Po en ial analysis, 19 (2003), 1-33.
Po en ial ope a o s, maximal unc ions, and
gene aliza ions o A∞
Ca los P´
e ez§and Richa d L. Wheeden
1 In oduc ion
The pu pose o his pape is o p o e wo kinds o weigh ed no m inequali ies o in eg al
ope a o s o po en ial ype and hei associa ed maximal ope a o s in spaces o homogeneous
ype. The i s kind o esul is an ex ension o a esul o Muckenhoup and Wheeden, which
showed ha o A∞weigh s in Euclidean space, a weigh ed Lpno m o he classical Riesz
ac ional in eg al ope a o Iα o a unc ion is equi alen o he same no m o he ac ional
maximal unc ion Mα . Ou ex ension o his esul in ol es ai ly gene al in eg al ope a o s
o po en ial ype and hei associa ed maximal ope a o s in spaces o homogeneous ype, and
classes o weigh s which a e mo e gene al han A∞. The second kind o inequali y ha we will
p o e gi es wo-weigh Lp, Lqno m es ima es o such maximal ope a o s, assuming again an
app op ia e (bu subs an ially weakened) e sion o A∞. An impo an poin he e is ha we
comple ely a oid using he “good-lambda inequali y” echnique o Bu kholde and Gundy.
Finally, we can combine hese wo esul s o ob ain wo-weigh Lp, Lqno m es ima es o
po en ial ope a o s ha imp o e simila ones de i ed by Sawye and Wheeden.
In he usual n-dimensional Euclidean space Rn, i 0 < α < n, le
Iα (x) = ZRn
(y)1
|x−y|n−αdy
§Pa ially suppo ed by DGICYT g an PB940192, Spain
AMS Subjec Classi ica ion 2000: 42B25, 44A15
deno e he Riesz ac ional in eg al o o o de α, and le
Mα (x) = sup
B:x∈B
(B)α−nZB
| (y)|dy
be he co esponding ac ional maximal unc ion o , whe e Bdeno es a Euclidean ball and
(B) is he adius o B. The poin wise inequali y
Mα (x)≤C Iα (x),
wi h Cindependen o xand , ollows easily om he de ini ions. On he o he hand, he
e e se poin wise inequali y is alse, bu i is shown in [MW] ha Iα and Mα a e o en
compa able in no m. To be mo e p ecise, i w∈A∞and 0 < p < ∞, hen i is shown in [MW]
ha ZRn
|Iα (x)|pw(x)dx ≤cZRn
Mα (x)pw(x)dx (1)
wi h cindependen o . He e A∞deno es he collec ion o weigh s won Rnwi h he
p ope y ha he e exis cons an s C, δ > 0 such ha i Eis a Lebesgue measu able subse o
a ball B, hen
w(E)
w(B)≤C|E|
|B|δ
,
whe e w(E) = REw(x)dx and |E|is he Lebesgue measu e o E.
This es ima e has had applica ions in po en ial heo y, such as in he p oo o he
Hedbe g–Wol heo em [AH] conce ning nonlinea po en ials. The co esponding heo em o
po en ial ope a o s o con olu ion ype plays a ole in he p oo o he cha ac e iza ion by
Ke man and Sawye o ace ype inequali ies, which in u n ha e applica ions o eigen alues
es ima es o Sch ¨odinge ope a o s (see [KS]). Inequali y (1) is also ela ed o he ac ha
he posi i e cone o he T iebel-Lizo kin space Fα,q
p,α < 0, is independen o q; see [JPW]
and [AH].
2
An example o he i s kind o esul ha we will p o e is an es ima e simila o (1) bu
wi h Rn eplaced by a space So homogeneous ype, Iα eplaced by a po en ial ope a o
T=TKo he o m
T (x) = T( dµ)(x) = ZS
(y)K(x, y)dµ(y) (2)
whe e µis he unde lying doubling measu e on S(see §3 o he exac de ini ions o a space o
homogeneous ype and a doubling measu e), and wi h Mα eplaced by
Mϕ (x) = Mϕ( dµ)(x) = sup
B:x∈B
ϕ(B)ZB
| (y)|dµ(y) (3)
whe e ϕ(B) = ϕK(B) is a unc ional which ac s on balls and is de ined by
ϕ(B) = sup
x,y∈B
d(x,y)≥c (B)
K(x, y) (4)
o a su icien ly small posi i e geome ic cons an c(see [SW1]). He e d(x, y) deno es he
quasime ic associa ed wi h S. Fo example, in he case o he Riesz po en ial we ha e
K(x, y) = |x−y|α−n,0< α < n, so ha ϕ(B)≈ (B)α−n, and hen Mϕ educes o he
ac ional maximal ope a o Mα. O he examples o ope a o s o he o ms (2) o (3) a ise
om impo an di e en ial ope a o s; see he nex sec ion o mo e de ails.
In ou gene aliza ion o (1), we will assume ha he weigh sa is ies a condi ion ha is
analogous o he A∞condi ion desc ibed abo e bu wi h Lebesgue measu e eplaced by a
no ion o con en , such as Hausdo con en , on he igh -hand side.
An example o he second kind o esul ha we will p o e is he wo-weigh es ima e
ZS
{Mϕ( dµ)w}qdµ1/q
≤CZS
(| | )pdµ1/p
wi h 1 < p ≤q < ∞and Cindependen o . Such es ima es ha e been s udied ex ensi ely,
bu he na u e o he condi ion ha we shall impose on he weigh s is di e en om
elsewhe e. In pa icula , in addi ion o he necessa y condi ion
ϕ(B)ZB
wqdµ1/q ZB
−p0dµ1/p0
≤c,
3
p0=p/(p−1), o all balls B, we shall assume ha −p0sa is ies an app op ia e A∞condi ion
o he con en ype. Fo compa ison pu poses, we no e ha he wo-weigh boundedness o
Mϕis s udied in [PW] unde a di e en kind o s eng hening o he necessa y condi ion, such
as a Fe e man–Phong condi ion o he ype
ϕ(B)µ(B)1/p0ZB
wqdµ1/q 1
µ(B)ZB
− p0dµ1/ p0
≤c
o some > 1. Fo example, weigh s o which jus e e se doubling condi ions a e alid may
sa is y an A∞condi ion o con en ype bu no a Fe e man–Phong condi ion.
As men ioned ea lie , i is possible o combine he wo kinds o esul s ha we will p o e.
In his way, we ob ain wo-weigh no m es ima es o po en ial ope a o s T assuming ha
bo h weigh s sa is y con en condi ions.
2 S a emen s o he main esul s
Following [SW1], we conside po en ial ope a o s o he o m
T( dσ)(x) = ZS
(y)K(x, y)dσ(y),(5)
whe e Sis a space o homogeneous ype wi h unde lying doubling measu e µ, and σis any
Bo el measu e on S. This de ini ion ag ees wi h (2) in case σ=µ. The exac de ini ion o a
space o homogeneous ype is gi en is §3; by a doubling measu e, we mean a Bo el measu e µ
wi h he p ope y ha he e is a cons an Csuch ha o e e y “ball” B⊂ S,
µ(2B)≤Cµ(B).
As usual, 2Bdeno es he ball wi h he same cen e as Bbu wice he adius. I d(x, y) is he
co esponding quasime ic in S, we will always assume ha he ke nel K(x, y) is nonnega i e
4
and sa is ies he ollowing g ow h condi ions: he e exis cons an s C1, C2s ic ly la ge han
1 such ha
K(x, y)≤C1K(x0, y) i d(x0, y)≤C2d(x, y),(6)
K(x, y)≤C1K(x, y0) i d(x, y0)≤C2d(x, y).
The main classical examples o such ope a o s a e he Riesz in eg als Iα men ioned in he
in oduc ion. An impo an class o examples o me ics o he han he usual Euclidean
me ic consis s o po en ial ope a o s ela ed o he egula i y o subellip ic di e en ial
equa ions. In pa icula , ec o ields o H¨o mande ype ([H]) as well as he classes o
nonsmoo h ec o ields s udied in [FL] and [SW2] lead o in eg al ope a o s o he ype we
will s udy. In addi ion, he di e en ial ope a o s o G ushin ype conside ed in [FGuW] (a
leas in he simples case o Lebesgue measu e) a e ela ed o in eg als o ype (2). In ac , o
all hese examples he associa ed po en ial ope a o has he o m
T (x) = ZS
(y)d(x, y)
µ(B(x, d(x, y))) dµ(y) (7)
whe e d(x, y) is a dis ance unc ion ha is na u ally ela ed o he ec o ields and B(x, )
deno es he co esponding ball wi h cen e xand adius .
Associa ed wi h Kis he unc ional ϕ=ϕKde ined in (4) by
ϕ(B) = sup
x,y∈B
d(x,y)≥c (B)
K(x, y)
o a su icien ly small posi i e geome ic cons an c. As men ioned ea lie , ϕ(B)≈ (B)α−n
in he case o he Riesz ac ional in eg al Iα. In he subellip ic case (7), no e ha
ϕ(B)≈ (B)/µ(B).
The condi ions (6) on Klead o use ul g ow h p ope ies o ϕ. I Bis a ball and θ > 0, le
θB deno e he ball concen ic wi h B whose adius is θ (B). I is shown in [SWZ, (4.2) and
5

(4.3)] ha i θ > 1, hen he e is a cons an Cdepending only on θ, C1, C2, he cons an cin
(4) and geome ic p ope ies o Sso ha
ϕ(B)≤Cϕ(θB) o all balls B⊂ S.(8)
Also, o such a cons an C(bu now one ha is independen o θ),
ϕ(B)≤Cϕ(B0) o all pai s o balls B0⊂B. (9)
We shall assume in some o ou esul s ha ϕsa is ies he ollowing addi ional condi ion
o some  > 0:
ϕ(B1)µ(B1)≤C (B1)
(B2)
ϕ(B2)µ(B2) i B1⊂B2.(10)
Fo example, in he case o he ac ional in eg als Iα, we can pick =αin (10); o he
ope a o in (7), we can choose = 1.
Fo any Bo el measu e σ, we de ine he maximal ope a o
Mϕ( dσ)(x) = sup
B:x∈B
ϕ(B)ZB
| |dσ. (11)
No e ha i σ=µ, his de ini ion ag ees wi h (3).
The poin wise es ima e T( dσ)(x)≥cMϕ( dσ)(x) is easy o show by using he
assump ions on K. Ou i s main heo em will show ha he opposi e inequali y o en holds
in no m. Thus he heo em gene alizes (1) o classical ac ional in eg als. In ac , he esul
imp o es (1) no only by ex ending i o spaces o homogeneous ype bu also by allowing a
la ge class o weigh s e en in he usual Euclidean case. In o de o s a e he esul , we i s
de ine a sui able class o measu es in a space o homogeneous ype. The de ini ion is
mo i a ed by a simila one in [SW1] o he usual Euclidean case, and i is mos con enien ly
ph ased in e ms o a g id Dmo dyadic cubes Q. He e mis a la ge nega i e in ege which
indexes he edgeleng hs `(Q) o he smalles cubes Q∈ Dm, namely, he smalles edgeleng hs
6
a e ρm o an app op ia e geome ic cons an ρ > 1, and each cube in he g id has edgeleng h
ρk o some k≥m(see §3 o de ails abou he g id). The cubes in Dmha e he impo an
nes edness p ope y ha i Q1, Q2∈ Dmand Q1∩Q26=∅, hen ei he Q1⊂Q2o Q2⊂Q1.
Also, each cube Q∈ Dmis con ained in a ball B(Q) whose adius is compa able o he
edgeleng h o Q;B(Q) is called he con aining ball o Q.
Gi en an in ege mand a nonnega i e unc ional τ(B) o balls B, we de ine he no ion o
he τcon en o a se E,||E||τ,m, as ollows. I E⊂ S, le
kEkτ,m = in {X
i
τ(B(Qi)) : Qi∈ Dm, E ⊂ ∪Qi}.
Thus kEkτ,m is a so o Hausdo con en o Eassocia ed wi h τand he g id Dm. Typical
choices o τa e τ(B) = (B)β o some β > 0, and also τ(B) = µ(B). The i s choice
co esponds o Hausdo con en o Eo dimension β.
By he nes edness p ope y o dyadic cubes, i E⊂Q∈ Dm hen in he de ini ion o
kEkτ,m we may assume ha all Qi⊂Q. We say ha a measu e ν∈Ady
∞(τ) i he e a e
posi i e cons an s C, δ independen o E, Q and msuch ha
ν(E)
ν(Q)≤CkEkτ,m
τ(B(Q))δ
i E⊂Q∈ Dm,(12)
whe e ν(E) deno es he ν-measu e o E.
In case νis absolu ely con inuous wi h espec o µ, say dν =w dµ, we say ha
w∈Ady
∞(τ) i w dµ ∈Ady
∞(τ). We also use he no a ion w(E) o he w dµ-measu e o E:
w(E) = REw dµ. I dν =w dx on Rn, i is no ha d o see ha when τis chosen so ha
τ(B) = (B)n(≈ |B|), hen w∈Ady
∞(τ) is he same as w∈A∞i wis a doubling weigh .
I we choose τ(B) = (B)β,β > 0, o all balls B, hen he class Ady
∞(τ) was de ined in
[SW1]. In his case, we will use he no a ion Aβ
∞ins ead o Ady
∞(τ), i.e.,
Aβ
∞=Ady
∞(τ) when τ(B) = (B)β.
7
Rela ed no ions based on balls ins ead o dyadic cubes can be ob ained by de ining
Hτ,B(E) = in {X
i
τ(Bi) : E⊂ ∪Bi⊂B}i E⊂B
and eplacing (12) by he assump ion
ν(E)
ν(B)≤CHτ,B(E)
τ(B)δ
i E⊂B. (13)
I νis a doubling measu e, hese wo no ions u n ou o be he same o many unc ionals τ;
see §3 o de ails. Recall ha we always assume ha he unde lying measu e µis a doubling
measu e.
We say ha a Bo el measu e νsa is ies he e e se doubling condi ion o o de β,β > 0, i
he e is a posi i e cons an csuch ha
(14)
We w i e ν∈RDβ o such a measu e ν. Any doubling measu e belongs o RDβ o some
β > 0 by [W, p.269]. We will show in §3 ha i νis a doubling measu e and ν∈RDβ, hen
ν∈Aβ
∞ o he same alue o β. In ac , he assump ion ha νis a doubling measu e is no
needed in o de o conclude ha ν∈Aβ
∞i νsa is ies he ollowing dyadic e e se doubling
condi ion uni o mly in m:
ν(Q2)≥c`(Q2)
`(Q1)β
ν(Q1) i Q1, Q2∈ Dm, Q1⊂Q2.(15)
We w i e ν∈RDdy
β o such ν.
We can now s a e ou i s main heo em. In i , we assume ha τ(B) is a unc ional o
which he e a e cons an s c,  > 0 wi h
a)τ(B1)≤cτ(B2) i B1⊂B2
b)τ(2B)≤cτ(B) o all balls B(16)
c)ϕ(B1)τ(B1)≤c (B1)
(B2)
ϕ(B2)τ(B2) i B1⊂B2.
8
Theo em 2.1 Le σand ωbe Bo el measu es on S,T( dσ)be de ined by (5) o a ke nel
which sa is ies (6), and Mϕ( dσ)be de ined by (11) wi h ϕas in (4). Le τbe a nonnega i e
unc ional which sa is ies (16). I ω∈Ady
∞(τ)and 1≤p < ∞, hen
ZS
|T( dσ)|pdω ≤CZS
(Mϕ( dσ))pdω (17)
wi h Cindependen o . In pa icula , i τsa is ies (16) and w∈Ady
∞(τ), hen
ZS
|T( dµ)|pw dµ ≤CZS
(Mϕ( dµ))pw dµ. (18)
In he impo an special case when τ(B) = (B)β o β > 0, Theo em 2.1 leads o he nex
co olla y assuming ha µsa is ies he doubling condi ion o o de D, i.e., ha
µ(B)≤C (B)
(˜
B)D
µ(˜
B) o all balls ˜
B⊂B.
Any doubling measu e sa is ies he doubling condi ion o o de D o some D > 0.
Co olla y 2.2 Le 1≤p < ∞,µbe a doubling measu e o o de D, and Kbe a ke nel such
ha ϕsa is ies (10) o some  > 0. I ωand σa e Bo el measu es on Sand ω∈Aβ
∞wi h
β+>D, hen ZS
|T( dσ)|pdω ≤CZS
|Mϕ( dσ)|pdω. (19)
In pa icula , (19) holds i ei he ω∈RDdy
βwi h β+ > D, o i ωis a doubling measu e and
ω∈RDβwi h β+ > D.
Fo example, in he usual n-dimensional Euclidean si ua ion wi h dµ =dx (so ha
D=n), Co olla y 2.2 includes he esul (1) om [MW] by choosing dω =w(x)dx and β=n
since A∞⊂An
∞. In ac , Co olla y 2.2 imp o es he esul in (1) by showing ha (1) is alid
i w∈Aβ
∞ o any β > n −α, since he alue o in (10) o he Riesz ope a o Iαis α.
Co olla y 2.2 hus ex ends he esul o [R] showing ha (1) is alid o any doubling weigh
wwhich sa is ies he e e se doubling condi ion o o de β o some β > n −α.
9
which implies (32).
We will use a g id o dyadic se s in Swhich a e “almos balls”, as cons uc ed in [SW1]. In
ac , he ollowing has been p o ed he e:
I ρ= 8κ5, hen o any (la ge nega i e) in ege m, he e a e poin s {xk
j}and a
amily Dm={Ek
j}o se s o k=m, m + 1,· · · and j= 1,2,· · · such ha
•B(xk
j, ρk)⊂ Ek
j⊂B(xk
j, ρk+1)
•Fo each k=m, m + 1,· · · , he amily {Ek
j}is pai wise disjoin in j, and
S=∪jEk
j.
•I m≤k < l, hen ei he Ek
j∩ El
i=∅o Ek
j⊂ El
i.
We call he amily D=∪m∈ZDma dyadic cube decomposi ion o Sand e e o he se s in
Das dyadic cubes. A dyadic cube will usually be deno ed by Q, and Q∗will deno e he
con aining ball desc ibed abo e wi h 1
ρQ∗⊂Q⊂Q∗; hus, i Q=Ek
j hen Q∗=B(xk
j, ρk+1).
We se `(Q) = (Q∗)/ρ and call `(Q) he “sideleng h” o Q. We no e ha while he cubes in
each Dmha e he dyadic p ope ies lis ed abo e, no nes edness p ope ies o he cubes in Dm1
ela i e o he cubes in Dm2a e assumed i m1and m2a e di e en .
Fo any Bo el measu e ω, de ine
Mω,mg(x) = sup
B:x∈B
(B)≥ρm
1
ω(B)ZB
|g|dω,
and also he dyadic e sion
Mdy
ω,mg(x) = sup
Q:x∈Q
Q∈Dm
1
ω(Q)ZQ
|g|dω.
I ωis absolu ely con inuous wi h espec o µ, i.e., dω =w dµ, hen we w i e Mw,mgand
Mdy
w,mgins ead o Mwdµ,mgand Mdy
wdµ,mg. As usual, we say ha wis a weigh i w(x) is a
16

nonnega i e locally in eg able unc ion wi h espec o µ, and o a measu able se E, we
w i e w(E) = REw(x)dµ(x).
Le us show, as men ioned in §2, ha (12) and (13) a e iden ical no ions o many
unc ionals τi νis a doubling measu e. In ac , we will show his is he case i τjus sa is ies
(16)(a),(b). Le νbe a doubling measu e. Fi s , suppose ha (13) holds o ν, and le
E⊂Q∈ Dm. Gi en η > 0, selec cubes {Qi}in Dmwi h E⊂ ∪Qi⊂Qand
Pτ(B(Qi)) <kEkτ,m +η, whe e B(Qi) deno es he con aining ball o Qi. No e ha
E⊂ ∪B(Qi)⊂cB(Q) o some geome ic cons an c which is independen o m, E, Q, and
{Qi}. By (13),
ν(E)
ν(cB(Q)) ≤CHτ,cB(Q)(E)
τ(cB(Q)) δ
≤CPτ(B(Qi))
τ(cB(Q)) δ
.
By he p ope ies o con aining balls, and since νis doubling and (16)(a) holds, we ob ain
ν(E)
ν(Q)≤CPτ(B(Qi))
τ(B(Q)) δ
≤CkEkτ,m +η
τ(B(Q)) δ
,
and (12) ollows by le ing η→0.
Con e sely, suppose ha (12) holds o a doubling measu e ν, and le E⊂ ∪Bi⊂B. We
wan o show ha
ν(E)
ν(B)≤CPτ(Bi)
τ(B)δ
o some δ > 0, i.e., ha (13) holds. Le mbe so la ge ha ρm<< (B). Co e Bby a ini e
numbe N1o disjoin dyadic cubes Q∈ Dmwi h `(Q)≈ (B), whe e N1and he cons an s o
equi alence a e independen o Band he g id Dm. By he doubling o ν,ν(B)≈ν(Q) o
each such Q, wi h simila cons an s o equi alence, assuming as we may ha each Q ouches
B. By conside ing he se s E∩Qindi idually and adding, we may assume ha E⊂Q o
one such Q. Also, by conside ing hose Biwi h (Bi)≥ρm, and e en ually le ing m→ −∞,
17
we may assume ha all (Bi)≥ρm. Co e each Biby a ini e numbe N2(independen o i
and m) o cubes {Qij}N2
j=1 in Dmwi h (Bi)≈`(Qij) uni o mly in i, j, m. I ollows easily
om he p ope ies o he quasime ic ha Qij ⊂cBi o a uni o m posi i e cons an c. Since
E⊂ ∪i,jQij and we may disca d any Qij no con ained in Q, i ollows om (12) ha
ν(E)
ν(Q)≤CPi,j τ(B(Qij))
τ(B(Q)) δ
=CPiPjτ(B(Qij))
τ(B(Q)) δ
≤CN2Piτ(Bi)
τ(B)δ
because he numbe o j’s is a mos N2and (16)(a),(b) hold. Since ν(B)≈ν(Q), a simila
es ima e holds wi h ν(Q) eplaced by ν(B) in he denomina o on he le , which gi es he
desi ed inequali y.
As usual, we say ha a measu e νbelongs o A∞(µ) i he e a e posi i e cons an s Cand
ηsuch ha
ν(E)≤Cµ(E)
µ(B)η
ν(B)
o e e y ball Band e e y measu able se E⊂B. I is no di icul o show ha i ν∈A∞(µ)
hen ν∈Ady
∞(µ) (in he sense o de ini ion (12)). In ac , le Ebe a se wi h E⊂ ∪Qi⊂Q o
Qi, Q ∈ Dm. I ν∈A∞(µ), i ollows easily ha νis doubling, and hen
ν(E)
ν(Q)≤cµ(E)
µ(Q)η
o some η > 0 since Qcan be included in a ball o compa able measu e. The e o e, since
E⊂ ∪B(Qi),
ν(E)
ν(Q)≤cXµ(B(Qi))
µ(Q)η
≤cXµ(B(Qi))
µ(B(Q)) η
since µis doubling. Consequen ly, ν∈Ady
∞(µ).
Le us now show ha any measu e νwhich sa is ies he dyadic e e se doubling condi ion
RDdy
β(see (15)) belongs o Aβ
∞ o he same alue o β. A simila ac was shown in [SW1] in
18
he Euclidean case. I E⊂ ∪Qi⊂Qwi h Qi, Q ∈ Dm, hen
ν(E)≤X
i
ν(Qi)≤cX
i`(Qi)
`(Q)β
ν(Q)
since ν∈RDdy
β. De ining τby τ(B) = (B)β o all B, i ollows ha
ν(E)≤cX
i
τ(B(Qi))
τ(B(Q)) ν(Q),
whe e B(Qi) and B(Q) a e he con aining balls o Qiand Q. Taking he in imum o e all
such co e ings {Qi}o E, we ob ain ha
ν(E)≤ckEkτ,m
τ(B(Q)) ν(Q),
so ha (12) holds wi h δ= 1 and hus ν∈Aβ
∞. The conclusion ha ν∈Aβ
∞also holds i he
hypo hesis ha ν∈RDdy
βis eplaced by he assump ions ha ν∈RDβand νis a doubling
measu e; in ac , hese assump ions a e easily seen o imply ha ν∈RDdy
β.
In passing, we no e ha i he unde lying measu e µ∈Ady
∞(τ) and i ν∈A∞(µ), namely,
o some η > 0,
ν(E)
ν(B)≤Cµ(E)
µ(B)η
i E⊂B,
hen also ν∈Ady
∞(τ) o he same τ(al hough he alue o δ o νequals η imes he alue o
δ o µ). This ollows easily om he de ini ions since µand νa e doubling measu es.
4 O licz spaces and O licz maximal unc ions
To p o e Theo em 2.3, we will use some ac s abou O licz spaces which we ecall he e,
e e ing o [RR] and [BS] o a comple e accoun .
A unc ion Φ : [0,∞)→[0,∞) is called a Young unc ion i i is con inuous, con ex,
inc easing and sa is ies Φ(0) = 0 and Φ( )→ ∞ as → ∞. I ollows ha Φ( )/ is inc easing,
19
and in pa icula ha
Φ(γ )≥γΦ( ) i γ≥1 and ≥0.
Some imes we will also assume ha Φ sa is ies he doubling condi ion Φ(2 )≤CΦ( ).
Fo O licz no ms we a e usually only conce ned abou he beha io o Young unc ions o
la ge. By de ini ion, he O licz space LΦconsis s o all measu able unc ions such ha
ZS
Φ| |
λdµ < ∞
o some posi i e λ. No e ha i 0 < λ1< λ2, hen
Φ| |
λ2≤λ1
λ2
Φ| |
λ1,
so ha
lim
λ→∞ ZS
Φ| |
λdµ = 0 i ∈LΦ.
The space LΦis a Banach unc ion space wi h he Luxembu g no m
k kΦ=k kΦ,µ = in {λ > 0 : ZS
Φ(| |
λ)dµ ≤1}.
Each Young unc ion Φ has an associa ed complemen a y Young unc ion ¯
Φ sa is ying
≤Φ−1( )¯
Φ−1( )≤2 (33)
o all > 0, whe e Φ−1s ands o he in e se unc ion o Φ. The unc ion ¯
Φ is called he
conjuga e o Φ, and he space L¯
Φis called he conjuga e space o LΦ. Fo example, i
Φ( ) = p o 1 < p < ∞ hen ¯
Φ( ) = p0, p0=p/(p−1), and he conjuga e space o Lp(µ) is
Lp0(µ). An example ha we will need is Φ( )≈ p(log )−1− o la ge , 1 < p < ∞, > 0,
wi h complemen a y unc ion ¯
Φ( )≈ p0(log )(p0−1)(1+) o la ge (c . [O], p.275).
A e y impo an p ope y o O licz spaces is he gene alized H¨olde inequali y
ZS
| g|dµ ≤ k kΦkgk¯
Φ.(34)
20
In o de o de ine ano he maximal unc ion which will play a ole in he p oo o Theo em
2.3, we need local e sions o O licz no ms. I Φ is a Young unc ion and Bis a ball, le
k kΦ,B =k kΦ,B,µ = in {λ > 0 : 1
µ(B)ZB
Φ(| |
λ)dµ ≤1}.
Fo his no m, he local e sion o he gene alized H¨olde inequali y (34) is
1
µ(B)ZB
g dµ ≤ k kΦ,Bkgk¯
Φ,B.(35)
De ine a maximal unc ion co esponding o Φ by
MΦ (x) = sup
B:x∈B
k kΦ,B.(36)
This maximal unc ion has been used in he usual Euclidean con ex in [P1] and also in he
case Φ( )≈ log in he wo k o T. Iwaniec and G eco [GI] and in [WW]. The no m beha io
o MΦ is closely ela ed o he nex de ini ion.
De ini ion 4.1 Le 1< p < ∞. A nonnega i e unc ion Φ( ), > 0,sa is ies he Bpcondi ion
i he e is a cons an c > 0such ha
Z∞
c
Φ( )
p
d
<∞.(37)
Simple examples o unc ions which sa is y Bpa e p−δand p(log(1 + ))−1−δ, bo h when
δ > 0.
The ele ance o condi ion Bps ems om i s ela ionship o he boundedness o MΦas
s a ed in he nex heo em om [PW].
Theo em 4.2 Le 1< p < ∞and Φbe a doubling Young unc ion. Then he ollowing
s a emen s a e equi alen .
i) Φ∈Bp, i.e., he e is a cons an c > 0such ha
Z∞
c
Φ( )
p
d
<∞.(38)
21

ii) The e is a cons an C > 0such ha
ZS
MΦ (x)pdµ(x)≤CZS
(x)pdµ(x) (39)
o all nonnega i e .
iii) The e is a cons an C > 0such ha
ZS
MΦ (x)pw(x)dµ(x)≤CZS
(x)pMw(x)dµ(x) (40)
o all nonnega i e and w, whe e Mw is he Ha dy–Li lewood maximal unc ion de ined in
(20).
Fo example, in he s anda d case when Φ( ) = wi h ≥1, so ha
k kΦ,B =µ(B)−1RB| | dµ1/ , he equi alence o (38) and (39) educes o he well-known
ac ha he mapping
→sup
B:x∈B1
µ(B)ZB
| | dµ1/
is bounded on Lp(S, µ) i and only i p> . The cha ac e iza ion o Bpgi en abo e was
p o ed in he Euclidean con ex in [P3] and used o de i e sha p wo weigh es ima es o he
classical Ha dy–Li lewood maximal unc ion. In he gene al case, he cha ac e iza ion o Bp
plays a main ole in some o he esul s in [PW]. Fo o he applica ions o di e en ope a o s
om ha monic analysis, see [P1], [P4] [P5], [P6], [CP1] and [CP2].
5 P oo s o Theo em 2.4 and Co olla y 2.5
Recall ha Mψ( dσ) is de ined by
Mψ( dσ)(x) = sup
B:x∈B
ψ(B)ZB
| (y)|dσ(y)
o any measu e σand any measu able , whe e ψ(B) is assumed o be nonnega i e and o
sa is y (23) and also he doubling condi ion. To p o e Theo em 2.4, we may assume ha is
22
nonnega i e, bounded and has bounded suppo . We may also assume by a limi ing a gumen
ha Mψ( dσ)(x) is o med by aking he sup emum only o e balls con aining xo adius a
leas ρm o ixed m, whe e ρis he cons an used in §3 o cons uc he dyadic g id Dm. Fo
each in ege kwe le
Ωk={x∈S:Mψ( dσ)(x)> γk}
whe e γ > 1 is a cons an o be chosen. Fo each x∈Ωk, he e is a ball Bxcon aining xwi h
ψ(Bx)ZBx
dσ > γk.
Now we claim he ollowing:
To each Bx, he e co esponds a dyadic cube Qx∈ Dm(Qxmay no con ain x,
al hough Bxdoes) o size compa able o Bxwi h Qx∩Bx6=∅and
ψ(B(Qx)) ZQx
dσ > γk
c,
whe e cis a geome ic cons an .
Recall he e ha i Qis any dyadic cube, hen B(Q) deno es he con aining ball o Q; he
adius o B(Q) is compa able o he edgeleng h o Q.
To p o e he claim, i s no e ha we can co e Bxby a ixed numbe N(independen o
x, γ, k) o disjoin dyadic cubes Qo size compa able o Bx. Indeed, le k0be he in ege such
ha ρk0≤ (Bx)< ρk0+1, and conside any mwi h m<k0(he e ρis he cons an used in §3)
o cons uc he dyadic g id Dm. We make he ollowing subclaim:
In Dm, he e a e a mos Ncubes {Ek0
j}jmee ing Bx(i.e., he e a e a mos N
cubes o sideleng h ρk0mee ing Bx), whe e Nis a s uc u al cons an which is
independen o Bxand m, p o ided ha m<k0.
23
To p o e he subclaim, ix m<k0and deno e hose cubes {Ek0
j}jin Dmwhich ha e nonemp y
in e sec ion wi h Bxby Qj,j= 1,· · · , N. The {Qj}a e disjoin , and hey a e con ained in
some ixed enla gemen o Bxsince hey ouch Bxand hei adii a e compa able o (Bx).
Thus, he sum o he µ(Qj) is a mos cµ(Bx). Since he size o each Qjis compa able o he
size o Bx, each µ(Qj) exceeds a ixed mul iple o µ(Bx) by doubling (in ac , he measu es a e
compa able). The e o e, he numbe No Qj’s mus be a mos a ixed geome ic cons an ,
which p o es he subclaim.
Consequen ly, o any ixed m<k0,i Qj, j = 1, . . . , N, a e he cubes in Dmmen ioned
abo e, hen χBx≤PN
j=1 χQj, and so
γk< ψ(Bx)ZBx
dσ ≤ψ(Bx)
N
X
j=1 ZQj
dσ.
Thus o some j0,
γk< N ψ(Bx)ZQj0
dσ.
Pick Qx o be Qj0. Since he sizes o Qx,Bxand B(Qx) a e compa able, he claim ollows by
using Qx∩Bx6=∅and he p ope ies o ψ.
Choosing γ > c we ha e
ψ(B(Qx)) ZQx
dσ > γk−1.(41)
De ine
e
Ωk=x: sup
Q∈Dm:x∈Q
ψ(B(Q)) ZQ
dσ > γk−1
and le {Qk
j}jbe he maximal cubes in Dmwi h
γk−1< ψ(B(Qk
j)) ZQk
j
dσ.
Then e
Ωk=∪jQk
j. Obse e ha i Qxis he dyadic cube om (41) hen Qx⊂Qk
j o some j,
and hence since he e exis s c0>1 such ha x∈c0B(Qk
j), we ob ain ha Ωk⊂ ∪jc0B(Qk
j).
24
Now i ˜
Qk
jis he nex la ges dyadic cube con aining Qk
j, hen
ψ(B(˜
Qk
j)) Z˜
Qk
j
dσ ≤γk−1
and he e o e, by using he doubling p ope y o ψ,
γk−1< ψ(B(Qk
j)) ZQk
j
dσ ≤cψγk−1.(42)
Le Mdy
σ,m be de ined as in Sec ion 3. To p o e he heo em i will enough o show ha
ZS
Mψ( dσ)qdω1/q
≤cZS
(Mdy
σ,m )pdσ1/p
(43)
wi h cindependen o mand , since Mdy
σ,m is bounded on Lp(dσ) uni o mly in m o
1< p < ∞(see o example [W], Lemma 3.8).
Now, we s a wi h
ZS
Mψ( dσ)qdω =X
kZΩk Ωk+1
Mψ( dσ)qdω
≤X
k
γ(k+1)qω(Ωk)≤γ2qX
k
γ(k−1)qX
j
ω(c0B(Qk
j))
≤γ2qX
k,j ψ(B(Qk
j)) ZQk
j
dσ!q
ω(c0B(Qk
j)).
Since ψ(B(Qk
j)) ≤ψ(c0B(Qk
j)) by (23)(a), i we use (25) o c0B(Qk
j), we can con inue wi h
≤CX
k,j
σ(B(Qk
j))−q/p0 ZQk
j
dσ!q
≤CX
k,j
σ(Qk
j)−q/p0 ZQk
j
dσ!q
,
≤C X
k,j
σ(Qk
j)1−p ZQk
j
dσ!p!q/p
,
whe e we ha e used he ac s ha Qk
j⊂B(Qk
j) and ha p≤q.
Recall ha e
Ωk=∪jQk
j. Now le Ek
j=Qk
j e
Ωk+1. Then Ek
j⊂e
Ωk e
Ωk+1 and he se s Ek
j
a e disjoin in bo h j, k. We wish o show ha σ(Qk
j)≤c σ(Ek
j) i γis su icien ly la ge. I is
25
by he maximali y o Ql
i. Then, since he Ql
ia e disjoin in i o ixed l, and since ω∈Ady
∞(τ),
X
i:Ql
i⊂Qk
j
ω(Ql
i) = ω(∪i:Ql
i⊂Qk
jQl
i)≤c



∪i:Ql
i⊂Qk
jQl
i

τ,m
τ(B(Qk
j)) 


δ
ω(Qk
j)
≤c Pi:Ql
i⊂Qk
jτ(B(Ql
i))
τ(B(Qk
j)) !δ
ω(Qk
j)≤cγ(k−l)δω(Qk
j).(50)
Now le
Ωk={x: sup
Q∈Dm:x∈Q
1
τ(B(Q)) ZQ
g dω > γk},
and se Ek
j=Qk
j Ωk+1. Then Ek
j⊂Ωk Ωk+1 and he se s Ek
ja e disjoin in bo h j, k. No e
ha Ωk=∪jQk
j. We wish o show ha ω(Qk
j)≤cω(Ek
j) i γis su icien ly la ge. I is enough
o show ha ω(Qk
j∩Ωk+1)<1
2ω(Qk
j). Bu by he dyadic s uc u e and ou ea lie
obse a ions,
ω(Qk
j∩Ωk+1) = X
i:Qk
j∩Qk+1
i6=∅
ω(Qk
j∩Qk+1
i) = X
i:Qk+1
i⊂Qk
j
ω(Qk+1
i)
≤cγ−δω(Qk
j) by (50) wi h l=k+ 1
<1
2ω(Qk
j) i γis la ge.
Nex , we ew i e he sum on he igh side o (49) as
X
j,k
ϕ(B(Qk
j)) Zκ(2κ+1)B(Qk
j)
dσ ω(Qk
j)1
ω(Qk
j)ZQk
j
g dω
and apply H¨olde ’s inequali y o ob ain
ZS
Tm( dσ)g dω ≤
c X
j,k "ϕ(B(Qk
j)) Zκ(2κ+1)B(Qk
j)
dσ#p
ω(Qk
j)!1/p 
X
j,k "1
ω(Qk
j)ZQk
j
g dω#p0
ω(Qk
j)

1/p0
.
32

We may eplace ω(Qk
j) by ω(Ek
j) in he nume a o s o bo h o hese sums and hen use he
disjoin ness o he Ek
j o majo ize he las exp ession by
cZS
Mϕ( dσ)pdω1/p ZS
(Mdy
ω,mg)p0dω1/p0
,
whe e Mdy
ω,mgis he dyadic maximal unc ion de ined in §3. As men ioned in §5, Mdy
ω,m is
bounded on Lp(dω) uni o mly in m o 1 < p < ∞. Thus, we ob ain ha
ZS
Tm( dσ)g dω ≤ckMϕ( dσ)kLp(dω)kgkLp0(dω),
wi h cindependen o m, and g, and consequen ly Theo em 2.1 ollows om duali y by
le ing m→ −∞.
2
Le us now p o e Co olla y 2.2. Le τbe de ined by τ(B) = (B)β o all B, wi h β > 0 o
be chosen. By hypo hesis, he e exis s  > 0 such ha
ϕ(B1)µ(B1)≤c (B1)
(B2)
ϕ(B2)µ(B2) i B1⊂B2.
The e o e, since µis assumed o sa is y he doubling condi ion o o de D,
ϕ(B1)≤c (B1)
(B2)−D
ϕ(B2) i B1⊂B2
=c (B1)
(B2)(β+−D)−β
ϕ(B2)
=c (B1)
(B2)β+−D
ϕ(B2)τ(B2)
τ(B1).
I ollows ha (16)(c) holds wi h  he e aken o be β+−D. Clea ly, (16)(a), (b) also hold
o any β > 0. Thus, i β+−D > 0, by applying Theo em 2.1 (wi h τ(B) = (B)βas
abo e), he conclusion o Co olla y 2.2 ollows immedia ely.
2
33
7 P oo o Theo em 2.3
We s a wi h he case p= 1, whe e he p oo will be a a ian o ha o Theo em 2.1. The
cases p > 1 and p < 1 will ollow om he case p= 1 using ex apola ion ideas and duali y
be ween Lpspaces.
7.1 The case p=1
By a limi ing a gumen we may assume ha wis bounded wi h compac suppo . Fo ≥0,
we s a wi h inequali y (49) wi h dω eplaced by w dµ and dσ eplaced by dµ:
ZS
Tm( dµ)w dµ ≤cγ X
k,j
ϕ(B(Qk
j)) Zκ(2κ+1)B(Qk
j)
dµ ZQk
j
w dµ,
whe e he dyadic cubes Qk
ja e now he maximal cubes in Dmwi h
γk<1
µ(Qk
j)ZQk
j
w dµ
and sa is y
γk<1
µ(Qk
j)ZQk
j
w dµ ≤cµγk.
De ine
Ωk=x: sup
Q∈Dm:x∈Q
1
µ(Q)ZQ
w dµ > γk,
and le Ek
j=Qk
j Ωk+1. No e ha Ωk=∪jQk
j. As usual, i Qk
j∩Qk+1
i6=∅, hen Qk+1
i⊂Qk
j.
I γis la ge enough, hen µ(Qk
j)≤2µ(Ek
j) since
µ(Qk
j∩Ωk+1) = X
i:Qk
j∩Qk+1
i6=∅
µ(Qk
j∩Qk+1
i)
=X
i:Qk+1
i⊂Qk
j
µ(Qk+1
i)≤γ−k−1X
i:Qk+1
i⊂Qk
jZQk+1
i
w dµ
≤γ−k−1ZQk
j
w dµ ≤γ−k−1cµγkµ(Qk
j)
34
=cµ
γµ(Qk
j)<1
2µ(Qk
j)
i γis la ge.
Thus he e a e se s Ek
j ha a e pai wise disjoin in bo h jand kwi h Ek
j⊂Qk
jand
µ(Qk
j)≤cµ(Ek
j) o a uni e sal cons an c. I we deno e ˜
Qk
j=κ(2κ+ 1)B(Qk
j), hen
ZS
Tm( dµ)w dµ
≤cX
k,j
ϕ(B(Qk
j)) Z˜
Qk
j
dµ 1
µ(Qk
j)ZQk
j
w dµ!µ(Qk
j)
≤cX
k,j
ϕ(˜
Qk
j)Z˜
Qk
j
dµ 1
µ(Qk
j)ZQk
j
w dµ!µ(Ek
j)
≤cX
k,j ZEk
j
Mϕ( dµ)Mw dµ ≤cZS
Mϕ( dµ)Mw dµ.
This concludes he p oo o (22) when p= 1.
7.2 The case p > 1
Now le p > 1. As we men ioned abo e, ou a gumen will be based on duali y and he case
p= 1. In ac we will p o e some hing sha pe han (21): i δ > 0, he e is a cons an C such
ha o any weigh wand all ,
ZS
|T( dµ)|pw dµ ≤CZS
(Mϕ( dµ))pML(log L)p−1+δ(w)dµ, (51)
whe e ML(log L)p−1+δdeno es he maximal unc ion MΦwi h
Φ( ) = (1 + log+ )p−1+δ, > 0.
The ac ha his es ima e is sha pe han (21) will be shown la e . By he case p= 1, he e
is a cons an cso ha o all , g ≥0 and all m,
ZS
Tm( dµ)g w1/p dµ ≤cZS
Mϕ( dµ)M(g w1/p)dµ.
35
Now by he gene alized H¨olde inequali y (35) o an app op ia e Young unc ion Ψ ha will
be chosen soon, we can con inue wi h
≤cZS
Mϕ( dµ)MΨgM¯
Ψ(w1/p)dµ
≤cZS
Mϕ( dµ)p(M¯
Ψ(w1/p))pdµ1/p ZS
(MΨg)p0dµ1/p0
.
To conclude he p oo o (51), we use Theo em 4.2 o an app op ia e Ψ ∈Bp0. Indeed, as
men ioned in §4, we can choose Ψ( )≈ p0(log )−1−, > 0 and la ge, wi h complemen a y
unc ion ¯
Ψ( )≈ p(log )(p−1)(1+) o la ge . Then by combining es ima es, we ob ain
ZS
Tm( dµ)g w1/p dµ ≤cZS
Mϕ( dµ)pML(log L)(p−1)(1+)(w)dµ1/p ZS
gp0dµ1/p0
.
Since he cons an cis independen o m, (51) ollows by duali y and le ing m→ −∞.
To show ha (51) implies (21), we ecall he ollowing lemma om [PW] (see also [P1],
[GI] and [WW] in he usual Euclidean case).
Lemma 7.1 Le k= 1,2,· · · .Then he e is a posi i e cons an csuch ha o any
measu able unc ion w,
kwkL(logL)k,B ≤c
µ(B)ZB
Mkw dµ, (52)
whe e Mkwdeno es he k- old i e a e o he Ha dy–Li lewood maximal unc ion de ined in
(20) and k·kL(logL)k,B deno es k·kΦ,B wi h Φ( ) = (1 + (log+ )k).
Fo k= 1,2,· · · , (52) implies ha
kwkL(logL)k,B ≤c
µ(B)ZB
Mkw dµ.
Thus, i we choose =[p]
p−1−1>0 in he p oo o (51), we ob ain
M¯
Ψ(w1/p)(x)p≈ ML(log L)[p](w)(x)≤c M[p]+1(w)(x),
and pa i) o Theo em 2.3 ollows.
36
7.3 The case p < 1
Assume now ha 0 < p < 1. We will p o e he inequali y
ZS
|T( dµ)|pw dµ ≤CZS
(Mϕ( dµ))pMw dµ (53)
by an ex apola ion a gumen , a e i s p o ing a s eng hened e sion o he case p= 1. We
begin wi h a de ini ion.
De ini ion 7.2 A weigh sa is ies he RH∞(µ)condi ion (i.e., he e e se H¨olde condi ion
o in ini e o de ) i he e is a cons an c > 0such ha o each ball B,
ess supB ≤c
µ(B)ZB
dµ
I is easy o check ha RH∞(µ)⊂A∞(µ); in ac we can ake η= 1 in he de ini ion o
A∞(µ).
We will p o e he ollowing e sion o he case p= 1:
Lemma 7.3 Le a weigh sa is ying he RH∞(µ)condi ion. Then he e is a cons an C
such ha o any weigh wand all ,
ZS
|T( dµ)| w dµ ≤CZS
Mϕ( dµ) Mw dµ. (54)
Fo he p oo o his lemma we p oceed as in he case = 1:
ZS
Tm( dµ) w dµ ≤cX
k,j
ϕ(˜
Qk
j)Z˜
Qk
j
dµ ZQk
j
w dµ
≤cX
k,j
ϕ(˜
Qk
j)Z˜
Qk
j
dµ ZQk
j
w dµ (ess supQk
j )
≤cX
k,j
ϕ(˜
Qk
j)Z˜
Qk
j
dµ 1
µ(Qk
j)ZQk
j
w dµ!ZQk
j
dµ since ∈RH∞(µ)
37

≤cX
k,j
ϕ(˜
Qk
j)Z˜
Qk
j
dµ 1
µ(Qk
j)ZQk
j
w dµ!ZEk
j
dµ since ∈A∞(µ)
≤cX
k,j ZEk
j
Mϕ( dµ)Mw dµ ≤cZS
Mϕ( dµ) Mw dµ.
2
Lemma 7.4 Le α > 0and gbe any unc ion such ha Mg is ini e a.e. Then
(Mg)−α∈RH∞(µ).
This obse a ion is due o C. Neugebaue [N] whe e he showed some hing be e :
w∈RH∞(µ)⇔w≈(Mg)−α o some gand some α > 0.
As usual we will deno e wB=1
µ(B)RBw dµ. To p o e (53), we i s obse e ha
w−1∈RH∞(µ) i w∈A1(µ), i.e., i wB≤C ess in Bw o all B, since hen
ess supBw−1= (ess in Bw)−1≤c(wB)−1≤c(w−1)B,
whe e in he las inequali y we ha e used H¨olde ’s inequali y. The cons an cis in ac he
in e se o he cons an Cin he de ini ion o A1(µ).
The second obse a ion we need is ha i w∈RH∞(µ) hen wλ∈RH∞(µ) when λ > 1:
ess supBwλ= (ess supBw)λ≤cλ(wB)λ≤cλ(wλ)B.
Ac ually his is also ue when 0 < λ < 1 bu is a bi ha de and we don’ need i ; see [N].
Le us now p o e (53) o 0 < p < 1. We will use an app op ia e duali y o he spaces
Lp(dν) when p < 1 and νis a measu e: i ≥0 hen
k kLp(dν)= in {Z u−1dν :ku−1kLp0(dν)= 1}=Z u−1dν
o some u≥0 such ha ku−1kLp0(dν)= 1, whe e p0=p
p−1<0. This ollows om he
“ e e se” H¨olde inequali y Z g dν ≥ k kLp(dν)kgkLp0(dν),
38
which is a consequence o he usual H¨olde inequali y. Combining his wi h he Lebesgue
di e en ia ion heo em and bo h Lemmas 7.3 and 7.4 o he weigh (M(gδ))−1/δ ∈RH∞(µ),
δ > 0 and g≥0, we ha e
ZS
Mϕ( dµ)Mw
gdµ ≥ZS
Mϕ( dµ)Mw
(M(gδ))1/δ dµ
≥cZS
T( dµ)w
(M(gδ))1/δ dµ ≥ckT( dµ)kLp(wdµ)
(M(gδ))−1/δ
Lp0(wdµ).
Howe e , since he in eg al on he igh side o (53) when aised o he powe 1/p equals
ZS
Mϕ( dµ)Mw
gdµ
o some g, e e y hing is educed o p o ing ha

(M(gδ))−1/δ
Lp0(wdµ)≥ckg−1kLp0(M(w)dµ).
Since p0<0, his is equi alen o saying ha
ZS
(M(gδ))−p0/δ w dµ ≤cZS
g−p0Mw dµ.
Bu i we choose δso ha 0 < δ < −p0, hen −p0/δ > 1 and he las es ima e ollows om he
known (see [FS]) weigh ed no m inequali y
ZS
(M )qw dµ ≤cZS
| |qMw dµ, q > 1.
2
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41