Embeddings of concave functions and duals of Lorentz spaces

Publ. Ma . 46 (2002), 489–515
EMBEDDINGS OF CONCAVE FUNCTIONS AND
DUALS OF LORENTZ SPACES
Go d Sinnamon
Abs ac
A simple exp ession is p esen ed ha is equi alen o he no m o
he Lp
→Lq
uembedding o he cone o quasi-conca e unc ions in
he case 0 <q<p<∞. The esul is ex ended o mo e gene al
cones and he case q= 1 is used o p o e a educ ion p inciple
which shows ha ques ions o boundedness o ope a o s on hese
cones may be educed o he boundedness o ela ed ope a o s on
whole spaces. An equi alen no m o he dual o he Lo en z
space
Γp( )= :∞
0
( ∗∗ )p 1/p
<∞
is also gi en. The exp ession is simple and conc e e. An applica-
ion is made o desc ibe he weigh s o which he Ha dy Li le-
wood Maximal Func ion is bounded on hese Lo en z spaces.
1. In oduc ion
The beha iou o he collec ion o non-nega i e, non-inc easing unc-
ions in weigh ed Lebesgue spaces is well unde s ood. Since [6] and
[9] in he ea ly 50’s, echniques in ol ing p ope ies o mono one unc-
ions ha e been used effec i ely o add ess a wide a ie y o ques ions
in weigh ed no m inequali ies, in e pola ion heo y, and unc ion space
heo y. Fo a ew o he many see [1], [3], [7], [8], [13], [14], [15],
[16], [17]. The s udy o he collec ion o conca e unc ions has also had
i s successes. See [4], [5], [10], [11] and e e ences he e. Conca e unc-
ions a ise na u ally in in e pola ion heo y and much o he ecen wo k
2000 Ma hema ics Subjec Classifica ion. P ima y: 26D15; Seconda y: 46E30,
42B25.
Key wo ds. Inequali ies, weigh s, Lo en z space, quasi-conca e unc ions, duali y,
maximal unc ion.
Suppo om he Na u al Sciences and Enginee ing Resea ch Council o Canada is
g a e ully acknowledged.
490 G. Sinnamon
shows ha hey a e o equal impo ance in weigh ed no m inequali ies
and unc ion spaces.
Ra he han wo king wi h he collec ion o non-inc easing, conca e
unc ions, i is common o s udy he cone o quasi-conca e unc ions.
This is he se o non-nega i e unc ions defined on (0,∞) such ha
(x) is non-dec easing and (x)/x is non-inc easing. Passing be ween
he wo collec ions is ou ine and he la e is mo e con enien o a ious
easons. The embedding ques ion o his cone is a key o effec i ely
using p ope ies o conca e unc ions: Fo which indices pand qand
which weigh s uand a e he quasi-conca e unc ions in Lp
also in Lq
u?
Va ious pa ial answe s o his ques ion a e a ailable. The case 0 <
p≤q<∞in pa icula has been simply cha ac e ized and in [10], [11]
e y igh bounds on he no m o he embedding ha e been gi en. Fo
he case 0 <q=1<p<∞sufficien condi ions which a e simila bu
no iden ical o he necessa y ones we e ob ained in [17].
A comple e answe o he embedding ques ion was gi en in [5] bu he
condi ions gi en a e complica ed and difficul o apply. Ou objec he e is
o gi e simple necessa y and sufficien weigh condi ions ha cha ac e ize
he embedding o he cone o quasi-conca e unc ions om Lp
o Lq
u.We
also gi e explici uppe and lowe bounds on he no m o he embedding.
This is accomplished in Theo em 2.6 and he embedding ques ion o
mo e gene al cones is answe ed in Theo em 2.7. In Sec ion 3, he esul s
a e applied o gi e a educ ion p inciple o ope a o s ac ing on such
cones. This shows he equi alence o he boundedness o an ope a o
on he cone wi h he boundedness o wo ela ed ope a o s on ela ed
spaces.
The dual o he Lo en z space Γp( ) is cha ac e ized in Sec ion 4.
Theo em 4.1 gi es a simple exp ession ha is equi alen o he no m
in he associa e space, he K¨o he dual. As an applica ion, in Sec ion 5
we gi e weigh condi ions o cha ac e ize he boundedness o he Ha dy-
Li lewood Maximal Func ion be ween Lo en z spaces.
To s udy quasi-conca e unc ions we need an ope a o on non-nega i e
unc ions whose images a e quasi-conca e unc ions. Al hough he gene -
alized S iel jes ans o ma ion h→ ∞
0
x
x+ h( )d is used o his pu pose
by some au ho s, we will adop he equi alen ope a o
h→ ∞
0
min(1, x/ )h( )d
which is also popula . The lack o smoo hness in he ke nel min(1, x/ )
will no bo he us. I is impo an o no e ha he esul s we ob ain
Conca e Func ions and Lo en z Duals 491
can easily be e-cas in e m o gene alized S iel jes ans o ma ions i
desi ed.
The weigh ed Lebesgue spaces al eady e e ed o a e defined as ol-
lows. I is a non-nega i e, Lebesgue measu able unc ion (a weigh )
on (0,∞) hen he weigh ed Lebesgue space Lp
is he collec ion o
Lebesgue measu able unc ions on (0,∞) o which
 p, ≡∞
0| |p 1/p ,0<p<∞
ess sup{x: (x)>0}| (x)|,p=∞
is fini e. I ≡1 we d op he weigh and w i e Lpand  p.
Th oughou he pape , p oduc s o he o m 0 ·∞ a e aken o be
ze o. Fo an index pwe define pby 1/p +1/p= 1. We say ha he
exp essions Cand Aa e equi alen and w i e C≈Ap o ided he e a e
posi i e cons an s kand Ksuch ha kA ≤C≤KA. The cons an s
depend only on he indices pand q. We keep ack o he cons an s
in he s a emen s o heo ems bu will o en a oid such de ails in he
p oo s, p e e ing o ocus on essen ial ea u es. In pa icula he ex-
ended Minkowski inequali y o 0 <s<∞,
min(1,21/s−1)( 1s+ 2s)≤ 1+ 2s
≤max(1,21/s−1)( 1s+ 2s)
(1.1)
will be used epea edly in he o m
 1+ 2s≈ 1s+ 2s.(1.2)
2. Ha dy inequali ies and conca e unc ions
In his sec ion we gi e necessa y and sufficien condi ions on indices p,
qand weigh s u, o he cone o quasi-conca e unc ions in Lp
o be
embedded in Lq
uwhen 0 <q<p<∞. We also gi e uppe and lowe
bounds o he no m o his embedding. This esul is in Theo em 2.6
while an analogue o mo e gene al cones may be ound in Theo em 2.7.
See also Theo em 3.1. Co esponding known esul s o he case 0 <p≤
q<∞a e s a ed in P oposi ion 2.8.
We begin by looking a he embedding in o Lq
uo a smalle cone
in L1
. Known weigh ed Ha dy inequali ies a e used o gi e a weigh
cha ac e iza ion in his si ua ion. F om he e we expand he cone o
include all quasi-conca e unc ions and hen use an in a iance p ope y
o he cone o quasi-conca e unc ions o pass om L1
o Lp
.
Le L+deno e he collec ion o non-nega i e, measu able unc ions
on (0,∞). We say ∈L+is quasi-conca e and w i e ∈Ω0,1p o ided
492 G. Sinnamon
(x) is non-dec easing and (x)/x is non-inc easing. Mo e gene ally, i
α+β>0 we w i e ∈Ωα,β p o ided xα (x) is non-dec easing and
x−β (x) is non-inc easing.
As men ioned we begin wi h weigh ed Ha dy inequali ies. Define he
Ha dy and dual Ha dy ope a o s Hαand Hβby
Hαh(x)=x−αx
0
αh( )d and Hβh(x)=xβ∞
x
−βh( )d .
The sum o he wo will a ise equen ly so o α+β>0 we in oduce
he ope a o
Hβ
αh(x)=Hαh(x)+Hβh(x)
=∞
0
min(( /x)α,(x/ )β)h( )d , h ∈L+.
(2.1)
Since we always suppose ha α+β>0, he second o m o Hβ
α
makes i clea ha xαHβ
αh(x) is non-dec easing and x−βHβ
ah(x) is non-
inc easing whene e h∈L+. Tha is, Hβ
αL+⊆Ωα,β. I also makes i
easy o check ha
∞
0
(Hβ
αh1)h2=∞
0
h1(Hα
βh2),h
1,h
2∈L+.(2.2)
P oposi ion 2.1. Suppose 0<q<1and U, V ∈L+.I Vis non-
inc easing and C0is he leas C o which
∞
0x
0
hq
U(x)dx1/q
≤C∞
0
hV, h ∈L+,
hen
(1−q)(1−q)/qC0≤∞
0
Vq/(q−1)(H0U)q/(1−q)U(1−q)/q
≤C0/(q(1−q)).
I Vis non-dec easing and C∞is he leas C o which
∞
0∞
x
hq
U(x)dx1/q
≤C∞
0
hV, h ∈L+,
hen
(1−q)(1−q)/qC∞≤∞
0
Vq/(q−1)(H0U)q/(1−q)U(1−q)/q
≤C∞/(q(1−q)).
P oo : The es ima e o C0is om [16, Theo em 3.3] and he one o C∞
ollows om he fi s by in e sion (x→1/x) on he hal line.
Conca e Func ions and Lo en z Duals 493
These weigh ed Ha dy inequali ies can be combined o gi e a weigh
cha ac e iza ion o he boundedness o he L1
→Lq
uembedding o a
sub-cone o he quasi-conca e unc ions. This sub-cone is he image L+
unde he map H1
0. No e ha
H1
0h(x)=∞
0
min(1, x/ )h(x)dx =x
0∞
y
h( )d
dy
is non-dec easing and conca e o all h∈L+. In pa icula , H1
0his
quasi-conca e.
Theo em 2.2. I 0<q<1and u, ∈L+ hen
sup
∈H1
0L+
 q,u
 1,
≈∞
0
(H0
1 )q/(q−1)(H0
qu)q/(1−q) (1−q)/q
.(2.3)
Mo e p ecisely, i he abo e equi alence is C≈A hen m(q)A≤C≤
M(q)Awhe e
m(q) = min(2−1,21−1/q)q(1 −q)and
M(q) = max(21/q−1,2)(1 −q)1−1/q.
(2.4)
P oo : We p o e only he equi alence and lea e he ca e ul acking o
cons an s o he in e es ed eade . The sup emum in (2.3) abo e is he
leas cons an C o which
∞
0
(H1
0h)qu1/q
≤C∞
0
(H1
0h) , h ∈L+.(2.5)
Since
∞
0
(H1
0h) =∞
0
h(H0
1 )
he inequali y (2.5) may be ew i en as
∞
0x
0
h( )d +x∞
x
h( )d
q
u(x)dx1/q
≤C∞
0
h( )H0
1 ( )d .
By (1.2),
C≈C0+C∞
(2.6)
whe e C0and C∞a e he leas cons an s o which
∞
0
0
h( )d q
u(x)dx1/q
≤C0∞
0
h( )H0
1 ( )d , h∈L+,(2.7)

494 G. Sinnamon
and
∞
0x
∞
x
h( )d
q
u(x)dx1/q
≤C∞∞
0
h( )H0
1 ( )d , h∈L+,(2.8)
hold, espec i ely. Since H0
1 is non-inc easing, he fi s pa o P opo-
si ion 2.1, wi h V=H0
1 and U=u, applied o (2.7) shows ha
C0≈∞
0
(H0
1 )q/(q−1)(H0u)q/(1−q)u(1−q)/q
.
To es ima e C∞we eplace h( )/ by h( ) in (2.8) and apply he second
pa o P oposi ion 2.1, wi h V( )= H0
1 ( ) and U(x)=xqu(x). No e
ha H0
1 ( ) is non-dec easing. We ge
C∞≈∞
0
(H0
1 )q/(q−1)(Hqu)q/(1−q)u(1−q)/q
.
Adding he las wo es ima es and appealing o (2.6) yields
C≈∞
0
(H0
1 )q/(q−1)(H0
qu)q/(1−q)u(1−q)/q
which comple es he p oo .
The connec ion be ween he cone o quasi-conca e unc ions and he
sub-cone H1
0L+is well unde s ood. The nex lemma se s ou he ea u es
o his ela ionship ha we equi e he e.
Lemma 2.3. Le be a quasi-conca e unc ion and le ˜
be he leas
conca e majo an o . Then 1
2˜
≤ ≤˜
and ˜
is he poin wise limi
o an inc easing sequence o unc ions in H1
0L+.
P oo : The defini ion o quasi-conca e in [2, Defini ion 2.5.6] is sligh ly
s onge han he one we gi e he e, equi ing ha also sa is y (x)=0
i and only i x= 0. Howe e , i is easy o see ha only he ze o unc ion
is los by his es ic ion. Thus, [2, P oposi ion 2.5.10] applies and we
see ha a quasi-conca e unc ion sa isfies 1
2˜
≤ ≤˜
.
Since ˜
is non-nega i e and conca e, we see ha a= limx→0 (x) and
b= limx→∞ (x)/x exis and a e non-nega i e. We may he e o e w i e
˜
(x)=a+bx+g(x) whe e gis a non-nega i e, conca e unc ion sa is ying
limx→0g(x) = limx→∞ g(x)/x = 0. I we ake hn( )=anχ(0,1/n)( )
hen H1
0hn(x) is a non-dec easing sequence which con e ges poin wise
o he cons an unc ion aas n→∞. I we ake hn( )=b χ(n,n+1)( )
hen H1
0hn(x) is a non-dec easing sequence which con e ges poin wise
o he unc ion bx as n→∞. To comple e he p oo i emains o
Conca e Func ions and Lo en z Duals 495
show ha gis also he poin wise limi o a non-dec easing sequence o
unc ions in H1
0L+.
The conca e unc ion g(x) has a de i a i e o almos e e y x,g(x)
is non-inc easing and since limx→0g(x) = limx→∞ g(x)/x = 0 we ha e
g(x)=x
0g( )d and limx→∞ g(x) = 0. Se
hn( )=(g( )−g((n+1) /n))/log((n+1)/n)
and check ha
∞
y
hn( )d
=(n+1)y/n
y
g( )d
(n+1)y/n
y
d
.
These a e ages o g o m a non-dec easing sequence indexed by nwhich
con e ges o g(y) o almos e e y y. I ollows ha he unc ions
H1
0hn(x)=x
0∞
y
hn( )d
dy
o m a non-dec easing sequence in H1
0L+which, by he Mono one Con-
e gence Theo em, con e ges o
x
0
g(y)dy =g(x).
This comple es he p oo .
Wi h his, Theo em 2.2 ex ends o he quasi-conca e unc ions.
Co olla y 2.4. Suppose 0<q<1and u, ∈L+.
sup
∈Ω0,1
 q,u
 1,
≈∞
0
(H0
1 )q/(q−1)(H0
qu)q/(1−q)u(1−q)/q
.
Mo e p ecisely, i he abo e equi alence is C≈A hen m(q)A≤C≤
2M(q)Awhe e mand Ma e gi en by (2.4).
P oo : The lowe bound equi es only he obse a ion ha H1
0L+⊆
Ω0,1. Fo he uppe bound we apply Lemma 2.3 o choose a non-
dec easing sequence no unc ions in H1
0L+which con e ges poin wise
o he leas conca e majo an ˜
o . By Theo em 2.2 and he Mono one
Con e gence Theo em,
 q,u ≤˜
q,u = lim
n→∞  nq,u ≈lim
n→∞  n1, =˜
1, ≤2 1, .
The main ad an age o wo king wi h Ω0,1 a he han H1
0L+is his
simple obse a ion: Suppose p>0.
I (x)p=g(xp) hen ∈Ω0,1i and only i g∈Ω0,1.(2.9)
496 G. Sinnamon
This gi es us he means o in oducing Lp-no ms in o he denomina o .
Lemma 2.5. Suppose p, q ∈(0,∞)and u, ∈L+. Then
sup
∈Ω0,1
 q,u
 p,
=sup
g∈Ω0,1
gq/p,U
g1,V 1/p
whe e Vand Ua e defined by
V(xp)dxp= (x)dx and U(xp)dxp=u(x)dx.(2.10)
P oo : The subs i u ion in (2.9) yields he equi alence. We no e ha U
and Vha e been defined so ha a change o a iable yields  p
q,u =
gq/p,U and  p
p, =g1,V .
Now we a e eady o gi e ou es ima e o he no m o he Lp
→Lq
u
embedding o he cone o quasi-conca e unc ions.
Theo em 2.6. Suppose ha 0<q<p<∞,1/ =1/q −1/p, and
u, ∈L+. Then
sup
∈Ω0,1
 q,u
 p,
≈∞
0
(H0
p )− /p(H0
qu) /pu1/
.(2.11)
Mo e p ecisely, i he equi alence is C≈A hen m(q/p)1/pA≤C≤
(2M(q/p))1/pAwhe e mand Ma e defined by (2.4).
P oo : Lemma 2.5 educes he p oo o an applica ion o Co olla y 2.4
wi h q eplaced by q/p and uand eplaced by he weigh s Uand V
om (2.10). Tha is,
sup
∈Ω0,1
 q,u
 p,
=sup
g∈Ω0,1
gq/p,U
g1,V 1/p
≈∞
0
H0
1V( )− /pH0
q/pU( ) /pU( )d 1/
.
No e ha (q/p)/(1 −q/p)= /p. We simpli y his by making he sub-
s i u ion → pand using (2.10) o ob ain
∞
0
H0
1V( p)− /pH0
q/pU( p)− /pu( )d 1/
.(2.12)
Conca e Func ions and Lo en z Duals 497
Now we make he subs i u ion x→xpin he in eg al o ms o H0
1Vand
H0
q/pUand use (2.10) again o ge
H0
1V( p)=∞
0
min(x/ p,1)V(x)dx
=∞
0
min((x/ )p,1) (x)dx =H0
p ( )
and
H0
q/pU( p)=∞
0
min((x/ p)q/p,1)U(x)dx
=∞
0
min((x/ )q,1)u(x)dx =H0
qu( ).
Replacing hese in (2.12) comple es he p oo o equi alence and we omi
he acking o cons an s.
Theo em 2.6 is eadily ex ended o a esul o mo e gene al cones han
he quasi-conca e unc ions. Recall ha Ωα,β is he collec ion o non-
nega i e unc ions such ha xα (x) is non-dec easing and x−β (x)is
non-inc easing.
Theo em 2.7. Suppose ha 0<q<p<∞,1/ =1/q −1/p, and
u, ∈L+.I α+β>0and Hβ
αL+⊆F⊆Ωα,β hen
sup
∈F
 q,u
 p,
≈∞
0
(Hpα
pβ )− /p(Hqα
qβ u) /pu1/
.(2.13)
Mo e p ecisely, i he abo e equi alence is C≈A hen (1/2)m(q/p)1/pA≤
C≤(2M(q/p))1/pAwhe e mand Ma e defined by (2.4).
P oo : Se ρ=1/(α+β) and o each ∈F define g by
g (x)=xαρ (xρ).
Se F0,1={g : ∈F}and no e ha o each ∈F,g (x) is non-
dec easing and g (x)/x is non-dec easing. Thus F0,1⊆Ω0,1. Also, i
=Hβ
αh o some h∈L+ hen he change o a iable → ρyields
g (x)=∞
0
min(1, x/ )[ αρh( ρ)ρ ρ−1]d
so g ∈H1
0L+.ThusH1
0L+⊆F
0,1⊆Ω0,1.
504 G. Sinnamon
Rema k. I is no difficul o see ha V0is non-ze o i and only i L1
λ⊆
Γp,λ( ) and V∞is non-ze o i and only i L∞
λ⊆Γp,λ( ). This explains
he appea ance o he e ms in ol ing g∗∞and g∗1and shows ha ,
despi e hei appea ance as echnical byp oduc s o in eg a ion by pa s
in Theo em 3.1, hey a e an essen ial ea u e o he heo y.
P oo : P o ing Theo em 4.1 will occupy us o he es o his sec ion.
The e a e ou s eps in he p oo :
1. Reduc ion o he case ha λis Lebesgue measu e on (0,∞).
2. P oo in he case ha g∗is an in eg al.
3. P oo in he case ha he associa e no m o gis fini e.
4. Elimina ion o he emaining case.
The fi s s ep is eadily accomplished by appealing o he Luxembu g
Rep esen a ion Theo em. Obse e ha Γp( ) ep esen s he no m Γp,λ( )
in he sense o [2, Theo em 2.4.10]. Tha is,
 Γp,λ( )= ∗Γp( ) o all ∈Γp,λ( ).
I ollows ha he associa e no m is ep esen ed in he same way so
gΓp,λ( )=g∗Γp( ) o all g∈Γp,λ( ).
In iew o his is i enough o p o e Theo em 4.1 in he case ha λis
Lebesgue measu e on (0,∞).
The second s ep is o p o e he heo em in he case ha g∗is an
in eg al, specifically ha
g∗( )=∞
u(x)dx
x
o some u∈L+. In his case we ha e
gΓp( )= sup
∈Γp( )∞
0 g
 Γp( )
= sup
∈L+∞
0 ∗g∗
 ∗∗p,
= sup
∈L+∞
0 ∗∗u
 ∗∗p,
= sup
F∈F ∞
0Fu
Fp,
whe e F={ ∗∗ : ∈L+}. Since x ∗∗(x)=x
0 ∗is non-dec easing
and ∗∗(x) is non-inc easing we see ha F⊆Ω1,0. On he o he hand,
le h∈L+and se (y)=∞
yh o see ha
H0
1h(x)=∞
0
min( /x, 1)h( )d =1
xx
0∞
y
h( )d dy = ∗∗(x).

Conca e Func ions and Lo en z Duals 505
I ollows ha H0
1L+⊆F⊆Ω1,0so we may apply Theo em 3.1 wi h
q=1, =p,α= 1, and β= 0 o ge
gΓp( )≈∞
0
H1u( )pHp
0 ( )−pH0 ( )d
1/p
+x−11,u
x−1p,
+∞
0
H0u( )pHp
0 ( )−pHp ( )d
1/p
+11,u
1p,
.
The e ms abo e in ol ing ucan all be w i en in e ms o g∗.
x−11,u =∞
0
u(x)dx
x=g∗(0) = g∗∞.
11,u =∞
0
u(x)dx =∞
0∞
u(x)dx
xd =∞
0
g∗( )d =g∗1.
H1u( )= ∞
u(x)dx
x= g∗( ).
H0u( )=
0
u(x)dx =
0
y
u(x)dx
xdy
=
0
g∗(y)−g∗( )dy = (g∗∗( )−g∗( )).
These subs i u ions gi e he desi ed esul in he case ha g∗is an
in eg al. The second s ep is comple e.
We now pass o he hi d s ep and assume ha gΓp( )<∞. The
fi s hing o es ablish is ha lim →∞ g∗( ) = 0. Fo each posi i e in e-
ge nse n=1
nχ(0,n)and no e ha ∗∗
n( ) = min(1/n, 1/ ). By (4.1)
and he Domina ed Con e gence Theo em,  nΓp( )→0asn→∞.
Since ghas fini e Γp( )-no m we see ha 1
nn
0g∗=∞
0 ∗
ng∗also
ends o ze o as n→∞. Because g∗is mono one his implies ha
lim →∞ g∗( ) = 0 as desi ed.
Now o γ>1 define
uγ(x)=(g∗(x)−g∗(γx))/log(γ) and gγ( )=∞
uγ(x)dx
x.
No e ha g∗
γ=gγ. The esul s o S ep 2 apply so we ha e
gγΓp( )≈g∗
γp, 0+g∗∗
γ−g∗
γp, ∞+V0g∗
γ∞+V∞g∗
γ1.(4.4)
506 G. Sinnamon
Using he ac ha lim →∞ g∗( ) = 0 we can exp ess gγas a mo ing
a e age o g∗:
gγ( )= 1
log(γ)∞
g∗(x)−g∗(γx)dx
x=γ
g∗(x)dx
xγ
dx
x.
I ollows ha o each ,gγ( ) is non-dec easing as γdec eases o 1 and
ha gγ( ) con e ges o g∗( ) o almos e e y .
By he Mono one Con e gence Theo em, we ha e
lim
γ↓1g∗
γp, 0+V0g∗
γ∞+V∞g∗
γ1=g∗p, 0+V0g∗∞+V∞g∗1.
Because Γp( )is a Banach Func ion Space we also ha e
lim
γ↓1gγΓp( )=g∗Γp( )=gΓp( ).
In o de o conclude ha (4.2) holds we s ill need o show ha
lim
γ↓1g∗∗
γ−g∗
γp, ∞=g∗∗ −g∗p, ∞.(4.5)
I is e iden ha he poin wise limi o g∗∗
γ−g∗
γis g∗∗ −g∗. By he
Domina ed Con e gence Theo em, (4.5) will ollow once we show ha
2 log(2)(g∗∗
2−g∗
2)isinLp
∞and domina es g∗∗
γ−g∗
γ o 1 <γ≤2.
Since g∗
2≤g∗and g∗Γp( )<∞we ha e g∗
2Γp( )<∞because
Γp( )is a Banach Func ion Space. In iew o (4.4) his implies ha
g∗∗
2−g∗
2p, ∞<∞and hence 2 log(2)(g∗∗
2−g∗
2)isinLp
∞.
To see ha 2 log(2)(g∗∗
2−g∗
2) domina es g∗∗
γ−g∗
γwe calcula e as
ollows:
log(γ)(g∗∗
γ( )−g∗
γ( ))
=1

0γy
y
g∗(x)dx
xdy −γ
g∗(x)dx
x
=1

0
g∗(x)x
x/γ
dy dx
x+1
γ
g∗(x)
x/γ
dy dx
x−γ
g∗(x)dx
x
=(1−1/γ)g∗∗( )−1
γ γ
g∗(x)dx
=(1−1/γ)g∗∗( )−1
γ − γ
g∗.
Conca e Func ions and Lo en z Duals 507
I 1 <γ≤2 hen 1 −1/γ ≤log(γ). Also, o each he mo ing
a e age 1
γ − γ
g∗is a non-inc easing unc ion o γ.Thus
g∗∗
γ( )−g∗
γ( )=1−1/γ
log(γ)g∗∗( )−1
γ − γ
g∗
≤g∗∗( )−1
2 − 2
g∗
= 2 log(2)(g∗∗
2( )−g∗
2( )).
This comple es S ep 3, showing ha (4.2) holds whene e i s le hand
side is fini e.
I bo h sides a e infini e hen (4.2) holds i ially. S ep 4 o he p oo
is o elimina e he emaining case by showing ha i he igh hand side
o (4.2) is fini e hen so is he le hand side. Fo each posi i e in ege n,
define gn= min(nχ(0,n),g∗) and no e ha g∗
n=gn. The sequence g∗
n
is non-dec easing and con e ges poin wise o g∗as n→∞so gn→g
in he Banach Func ion Space Γp( ). To show ha gΓp( )<∞we
show ha he no ms gnΓp( )a e bounded independen ly o n.Todo
his we no e ha (4.1) implies ha
gnΓp( )≤nχ(0,n)Γp( )<∞
so he esul s o S ep 3 apply and we ha e
gnΓp( )≈g∗
np, 0+g∗∗
n−g∗
np, ∞+V0g∗
n∞+V∞g∗
n1.
Again i is easy o handle h ee o he e ms. Since g∗
n≤g∗we ha e
g∗
np, 0≤g∗p, 0,g∗
n∞≤g∗∞, and g∗
n1≤g∗1. The e o e,
he sum o hese h ee e ms is bounded independen ly o nby he igh
hand side o (4.2) which is assumed o be fini e.
The ou h e m, g∗∗
n−g∗
np, ∞, is also bounded by a mul iple o he
igh hand side o (4.2) bu a li le mo e wo k is equi ed o demons a e
his. The unc ion g∗
nis non-inc easing and bounded by n. The e o e
i akes he alue non an in e al o he o m (0,
n) o some n≥0.
When 0 < <
nwe ha e g∗∗
n( )−g∗
n( ) = 0. When n< <nwe ha e
g∗∗
n( )−g∗
n( )=g∗∗
n( )−g∗( )≤g∗∗( )−g∗( ). When >nwe ha e
g∗∗
n( )−g∗
n( )=g∗∗
n( )=1

0
g∗
n≤1
n
0
g∗=n
g∗∗(n).
Thus
g∗∗
n−g∗
np, ∞≤g∗∗ −g∗p, ∞+ng∗∗(n)∞
n
∞( )d
p1/p
508 G. Sinnamon
and ou objec is o show ha he las wo summands a e bounded by
he igh hand side o (4.2). The fi s is i ially so and we w i e he
second as
n(g∗∗(n)−g∗(n))∞
n
∞( )d
p1/p
+ng∗(n)∞
n
∞( )d
p1/p
.(4.6)
Obse e ha (g∗∗( )−g∗( )) = 
0g∗∗(y)−g∗( )dy is non-dec easing so
n(g∗∗(n)−g∗(n)) ∞
n
∞( )d
p1/p
≤∞
n
(g∗∗( )−g∗( ))p ∞( )d 1/p
≤g∗∗ −g∗p, ∞.
The second e m in (4.6) equi es some in eg a ion using (4.3).
np∞
n
∞( )d
p=−np
p
0
(x)dx + p∞
(x)dx
xp1−p
∞
n
≤1
p1
npn
0
(x)dx +∞
n
(x)dx
xp1−p
=n
0
0( )d +1
p∞
0
(x)dx
xp1−p
.
The e o e,
ng∗(n)∞
n
∞( )d
p1/p
≤n
0
g∗( )p 0( )d +1
pg∗p
∞∞
0
(x)dx
xp1−p1/p
≤g∗p
p, 0+1
pg∗p
∞Vp
01/p
.
Which is bounded by (a mul iple o ) he igh hand side o (4.2). This
comple es S ep 4 and he p oo o Theo em 4.1.
Conca e Func ions and Lo en z Duals 509
We ema k ha he e m g∗∗ −g∗p, ∞in (4.2) may be eplaced by
sup
h∗≤g∗
h∗∗ −h∗p, ∞.
Al hough his new e m may be subs an ially la ge han g∗∗ −g∗p, ∞
o example when g∗is cons an , he equi alence (4.2) is no affec ed due
o he p esence o he o he e ms. Indeed, he p oo o Theo em 4.1 is
simple wi h he new e m in place.
5. The Ha dy-Li lewood Maximal Func ion
The educ ion p inciple in Theo em 3.2 can be used o gi e c i e ia
o de e mine whe he o no he Ha dy-Li lewood Maximal Func ion
is bounded be ween Lo en z spaces. I is a locally in eg able unc ion
on Rnwe define M o be
M (x) = sup 1
µn(Q)Q
| |dµn
whe e he sup emum is aken o e all cubes Qcon aining xwhose sides
a e pa allel o he axes. He e µndeno e Lebesgue measu e on Rn.
Theo em 5.1. Suppose p, q ∈(1,∞)and u, ∈L+. Define Vby
V( )= 1
p
0
(x)dx +∞
(x)dx
xp.
Then M:Γ
p,µn( )→Γq,µn(u)i and only i : Ei he 1<p≤q<∞,
and all o
sup
y>0∞
y
u(x)dx
xq1/q y
0
(log(y/ ))p−1V( )1−pd
1/p
,
sup
y>0∞
y
(log(x/y))qu(x)dx
xq1/q
V(y)−1/p,
sup
y>0∞
y
u(x)dx
xq1/q py
0
V( )1−pd
−V(y)1−p1/p
,and
sup
y>0y
0
u(x)dx1/q
(ypV(y))−1/p ,

510 G. Sinnamon
a e fini e; o 1<q<p<∞,1/ =1/q −1/p, and all o
∞
0∞
y
u(x)dx
xq /p y
0
(log(y/ ))p−1V( )1−pd
 /p
u(y)dy
yq,
∞
0∞
y
(log(x/y))qu(x)dx
xq /q
V(y)− /q d(−V(y)),
∞
0∞
y
u(x)dx
xq /qp
y
0
V( )1−pd
−V(y)1−p /q
d(ypV(y))
ypV(y)p,and
∞
0y
0
u(x)dx /p
(ypV(y))− /p u(y)dy
a e fini e.
P oo : We ci e [2, Theo em 3.8] o he well known equi alence (M )∗≈
∗∗. I implies ha M:Γ
p,µn( )→Γq,µn(u) i and only i
sup
∈L+∞
01
xx
0 ∗∗qu(x)dx1/q
∞
0( ∗∗)p 1/p <∞.
Tha is,
T:F∩Lp
→Lq
u
(5.1)
whe e Tis he ope a o TF(x)=1
xx
0Fand F={ ∗∗ : ∈L+}.As
we obse ed in Pa 2 o he p oo o Theo em 4.1, H0
1L+⊆F⊆Ω1,0.
Thus, we can apply Theo em 3.2 wi h α=1,β= 0, and Ls
w=Lq
u o
see ha (5.1) holds i and only i
TH1:Lp
1→Lq
u,(5.2)
TH0:Lp
2→Lq
u,(5.3)
i x−1∈Lp
hen T(x−1)∈Lq
u,and
i 1 ∈Lp
hen T(1) ∈Lq
u.
Conca e Func ions and Lo en z Duals 511
Since T(x−1)≡∞and T(1) ≡1 he la e wo condi ions educe o
u≡0o ∞
0
(x)dx
xp=∞,(5.4)
and
∞
0
<∞=⇒∞
0
u<∞.(5.5)
The condi ions (5.2) and (5.3) educe o weigh ed no m inequali ies o
which necessa y and sufficien condi ions a e known. Ou ask now is
o simpli y he known condi ions using he defini ions o 1and 2 om
Theo em 3.2. We ha e
1( )= p−1
0
(x)dx + p∞
(x)dx
xpp
0
(x)dx1−p
and
2( )= p−1
0
(x)dx + p∞
(x)dx
xpp p∞
(x)dx
xp1−p
.
In e ms o V hese become
p p 1( )1−p=V( )−pd
d (−V( )) and(5.6)
p 2( )1−p=( pV( ))−pd
d ( pV( )).(5.7)
The ope a o in (5.2) is
TH1 (x)= 1
xx
0
1
yy
0
( )d dy =1
xx
0
log(x/ ) ( )d
so, wi h g( )= ( ), we see ha (5.2) holds i and only i he inequali y
∞
0x
0
log(x/ )g( )d q
u(x)dx
xq1/q
≤C∞
0
g( )p 1( )d
p1/p
(5.8)
holds o some C>0 and all g∈L+.By[18, Theo ems 1 and 2], (5.8)
holds i and only i : Ei he 1 <p≤q<∞,
sup
y>0∞
y
u(x)dx
xq1/qy
0
(log(y/ ))p p 1( )1−pd 1/p
<∞,(5.9)
and
sup
y>0∞
y
(log(x/y))qu(x)dx
xq1/qy
0
p 1( )1−pd 1/p
<∞;(5.10)
512 G. Sinnamon
o 1 <q<p<∞,1/ =1/q −1/p,
∞
0∞
y
u(x)dx
xq /p y
0
(log(y/ ))p p 1( )1−pd  /p
u(y)dy
yq<∞,
(5.11)
and
∞
0∞
y
(log(x/y))qu(x)dx
xq /q
y
0
p
1( )1−pd  /q
yp
1(y)1−pdy<∞.
(5.12)
The ope a o in (5.3) is
TH0 (x)= 1
xx
0∞
y
( )d dy =1
xx
0
( )d +∞
x
( )d ,
a sum o wo Ha dy ope a o s. Thus (5.3) holds i and only i he wo
weigh ed Ha dy inequali ies
∞
0x
0
g( )d q
u(x)dx
xq1/q
≤C∞
0
g( )p 2( )d
p1/p
and
∞
0∞
x
( )d q
u(x)dx1/q
≤C∞
0
( )p 2( )d 1/p
hold o some cons an C>0 and all g∈L+and ∈L+ espec i ely.
The condi ions (see [12]) unde which hese hold a e: Ei he 1 <p≤
q<∞,
sup
y>0∞
y
u(x)dx
xq1/q y
0
p 2( )1−pd 1/p
<∞,(5.13)
and
sup
y>0y
0
u(x)dx1/q ∞
y
2( )1−pd 1/p
<∞;(5.14)
o 1 <q<p<∞,1/ =1/q −1/p,
∞
0∞
y
u(x)dx
xq /q
y
0
p
2( )1−pd  /q
yp
2(y)1−pdy<∞,(5.15)
Conca e Func ions and Lo en z Duals 513
and
∞
0y
0
u(x)dx /p
∞
y
2( )1−pd  /p
u(y)dy<∞.(5.16)
Using he p ope ies (5.4) and (5.5) and he subs i u ions (5.6) and
(5.7) o elimina e 1and 2, (5.9), (5.10), (5.13), and (5.14) can be
simplified o yield he ou weigh condi ions gi en in he case 1 <p≤
q<∞. Simila ly, (5.11), (5.12), (5.15), and (5.16) simpli y o yield he
ou weigh condi ions gi en in he case 1 <q<p<∞.
We ha e shown ha he weigh condi ions gi en in he s a emen o
he heo em, oge he wi h (5.4) and (5.5), a e necessa y and sufficien
o he boundedness o M. All ha emains is o show ha (5.4) and
(5.5) a e consequences o he weigh condi ions.
W i e V( )=∞
0max( , x)−p (x)dx o see ha V( )≤V(0) =
∞
0 (x)dx/xp.I V(0) <∞ hen o any y>0,
y
0
log(y/ )p−1V( )1−pd
≥V(0)1−py
0
log(y/ )p−1d
=∞.
In iew o his, he fi s weigh condi ion in ei he he case 1 <p≤q<∞
o he case 1 <q<p<∞can hold only i uis almos e e ywhe e 0.
Thus (5.4) holds.
I ∞
0 <∞i ollows ha ypV(y) is bounded abo e and hence
he ou h weigh condi ion in ei he he case 1 <p≤q<∞o he
case 1 <q<p<∞would ail unless ∞
0u<∞. Thus (5.5) also holds.
This comple es he p oo .
We would like o hank he e e ee o poin ing ou ha he weigh
condi ions (5.4) and (5.5) ollow om he o he s in Theo em 5.1.
Since (M )∗≈ ∗∗ he boundedness o M:Γ
p,µn( )→Λq,µn(u) e-
duces o a s aigh o wa d applica ion o Theo ems 2.7 and 2.8 wi h
α= 1 and β= 0. He e Λq,µn(u)={ : ∗q,u <∞}.
Theo em 5.2. Le p, q ∈(1,∞)and u, ∈L+. Then M:Γ
p,µn( )→
Λq,µn(u)i and only i : Ei he 1<p≤q<∞and
sup
y>0y
0
u(x)dx+yq∞
y
u(x)dx
xq1/qy
0
(x)dx+yp∞
y
(x)dx
xp−1/p
is fini e; o 1<q<p<∞,1/ =1/q −1/p, and
∞
0y
0
u(x)dx+yq∞
y
u(x)dx
xq /p
y
0
(x)dx+yp
∞
y
(x)dx
xp− /p
u(y)dy
is fini e.