ma 2.1). By Lemma 3.8 we hus see ha we can spli T0 Pin o a
collec ion o disjoin con ex sub ees o T0, whose ops o m a uni o m
2C0/δ + 1-packing o T0. On each such sub ee (22) holds. Thus i we
apply Theo em 3.5 o each sub- ee and hen combine all he decom-
posi ions, we ob ain he desi ed decomposi ion o he o iginal ee T0
(wi h he cons an s C(C0,δ) wo sened by a ac o o 2C0/δ + 1).
Hence o h we assume (22). Unde his assump ion we will no need P
any mo e, and will se i equal o he emp y se .
288 Ausche , Ho mann, Muscalu, Tao, Thiele
I T ee(I) is a comple e ee, we define a (dyadic) Hpa om on T ee(I)
o be a unc ion a∈S0(I) such ha a2≤|I|1/2−1/p. Equi alen ly,
a∈S0is an Hpa om on T ee(I) i and only i he wa ele ans o m Wa
o ais suppo ed on T ee(I) and |Wa|2size(T ee(I)) ≤|I|−2/p; his is
because o (6).
In his sec ion we show
Theo em 4.3 (Equi alen defini ions o Hp).Le ∈S0and 0<p≤
1. Then he ollowing s a emen s a e equi alen .
(i) S p1.
(ii) ˜
M p1.
(iii) The e exis s a collec ion Io dyadic in e als, and o each I∈
I he e exis s a non-nega i e numbe cIand an Hpa om aIon
T ee(I)such ha =IcIaIand Icp
I1.
P oo : We fi s show ha (iii) implies (i) and (ii). F om he quasi-
iangle inequali y
 +gp
p≤ p
p+gp
p
we see ha i suffices o e i y his on a oms, i.e. o show ha SaIp,
˜
MaIp1 whene e aIis a Hpa om on T ee(I).
Fix I,a. By cons uc ion ˜
MaIand SaIa e suppo ed on I,soby
H¨olde i suffices o show SaI2,˜
MaI2|I|1/2−1/p. Bu his ollows
om he L2no maliza ion o aIand he ac ha S,˜
Ma e bounded
on L2.
I emains o show ha ei he one o (i) o (ii) a e enough o im-
ply (iii). Le be any elemen o S0, hus =P∈P+W (P)φP.
Se a:= |W |2. We apply Lemma 4.1 epea edly, s a ing wi h a
sufficien ly la ge nand se ing Pn:= P+, and hen dec emen ing n
indefini ely. E en ually one ob ains a pa i ion
P+=
n∈Z
T∈Tn
T∪P−∞
whe e he Tna e as in Lemma 4.1, and |W |2size∗(P−∞ )=0. Thus
W anishes on P−∞, and only a fini e numbe o Tna e non-emp y.
We hus ha e =n∈ZT∈TnΠT . I we hen se cIT:= 2n/2|IT|1/p
and aIT:= ΠT /cIT hen we ha e
=
n∈Z
T∈Tn
cITaIT.

T ees, Ex apola ion, T(b)289
By (32) (wi h p= 2) we see ha each aITis an Hpa om. To show (iii)
i hus emains o show ha

n
T∈Tn
cp
IT=
n
2np/2
T∈Tn
|IT|1.
Fi s suppose ha (i) holds. Fo each nand each T∈Tn, we see ha
IT
|SΠT |q|IT|SΠT q
BMO =|IT|ΠT q
BMO ∼2nq/2|IT|
o all 2 <q<∞by he John-Ni enbe g inequali y (Lemma 3.4), (41),
and (31). Also, we ha e
IT
|SΠT |2=IT
|ΠT |2∼2n|IT|.
By H¨olde we hus ha e
IT
|S | ≥IT
|SΠT | 2n /2|IT|,
o any 0 < <p, wi h he implici cons an depending on . This
clea ly implies
x∈IT:|S (x)|2n/2
|S | 2n /2|IT|.
Summing o e all T∈Tnand using he disjoin ness o he ITwe ob ain
|S (x)|2n/2
|S | 2n /2
T∈IT
|IT|.
Mul iplying by 2n(p− )/2and summing o e nwe ob ain

n:|S (x)|2n/2
2n(p− )/2|S | 
n
2np/2
T∈IT
|IT|.
Since he le -hand side is compa able o S p
p, he claim ollows.
Now suppose ins ead ha (ii) holds. By (32), (42) we ha e IT|˜
M | 
2n /2|IT| o all 0 < <p. Now we a gue as wi h S o ob ain (iii)
om (ii).
F om he abo e p oo we see ha he a oms in ac obey he BMO
bound aITBMO |I|−1/p. One can imp o e his BMO con ol o L∞
con ol by epea ing he a gumen in he John-Ni enbe g inequali y
(Lemma 3.4). Namely, one loca es he maximal sub-in e als whe e he
a e ages o aITa e la ge and sepa a es off hose ees, lea ing behind a
290 Ausche , Ho mann, Muscalu, Tao, Thiele
bounded a om. One hen epea s he p ocess un il only bounded a oms
emain, in he spi i o Lemma 3.1. We omi he de ails.
Le be in BMO. F om (7), (6), we ha e
 BMO = sup
T
1
|IT|1/2ΠT 2.
Clea ly i suffices o ake sup ema o e comple e ees T, hus by (4)
 BMO = sup
I
1
|I|1/2I
| −[ ]I|21/2
.
By duali y we hus ha e
 BMO = sup
I
sup
a∈S0(I):a2=1
|I|−1/2| ,a|(43)
o equi alen ly
 BMO = sup{| ,a| :ais a H1a om}.
Thus, as is well known, BMO is he dual o H1.
5. The Ca leson embedding heo em and pa ap oduc s
We now gi e a sligh a ian o he abo e me hod, in which one selec s
ees using he a e ages [ ]Iins ead o he sizes. This ype o a gumen
is o cou se e y old, and he a gumen s he e a e by no means new.
On he o he hand, his ype o ee selec ion me hod is a special case
o he “mean selec ion” algo i hm used ( oge he wi h a size selec ion
algo i hm) in he p oo o Ca leson’s heo em in [41].
We begin wi h
Lemma 5.1 (Ca leson embedding heo em).Le Pbe a collec ion o
lacuna y iles, a:P→R+be a unc ion, and 1<p<∞. Then we
ha e

P∈P
a(P)|[ ]IP|pasize∗(P) p
p
o all locally in eg able unc ions , wi h he implici cons an s depend-
ing on p.
T ees, Ex apola ion, T(b)291
P oo : We apply Lemma 4.2 epea edly, s a ing wi h a sufficien ly
la ge nand dec emen ing n epea edly. This gi es us a pa i ion P=
n∈ZT∈TnT∪P−∞ whe e he Tna e as in Lemma 4.2, and
 mean∗(P−∞)= 0. The con ibu ion o P−∞ is ze o, so i suffices
o con ol

n
T∈Tn
P∈T
a(P)|[ ]IP|p.
I P∈T∈Tn, hen |[ ]IP|≤ mean(P)≤ mean∗(T)2n. F om
his and (2), (3), (39) we may es ima e he p e ious by

n
T∈Tn
P∈T
a(P)2np

n
T∈Tn
asize∗(P)|IT|2np

n
asize∗(P)2np2−n| |2n
| |
∼asize∗(P)| |p
as desi ed.
We can apply his heo em o a ious linea and bilinea ope a o s.
To do his we shall need some no a ion.
Fo any sequence (aP)P∈P+o eal numbe s, we define he wa ele
mul iplie W−1aPW om S0 o S0by
W−1aPW := 
P∈P+
aPW (P)φP.
Wa ele mul iplie s a e he disc e e analogue o pseudo-diffe en ial ope -
a o s, wi h aPbeing he disc e e analogue o a symbol a(x, ξ). Obse e
ha i aPis bounded, hen W−1aPWis bounded on L2and also bounded
on BMO. One can o cou se ex end he domain W−1aPW om S0 o S,
al hough some o he algeb a p ope ies a e los in doing so (since Wis
no injec i e on S).
292 Ausche , Ho mann, Muscalu, Tao, Thiele
Le ,gbe elemen s o S. We define he “high-low”, “low-high”, and
“high-high” pa ap oduc s14
πhl( ,g):= 
P∈P+
W (P)[g]PφP
πlh( ,g):= 
P∈P+
[ ]PWg(P)φP
πhh( ,g):= 
P∈P+
W (P)Wg(P)χIP
|IP|.
These pa ap oduc s ha e he symme ies
πhh( ,g)h=πhl(g,h) =πlh(h, )g
=πhh(g, )h=πhl( ,h)g=πlh(h, g)
=
P∈P+
W (P)Wg(P)[h]P
(44)
and can be exp essed in e ms o he Li lewood-Paley squa e unc ion S:
πhl( ,g)=S∗(gS ); πlh( ,g)=S∗( Sg); πhh( ,g)=S ·Sg.
When ,gha e mean ze o (i.e. ,g ∈S0), hen he pa ap oduc s de-
compose he poin wise p oduc ope a o :
g =πhl( ,g)+πlh( ,g)+πhh( ,g).(45)
To see his, i suffices by bilinea i y o educe o he case when =φP
and g=φQ o some P,Q ∈P+.I IPand IQa e disjoin hen bo h
sides a e ze o. Thus he e a e only h ee cases: P>
Q,P<
Q, and
P=Q. In hese h ee cases he eade may easily e i y ha g is
equal o he high-low, low-high, o high-high pa ap oduc o and g
espec i ely, and ha he o he wo pa ap oduc s anish.
We obse e ha he high-low and low-high pa ap oduc s can be w i -
en as wa ele mul iplie s:
πhl( ,g)=W−1[g]PW ;πlh( ,g)=W−1[ ]PWg.(46)
14The con inuous coun e pa s would be some hing like (Q )(P g)d
,
(P )(Q g)d
, and (Q )(Q g)d
, whe e Q is as be o e and P is a sui -
able app oxima ion o he iden i y a wid h 1/ , e.g. P := e 2∆. The p ecise
defini ion o a pa ap oduc is no s anda dized, o ins ance πhh is no conside ed a
pa ap oduc in some ex s.
T ees, Ex apola ion, T(b)293
The high-high pa ap oduc canno be w i en in his way, bu we ha e
he use ul ela ionship
πhh(W−1aPW ,g)=πhh( ,W−1aPWg).(47)
One can also w i e pa ap oduc s using bo h lacuna y and non-
lacuna y iles, o ins ance
in πhh( g)h=
Idyadic
|I|−1/2 ,φP+(I)g,φP+(I)h, φP0(I).(48)
The bilinea Hilbe ans o m u ns ou o ha e a simila expansion,
bu wi h he sum anging o e a la ge collec ion o iples o iles han
he ones o pa ap oduc s (specifically, he iles need no be lacuna y
o non-lacuna y, and ange o e a h ee-pa ame e amily a he han a
wo-pa ame e one). See e.g. [39], [40], [52], [45], [53], [46].
F om he Ca leson embedding heo em we ha e pa ap oduc es i-
ma es:
Co olla y 5.2 (L2×BMO →L2pa ap oduc es ima es).We ha e
πhl( ,g)2 2g∞
and
πhh( ,g)2,πlh( ,g)2 2gBMO
o all ,g ∈S.
In o he wo ds, pa ap oduc s map L2×L∞ o L2(jus as he poin wise
p oduc does), and he L∞ ac o can be elaxed o BMO as long as one
only conside s high equencies o a BMO unc ion. No e ha he low
equency po ion o a BMO unc ion is somewha ill-defined since a
BMO unc ion migh only be de e mined up o a cons an .
P oo : The fi s bound ollows om (46) since [g]Pis bounded by g∞.
To p o e he second bound, i suffices by (44) o conside πlh. By o -
hogonali y we ha e
πlh( ,g)2=
P∈P+
|Wg(P)|2|[ ]IP|21/2
.
The claim now ollows om Ca leson embedding (Lemma 5.1).
Lemma 5.3 (BMO ×BMO →BMO pa ap oduc es ima e).We ha e
πhh( ,g)BMO  BMOgBMO
o all ,g ∈S.

294 Ausche , Ho mann, Muscalu, Tao, Thiele
Fo he o he pa ap oduc s πhl,πlh one mus place he “low” ac-
o in L∞ a he han BMO, as in Lemma 5.2; his is again an easy
consequence o (46).
P oo : By Lemma 3.3 wi h p=1 i suffices o show |ΠT ee(I)πhh( ,g)|
|I| o all dyadic in e als I.
Fix I, and expand he le -hand side as

P∈P+
W (P)Wg(P)ΠT ee(I)χIP
|IP|
.
The summand anishes unless P∈T ee(I). Thus we can w i e he abo e
as

ΠT ee(I)
P∈T ee(I)
W (P)Wg(P)χIP
|IP|
.
Since ΠT ee(I)is bounded on L1, we can bound his by

P∈T ee(I)
W (P)Wg(P)χIP
|IP|
.
Pu ing he absolu e alues inside and pe o ming he in eg a ion, we
can bound his by

P∈T ee(I)
|W (P)||Wg(P)|.
The claim hen ollows om Cauchy-Schwa z and (7).
5.1. Weak- ype es ima es. We now show how o use he abo e ma-
chine y o p o e Lp,∞pa ap oduc es ima es, whe e Lp,∞is he weak Lp
(quasi-)no m
 Lp,∞:= sup
λ>0
λ|{x:| (x)|≥λ}|1/p.
We need he ollowing basic cha ac e iza ion o weak Lp o 0 <p<∞:
Lemma 5.4. Le 0<p<∞and A>0. Then he ollowing s a emen s
a e equi alen up o cons an s:
(i)  p,∞A.
(ii) Fo e e y se Ewi h 0<|E|<∞, he e exis s a subse E⊂E
wi h |E|∼|E|and | ,χE| A|E|1/p.
He e pis defined by 1/p+1/p =1(no e ha pcan be nega i e!).
T ees, Ex apola ion, T(b)295
P oo : To see ha (i) implies (ii), se
E:= E {x:| (x)|≥CA|E|−1/p}.
I Cis a sufficien ly la ge cons an , hen (i) implies |E|∼|E|, and he
claim ollows.
To see ha (ii) implies (i), le λ>0 be a bi a y and se E:= {x:
Re( (x)) >λ}. Then by (ii) we ha e
λ|E|∼λ|E|A|E|1/p,
and (i) easily ollows ( eplacing Re by −Re, Im, −Im as necessa y).
When p>1 we can always se E=E, and he abo e lemma hen
eflec s he duali y be ween Lp,∞and Lp,1. Howe e o p≤1 he
eedom o se E o be smalle han Eis necessa y (since need no be
locally in eg able).
A ypical applica ion o Lemma 5.4 is
P oposi ion 5.5 (Lp×Lq→L ,∞pa ap oduc es ima es).We ha e
πhl( ,g) ,∞ pgq
whene e 1<p,q<∞and 1/p +1/q =1/ . Simila ly o πlh,πhh.
No e ha can be less han 1. One can s eng hen he weak L
o s ong L by mul ilinea in e pola ion (see e.g. [6], [45], [33]). The
con inuous e sion o hese dyadic pa ap oduc es ima es can be ound
in, e.g. [14]–[19]; he e sion o <1 was fi s p o en in [30] (wi h
some special cases in [9], [14]). I is possible o ob ain he con inuous
es ima es om he dyadic ones ia a e aging a gumen s, bu we shall
no do so he e.
P oo : We fi s conside πhl. We may no malize  p=gq=1;
we may assume ha and ga e dyadic es unc ions. Le Ebe a
measu able se wi h 0 <|E|<∞. We need o find a se E⊂Ewi h
|E|∼|E|such ha
|πhl( ,g),χ
E| |E|1/ .(49)
By escaling (using he hypo hesis 1/p+1/q =1/ ) we may ake |E|∼1.
We choose Eas
E:= E {x:M| |p(x)+M|g|q(x)≥C}.
I Cis la ge enough, hen |E|∼|E|by he Ha dy-Li lewood maximal
inequali y (see e.g. [51]).
We wish o show (49) wi h |E|∼1. By (44) i suffices o show ha
|P∈PW (P)[g]IPWχE(P)|1 o all con ex collec ions Po iles.
296 Ausche , Ho mann, Muscalu, Tao, Thiele
We may emo e all iles in P o which IP∩E=∅, since WχE
anishes on hese iles. Fo any emaining ile Pwe hen ha e
IP
| |p|IP|(50)
by cons uc ion o E. Simila ly, we ha e
gmean∗(P)= sup
P∈P
[|g|]IP[|g|q]1/q
IP1.
Thus we educe o showing

P∈P
|W (P)||WχE(P)|1.(51)
F om (50) and he Lpboundedness o he Li lewood-Paley squa e unc-
ion (see e.g. [51]) we ha e
IP
|SΠT ee(IP) |pIP
|ΠT ee(IP) |pIP
| |p|IP|,
o all P∈P, hus
IP

P∈T ee(IP)
|W (P)|2χIP(x)
|IP|

p/2
|IP|.
Applying Chebyshe ’s inequali y and Co olla y 3.2 we hus see ha
|W |2size∗(P)1. Also, we ha e
|WχE|2size∗(P)≤χE2
BMO ≤χE2
∞≤1.
Thus we can find an n=O(1) such ha
|W |2size∗(Pn)≤22nand |WχE|2size∗(Pn)≤22np/s,
whe e ha e se Pn:= P, and s>1 is an exponen close o 1 o be chosen
la e .
By a fini e numbe o applica ions o Lemma 4.1 wi h a:= |W |2o
a:= |WχE|2we may pa i ion Pn=T∈TnT∪Pn−1whe e Pn−1is a
con ex collec ion o iles such ha
|W |2size∗(Pn−1)≤22(n−1) and |WχE|2size∗(Pn−1)≤22(n−1)p/s
and Tnis a collec ion o con ex ees wi h disjoin spa ial suppo s such
ha ei he
IT
|ΠT |p2np|IT|
T ees, Ex apola ion, T(b)297
o
IT
|ΠTχE|s2np|IT|
o each T∈Tn. F om (42) we hus ha e
(˜
M )p+(˜
MχE)s2np 
T∈Tn
|IT|;
om ou assump ions on ,χEand he Ha dy-Li lewood maximal
inequali y (see e.g. [51]) we hus ha e T∈Tn|IT|2−np.Wenow
e u n o (51), and es ima e he con ibu ion o he ees in Tnby

T∈Tn
P∈T
|W (P)||WχE(P)|.
We apply Cauchy-Schwa z ollowed by he size con ol on W and WχE
we may bound his by

T∈Tn
|IT||W |21/2
size(T)|WχE|21/2
size(T)

T∈Tn
|IT|2n2np/s 2−np2n2np/s.
We now u n o he con ibu ion o he iles in Pn−1. We may i e -
a e he abo e p ocedu e, decomposing Pn−1in o Tn−1and Pn−2, and
con inue in his ashion un il we a e le wi h a collec ion o iles P−∞
wi h size ze o, which we can disca d. Summing up, we can hus con ol
he le -hand side o (51) by n≤O(1) 2−np2n2np/s. I one chooses s
sufficien ly close o 1, hen his sum con e ges, and we a e done.
A simila a gumen handles πhl. The emaining pa ap oduc πhh
hen ollows om (45) and H¨olde ’s inequali y (which is s ill alid o
<1).
One can modi y he abo e a gumen o ob ain he co esponding es-
ima e o he bilinea Hilbe ans o m (in he Walsh model, a leas );
see e.g. [39], [53], [46]. A difficul y in ha case is ha he iles a e
no longe lacuna y, and one canno gua an ee he spa ial disjoin ness o
he ees Tin Tn. Howe e one can s ill make he ees essen ially dis-
join in phase space, bu hen one can only use L2es ima es o con ol
T∈Tn|IT|ins ead o Lpes ima es. Because o his, he abo e s a egy
only seems o wo k o he bilinea Hilbe ans o m when >2/3; i
appea s ha one needs e y diffe en echniques o handle he emaining
case 1/2< ≤2/3.
304 Ausche , Ho mann, Muscalu, Tao, Thiele
The weak boundedness condi ion (56) can ac ually be emo ed; see
he ema ks a e Theo em 6.8. In o mally, his heo em asse s ha o
p o e he L2boundedness o an ope a o T i ac ually suffices o es ab-
lish boundedness o a single unc ion bP o each in e al IP, p o ided
ha bPis no degene a e (in he sense ha i s mean is la ge) and p o-
ided ha T∗(1) is unde con ol. In he nex sec ion we shall emo e
he condi ion on T∗(1), ob aining a “ wo-sided” e sion o his heo em.
P oo : Again i suffices o show ha T(1) is in BMO. By Lemma 3.1 i
suffices o show ha o e e y comple e ee Twe ha e

P∈T T∈TT
|W(T(1))(P)|2|IT|(64)
o some collec ion To disjoin con ex ees in Twhose ops o m a
(1 −ε)-packing o T o some ε>0.
Fix T. By Lemma 6.5 we can indeed find such a collec ion Twi h
he addi ional p ope y ha bis pseudo-acc e i e on T T∈TT.The
claim (64) hen ollows om he a gumen used o p o e Co olla y 6.4.
The abo e a gumen s do no ex end well o wo-sided si ua ions in
which one con ols T(b1) and T∗(b2) (unless one o b1,b2is close o a
cons an , e.g. in BMO no m). In o de o handle he gene al case we
need adap ed Haa bases, o which we now u n.
6.2. Adap ed Haa bases, and wo-sided T(b) heo ems. Le P
be a collec ion o iles, and le bbe a unc ion which is s ongly pseudo-
acc e i e on P. Fo each P∈P, we define he adap ed Haa wa ele φb
P
(in oduced in [20]; see also [4]) by
φb
P:= |IP|−1/2[b]P
[b]P
χIPl−|IP|−1/2[b]Pl
[b]P
χIP .(65)
Obse e ha his collapses o φPi bis cons an on IP. Fo non-
cons an b,φb
Pis no longe mean ze o, bu one can easily e i y ha φb
P
s ill obeys he weigh ed mean ze o condi ion
bφb
P=0.(66)
As a consequence we ha e he o hogonali y p ope y
φb
Pbφb
Q= 0 o all dis inc P,Q ∈P.(67)

T ees, Ex apola ion, T(b)305
F om (65) we see ha
φb
Pbφb
P=[b]P [b]Pl
[b]P
=2
[b]−1
Pl+[b]−1
P
.(68)
In pa icula , om he s ong pseudo-acc e i i y condi ion (59) we ha e
he bound
φb
Pbφb
P
1.(69)
I is in e es ing ha his bound uses only he s ong pseudo-acc e i i y
o b, and in pa icula does no equi e L∞con ol on b.
Define he dual adap ed Haa wa ele ψb
Pby
ψb
P:= φb
Pb
φb
Pbφb
P
.
By (67), (69) we hus ha e ha ψb
P,φ
b
Q=δPQ whe e δis he K onecke
del a. In pa icula we ha e he ep esen a ion o mula
=
P∈P
Wb (P)ψb
P
(70)
whene e gis in he span o {ψb
P:P∈P}, whe e he adap ed wa ele
coefficien s Wb (P) a e defined by
Wb :=  ,φb
P.
We ha e he ollowing basic o hogonali y p ope y:
Lemma 6.7. Le Tbe a con ex ee, and le bbe a unc ion which is
pseudo-acc e i e on Tand obeys he mean bound
|b|2mean∗(T)1.(71)
Then o any unc ion19 ∈Swe ha e

P∈T
|Wb (P)|21/2
 2.(72)
In ac , he mo e gene al es ima e

P∈T
|Wb(b )(P)|21/2
 2|b|21/2
mean∗(T)
(73)
holds o any ,b∈S.
19I can easily be seen, by aid o (70), ha he es ima e (72) can be e e sed o all
in he span o he φb
P, bu we will no use his.
306 Ausche , Ho mann, Muscalu, Tao, Thiele
P oo : F om (71) and (6) we obse e ha
|Wb|2size∗(T)1.(74)
We fi s p o e (72). F om (65) and he iden i y |IP|−1/2Wb(P)=
[b]Pl−[b]P=[b]P−[b]P , we ob ain he iden i y
Wb (P)=W (P)−Wb(P)
[b]P
[ ]P.
I we eplace Wbby Wb hen (72) ollows om Bessel’s inequali y and
he o hono mali y o he Haa wa ele s φP. By he p e ious iden i y
and he iangle inequali y i hus suffices o show

P∈T
Wb(P)
[b]P
[ ]P
21/2
 2.
We may disca d [b]Pby pseudo-acc e i i y (58). The claim hen ollows
om Ca leson embedding (Lemma 5.1) and (74).
Now we p o e (73). Le Qdeno e he collec ion o iles in P+which
a e child en o iles in T, bu a e no in Ti sel . In o de o ensu e ha
he in e als {IQ:Q∈Q}pa i ion ITwe will allow he iles Q o ha e
spa ial in e als |IQ|=2
−M−1; he pa i ion p ope y hen ollows om
he con exi y o T.
The unc ion b −Q∈Q[b ]QχIQhas mean ze o on e e y in e -
al IQ, and is hus o hogonal o φb
P o e e y P∈T. We may hus eely
eplace b by he a e aged unc ion Q∈Q[b ]QχIQin (73). By (72)
i hus suffices o show ha






Q∈Q
[b ]QχIQ





2
2
 2
2|b|2mean∗(T).
Bu om Cauchy-Schwa z we ha e
|[b ]Q|2|IQ| 2
L2(IQ)|b|2mean(2Q)≤ 2
L2(IQ)|b|2mean∗(T),
and he claim ollows by summing in Q.
We can now gi e ou main esul , namely a dyadic local T(b) heo em.
T ees, Ex apola ion, T(b)307
Theo em 6.8 (Dyadic local T(b) heo em).Le Tbe a pe ec Calde-
´on-Zygmund ope a o , and suppose ha o each P∈P+we can find
unc ions b1
P,b2
Pin S(IP)obeying he no maliza ion
[b1
P]P=[b2
P]P=1(75)
and he bounds
IP
|b1
P|2+|Tb1
P|2+|b2
P|2+|T∗b2
P|2|IP|.(76)
Then Tis bounded on L2.
This heo em is a s onge e sion o he local T(b) heo em in [11]
(bu o he dyadic se ing wi h pe ec cancella ion), which equi ed
L∞con ol in (76) ins ead o L2con ol. (This was gene alized o BMO
con ol and o non-doubling si ua ions in [47].) Also i equi ed he
global T(b) heo em o Da id, Jou n´e and Semmes [23] (which we ins ead
deduce as a co olla y o Theo em 6.8). We make some u he ema ks
a e he p oo o he heo em.
P oo : This p oo is somewha leng hy and so we spli he a gumen in o
se e al s ages.
S ep 0. P elimina y es ima es: We begin wi h a basic lemma which al-
eady shows he impo ance o he no maliza ion (75).
Lemma 6.9 (b1
Pspans S(IP)/S0(IP)).Fo any ile P∈P+and any
∈S(IP), we ha e
 L2(IP) −[ ]PL2(IP)+|IP|−1/2| ,b1
P|.
Simila ly o b2
P.
P oo : Le hbe an a bi a y elemen o S(IP) wi h h2= 1. Then
 ,h= ,h−[h]Pb1
P+[h]P ,b1
P= −[ ]P,h−[h]Pb1
P+[h]P ,b1
P.
By Cauchy-Schwa z and (76) we hus ha e
| ,h|  −[ ]PL2(IP)+|IP|−1/2| ,b1
P|.
Taking sup ema o e all h, he claim ollows.
A use ul applica ion o he abo e lemma is he ollowing con enien
unca ion p ope y o he b1
Pand b2
P(al eady obse ed in [11]):
308 Ausche , Ho mann, Muscalu, Tao, Thiele
Co olla y 6.10. Le P,Qbe lacuna y iles wi h Q≤P. I we ha e
he es ima e
IQ
|Tb1
P|2+|b1
P|2K|IQ|(77)
o some K1, hen we ha e
2IQ
|T(b1
PχIQ)|2K|IQ|.
Simila ly o b2
P(bu wi h T eplaced by T∗).
P oo : By (52) and (77) he po ion o he in eg al on 2IQ IQis accep -
able, so i suffices o bound he in eg al on IQ. F om Cauchy-Schwa z,
(76) and (77) we ha e
|T(b1
PχIQ),b
2
Q|=|b1
PχIQ,T∗b2
Q|≤b1
PL2(IQ)T∗b2
QL2(IQ)K1/2|IQ|.
By Lemma 6.9 i hus suffices o show ha
T(b1
PχIQ)−[T(b1
PχIQ)]QL2(IQ)K1/2|IQ|1/2.
Now obse e ha o e e y h∈S0(IQ)weha e
T(b1
PχIQ),h=b1
PχIQ,T∗h=b1
P,T∗h=Tb1
P,h.
By duali y his implies ha
T(b1
PχIQ)−[T(b1
PχIQ)]Q=T(b1
P)−[T(b1
P)]Q
on IQ. The claim hen ollows om (77).
This co olla y will be use ul in es ima ing he ope a o T when ac ing
on objec s such as ψb1
P
Qwhich can be exp essed as linea combina ions o
unca ed e sions o b1
P. Simila ly when es ima ing T∗on objec s such
as ψb2
P
Q.
S ep 1. O e iew o main a gumen : We now begin he main a gumen .
Le Abe he bes cons an such ha
T∗χIPL1(IP)≤A|IP|
o all iles P∈P+. We claim ha A=O(1); om his and he
co esponding claim o TχIP(which is o cou se symme ic) he heo em
will ollow om he local T(1) heo em (Co olla y 6.3).
In ac we will show
IP
T 
≤((1 −ε)A+O(1))|IP| ∞
(78)
T ees, Ex apola ion, T(b)309
o all iles P∈P+and ∈S(IP), and some 0 <ε1 depending
only on he implici cons an in (76). By duali y his implies ha A≤
(1 −ε)A+O(1), which will p o e he desi ed bound on A.
Fix P, . We shall p o e he es ima e (78) in h ee s eps. Fi s ly
(in S ep 2), we decompose and educe ma e s o p o ing a Ca leson
measu e ype es ima e on he wa ele coefficien s |T∗χIP,ψb1
P
Q|2; his
a gumen shall use s opping- ime a gumen s (which we encapsula e as
Lemma 6.11) based on b1
Pbu no on b2
P. Then (in S ep 3), we decom-
pose χIPand use s opping ime a gumen s (again using Lemma 6.11)
based on b2
Pbu no on b1
P. I will be impo an no o y o handle
b1
Pand b2
Pa he same ime as we will lose he c ucial (1 −ε) packing
p ope y o he ees le ou by he s opping ime algo i hm i we do so.
The pu pose o hese s opping a gumen s is o impose some pseudo-
acc e i i y and o he egula i y p ope ies on he b1
Pand b2
P. Once we
ha e enough egula i y p ope ies, we can hen (in S ep 4) do an ele-
men a y compu a ion o es ima e he wa ele coefficien s |T∗χIP,ψb1
P
Q|
poin wise by he quan i ies which we know o be con olled by hypo hesis
(see (76) below).
S ep 2. P uning he bad iles o b1
P:We now begin he fi s o he
h ee s eps ou lined abo e. We would like o b eak up in o linea
combina ions o he wa ele s ψb1
P
Q, bu we canno do his o all Qbe-
cause we do no con ol he s ong pseudo-acc e i i y o b1
P. Howe e ,
by using Lemma 6.5 and some o he selec ion algo i hms we can find
a la ge sub ee o T ee(P) o which we can decompose as desi ed,
modulo accep able e o s:
Lemma 6.11. Le P∈P+be a ile. Then we can pa i ion
T ee(P)=T1∪Pbuffe ∪
T∈T
T
whe e
•Tis a collec ion o disjoin comple e ees in T ee(P)whose ops
o m a (1 −ε)-packing o T ee(P) o some 0<ε1(depending
only on he implici cons an in (76));
•T1is a ee wi h op Psuch ha b1
Pis s ongly pseudo-acc e i e
on T1(wi h cons an s pe haps depending on ε);
•Pbuffe is a 2-packing o T ee(P),T1∪Pbuffe is con ex, b1
Pis
pseudo-acc e i e on T1∪Pbuffe and we ha e he mean bounds
|b1
P|2+|Tb1
P|2mean∗(T1∪Pbu e )1(79)
(wi h he implici cons an depending on ε).

310 Ausche , Ho mann, Muscalu, Tao, Thiele
•We ha e he decomposi ion
=[ ]Pb1
P+
Q∈T1
Wb1
P (Q)ψb1
P
Q+
T∈T
( χIT−[ ]PTb1
PT)+ 
Q∈Pbu e
ϕQ
(80)
whene e ∈S(IP), whe e he “buffe unc ions” ϕQa e suppo ed
on IQ, ha e mean ze o, and ake he o m
ϕQ=aQb1
PχIQl+a
Qb1
PχIQ +a
Qb1
Ql+a
Qb1
Q
whe e he coefficien s aQ,a
Q,a
Q,a
Qdepend on and he b1
Pand
(when |IQ|=2
−M) obey he bounds
|aQ|+|a
Q|+|a
Q|+|a
Q| ∞.(81)
A simila s a emen holds wi h b1
Pand Tb1
P eplaced by b2
Pand T∗b2
P
(bu he se s T1,Pbuffe and Ta e diffe en hen).
The ee T1 ep esen s he “good” po ion o he ee T ee(P), in
which b1
Pis nei he oo la ge no oo small (so in pa icula he Wb1
P
wa ele sys em is well-beha ed on T1). The buffe iles Pbuffe a e hose
iles immedia ely abo e T1(and a e hus sligh ly less “good”), while he
emaining ees Tha e no good p ope ies a all, excep ha hey only
occupy a mos (1 −ε) o he ee T ee(P). This decomposi ion sha es
many ea u es in common wi h Lemma 3.7 ( o ins ance, he ees Ta e
o med om hose in e als whe e bis oo “hea y” o oo “ligh ”).
In e ms o he phase plane (bu adap ed o he Wb1
Pwa ele sys em
ins ead o he Haa wa ele sys em), one can in e p e he igh -hand
side o (80) as ollows. The fi s e m co esponds o he egion o phase
space below he ee T1. The second e m co esponds o T1i sel . The
hi d e m co esponds o he egion abo e T1∪Pbuffe , while he las
e m is an e o e m co esponding o he egion Pbuffe . In he model
case b1
P=χIP, (80) simplifies o
=[ ]PχIP+Π
T1 +
T∈T
ΠT +
Q∈Pbu e
W (Q)φQ.
P oo : We begin by applying Lemma 6.5 o T ee(P) o find a p elimi-
na y collec ion T0o disjoin con ex ees in T ee(P) such ha he ops
o T0a ea(1−2ε)-packing, and such ha b1
Pis pseudo-acc e i e (wi h
cons an s depending on ε) on he ee
T2:= T ee(P) 
T∈T0
T.
T ees, Ex apola ion, T(b)311
Howe e we do no ye ha e (79). To ob ain hese bounds we le Q
deno e he se o all iles Q∈T2 o which
|b1
P|2+|Tb1
P|2mean(Q)≥C/ε
and which a e maximal wi h espec o ≤. I he cons an Cis chosen
la ge enough, hen Qis a ε-packing o T ee(P). Thus i we define
T:= T0∪
Q∈Q
(T ee(Q)∩T2)
hen we see ha (79) holds on he ee
T3:= T ee(P) 
T∈T
T,
while he ops o Ta e s ill a (1 −ε)-packing o T ee(P).
We now pe o m one mino modifica ion o T o make Tsibling-
ee. I Tcon ains wo ees whose ops PT,PT a e siblings, we can
conca ena e hese ees and add a new ile 2PT=2PT o join hese
ees o a la ge ee wi hou affec ing he (1 −ε)-packing na u e o he
ee ops. Repea ing his p ocess as o en as necessa y (i mus e mina e
since T ee(P) only has a fini e numbe o iles) we can make Tsibling-
ee.
Fo simila easons we may assume ha he ees20 in Ta e comple e,
since we can always eplace an incomple e ee by he comple ion o ha
ee, abso bing any sub- ees ha we e also in Ti necessa y.
We now define21 Pbuffe o be he se o iles Qin T3such ha one
o bo h22 o he child en Ql,Q o Qa e no in T3. Since T3is a con ex
ee, he child en o Qwho a e no in T3mus ha e disjoin spa ial
suppo s as Q a ies in Pbuffe . This implies ha Pbuffe is a 2-packing.
We now se T1:= T3 Pbuffe . No e ha all child en o iles in T1lie in
T1∪Pbuffe so ha b1
Pis s ongly pseudo-acc e i e on T1, bu is me ely
pseudo-acc e i e on T1∪Pbuffe .
20Al e na i ely, we could a oid hese modifica ions by combining he s opping ime
a gumen he e wi h he one in Lemma 6.5.
21The algo i hm he e is ex emely simila o he one used o p o e Theo em 3.5.
Indeed, one can e en e-use Figu e 3. The ees T0a e he “ligh ” ees whe e b1
P
has oo small a mean; he iles Qco espond o he ci cled “hea y” iles, whe e b1
P
o Tb1
Phas oo la ge an L2no m. The iles Pbuffe a e hus he buffe iles, which
a e he ones jus below he hea y o ligh iles, as well as he iles a he e y fines
scale.
22O cou se, because we made Tsibling- ee, he only way bo h he child en o Q ail
o be in T3is i Qis a he fines scale, i.e. i |IQ|=2
−M.
312 Ausche , Ho mann, Muscalu, Tao, Thiele
The only p ope y le o e i y is he decomposi ion (80). I is
a cons an mul iple o b1
P hen only he fi s e m is non-ze o ( hanks
o (66)) and he claim is easily e ified. By sub ac ing off a cons an
mul iple we may hus assume ha has mean ze o on IP.
I will suffice o p o e he iden i y assuming ha
[b1
P]Q,[b1
P]Ql,[b1
P]Q = 0 o all Q∈T ee(P),
since he gene al case hen ollows by an ob ious limi ing a gumen ( he
bounds (81) will no depend quan i a i ely on he abo e condi ion). In
his case (70) applies23. Compa ing his wi h (80) and using he mean
ze o condi ion, we educe o showing ha

Q∈Pbu e
Wb1
P (Q)ψb1
P
Q+
T∈T
Q∈T
Wb1
P (Q)ψb1
P
Q
=
T∈T
( χIT−[ ]PTb1
PT)+ 
Q∈Pbu e
ϕQ
o sui able ϕQ.
Le Q∈Pbuffe . Fi s suppose ha nei he child o Qis a op o a ee
in T(since T ee(P) is comple e, his can only happen when |IQ|=2
−M).
In his case we simply se ϕQ:= Wb1
P (Q)ψb1
P
Q.
Now suppose ha one child o Qis a op o a ee Tin T; wi hou
loss o gene ali y we assume Qlis such a op. Since Tis sibling- ee, Q
is no a op and mus he e o e lie in T1∪Pbuffe . In pa icula we ha e
he lowe bounds
|[b1
P]Q |,|[b1
P]Q|1.(82)
We do no ha e good lowe bounds on |[b1
P]Ql|, bu o una ely we can
ake ad an age o some “wiggle oom” in he buffe , and a oid using his
in ou compu a ions by exploi ing he iden i y
Wb1
P (Q)ψb1
P
Q+
Q∈T
Wb1
P (Q)ψb1
P
Q
=
Q∈T ee(Q)
Wb1
P (Q)ψb1
P
Q−
Q∈T ee(Q )
Wb1
P (Q)ψb1
P
Q.
23One can easily e i y (ei he by a dimension coun ing a gumen , o by induc-
i ely wo king om he fines scale upwa ds) ha he wa ele s ψb1
P
Q o Q∈T ee(P)
span S0(P).
T ees, Ex apola ion, T(b)313
The unc ion Q∈T ee(Q)Wb1
P (Q)ψb1
P
Qclea ly is suppo ed on IQ
and has mean ze o, while
−
Q∈T ee(Q)
Wb1
P (Q)ψb1
P
Q=
Q∈T ee(P) T ee(Q)
Wb1
P (Q)ψb1
P
Q
is a cons an mul iple o b1
Pon IT. Thus we ha e

Q∈T ee(Q)
Wb1
P (Q)ψb1
P
Q= χIQ−[ ]Q
[b1
P]Q
b1
PχIQ.
Simila ly we ha e

Q∈T ee(Q )
Wb1
P (Q)ψb1
P
Q= χIQ −[ ]Q
[b1
P]Q
b1
PχIQ .
Sub ac ing he wo we hus see ha
Wb1
P (Q)ψb1
P
Q+
Q∈T
Wb1
P (Q)ψb1
P
Q= χIT−[ ]Q
[b1
P]Q
b1
PχIQ+[ ]Q
[b1
P]Q
b1
PχIQ .
I we hus define
ϕQ:= [ ]PTb1
PT−[ ]Q
[b1
P]Q
b1
PχIQ+[ ]Q
[b1
P]Q
b1
PχIQ
we see ha (81) ollows; he mean ze o condi ion can be seen by (75)
and inspec ion. This comple es he p oo o Lemma 6.11.
Roughly speaking, he abo e lemma s a es ha we can find a la ge
ee T1on which b1
Pis pseudo-acc e i e, on which b1
Pand Tb1
Pa e effec-
i ely bounded, and o which we ha e a ep esen a ion o he o m (70).
We now un an a gumen in he spi i o Lemma 3.1 o localize ma e s
exclusi ely o his ee T1.
We apply he abo e lemma fi s wi h he b1
P. We decompose us-
ing (80), hus es ima ing he le -hand side o (78) by he sum o he
e m below T1
IP
[ ]PTb1
P
,(83)
he e ms coming om T1

Q∈T1
Wb1
P (Q)IP
Tψb1
P
Q
,(84)
320 Ausche , Ho mann, Muscalu, Tao, Thiele
No ice ha b1,b2a e only assumed o be in BMO27 a he han L∞.
This gene aliza ion o he s anda d T(b) heo em appea s o be
new. Also obse e ha he abo e a gumen also wo ks in he special
case T(b1)=T
∗(b2) = 0 i we d op he pa a-acc e i i y and BMO hy-
po heses on b1,b2and ins ead impose he e e se H¨olde condi ions
(1
|IP|IP|b1|2)1/2
|[b1]P|,(1
|IP|IP|b2|2)1/2
|[b2]P|1
on b1,b2. I is in ac likely ha we can ob ain a T(b)- ype heo em o
a bi a y (complex) dyadic A∞weigh s b1,b2(see [28]), bu we will no
a emp o gi e he mos gene al s a emen s he e.
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P. Ausche :
LAMFA, CNRS UMR 6140
Facul ´edeMa h´ema iques e d’In o ma ique
Uni e si ´e de Pica die-Jules Ve ne
80039 Amiens CEDEX 1
F ance
E-mail add ess:[email p o ec ed]
S. Ho mann:
Ma hema ics Depa men
Uni e si y o Missou i
Columbia MO, 65211
U.S.A.
E-mail add ess:[email p o ec ed]
C. Muscalu:
Depa men o Ma hema ics
UCLA
Los Angeles CA 90095-1555
U.S.A.
E-mail add ess:[email p o ec ed]
T. Tao:
Depa men o Ma hema ics
UCLA
Los Angeles CA 90095-1555
U.S.A.
E-mail add ess:[email p o ec ed]
C. Thiele:
Depa men o Ma hema ics
UCLA
Los Angeles CA 90095-1555
U.S.A.
E-mail add ess:[email p o ec ed]
T ees, Ex apola ion, T(b)325
P ime a e si´o ebuda el 20 de se emb e de 2001,
da e a e si´o ebuda el 25 de juny de 2002.