Hardy spaces of Dirichlet series and pseudomoments of the Riemann zeta function

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by  Andriy Bondarenko

HARDYSPACES OFDIRICHLETSERIESAND
PSEUDOMOMENTS OF THE RIEMANNZETAFUNCTION
ANDRIYBONDARENKO,OLEFREDRIKBREVIG,EEROSAKSMAN,KRISTIANSEIP,ANDJINGZHAO
7
1
0 ABSTRACT. WestudyHp spacesofDirichletseries,calledHp,for0 p .Webeginbyshow-
2 ingthatHp maybedefinedeitherbytakinganappropriateLp clo<sure<o∞fallDirichletpolyno-

n mials orby requiringthe sequence of“mteAbschnitte” to beuniformly bounded inLp. After
a showingthatthesedefinitionsareequivalent,weproceedtoestablishupperandlowerweighted
J
ℓ2estimates(calledHardy–Littlewoodinequalities)aswellasweightedℓ∞estimatesforthecoef-
4 ficientsoffunctionsinHp.Wediscusssomeconsequencesoftheseestimatesandobservethat
2
theHardy–LittlewoodinequalitiesdisplaywhatwewillcallacontractivesymmetrybetweenHp

] andH4/p. TherelevanceoftheHardy–Littlewoodinequalitiesforthestudyofthedualspaces
A (Hp) isillustratedbyaresultaboutthelinearfunctionalsgeneratedbyfractionalprimitivesof
∗
F theRiemannzetafunction.Wededucegeneralestimatesofthenormofthepartialsumoperator
h. ∞n 1ann−s 7→ nN 1ann−s onHp with0<p≤1,supplementingaclassicalresultofHelsonfor
t the=range1 p =.Finally,wediscusstherelevanceofourresultsforthecomputationoftheso-
a P < P<∞
calledpseudomomentsoftheRiemannzetafunctionζ(s)(inthesenseofConreyandGamburd).
m
WeapplyourupperHardy–Littlewoodinequalitytoimproveonanearlierasymptoticestimate
[
whenp ,butatthesametimeweshow,usingideasfromrecentworkofHarper,Nikeghbali,
→∞
1 andRadziwiłłandsome probabilisticestimates ofHarper, thattheHardy–Littlewoodestimate
v forp 1failstogivetherightasymptoticsforthepseudomomentsofζα(s)forα 1.
2 < >
4
8
6 1. INTRODUCTION
0
. Hp spacesofDirichletseries,tobecalledHp inwhatfollows,havebeenstudiedextensively
1
0 in recentyears butmostlyin theBanach spacecase p 1, with a view totheoperatorsacting
≥
7 on them. In thepresent paper, we exploreHp in the full range0 p , which in part can
1 < <∞
begivenanumbertheoreticmotivation:Theinterplaybetweentheadditiveandmultiplicative
:
v
structureoftheintegersisdisplayedinamoretransparentwaybytheresultsobtainedwithout
i
X anya priorirestrictionontheexponent p. As an example,wementionthatthemultiplicative
r estimatesofSection3ofthispaperexhibitwhatwewillcallacontractivesymmetrybetweenHp
a
and H4/p, whichis particularlysignificantforthestudyofHp. Werefer to theseestimatesas
multiplicativebecausetheyareobtainedbymultiplicativeiterationviatheBohrlift(seebelow)
ofestimatesfor Hp spacesoftheunitdisc. Wenoteinpassingthat,surprisingly,thereremain
basic problems related to the contractive symmetry that are still open in the case of the unit
disc.
By a basicobservationofBohr, themultiplicative structure oftheintegersallowsus to view
anordinaryDirichletseriesoftheform
f(s) ∞ ann−s
=
n 1
X=
TheresearchofBondarenko,Brevig,Seip, andZhaoissupportedbyGrant227768oftheResearchCouncilof
Norway.
1
2 ANDRIYBONDARENKO,OLEFREDRIKBREVIG,EEROSAKSMAN,KRISTIANSEIP,ANDJINGZHAO
asafunctionofinfinitelymanyvariables.Indeed,bythetransformationz p s (herep isthe
j = −j j
jthprimenumber)andthefundamentaltheoremofarithmetic,wehavetheBohrcorrespon-
dence,
(1) f(s): ∞ a n s Bf(z): ∞ a zκ(n),
n − n
= ←→ =
n 1 n 1
X= X=
where we use multi-index notation and κ(n) (κ ,...,κ ,0,0,...) is the multi-index such that
1 j
=
n pκ1 pκj. This transformation—theso-called Bohr lift—givesan isometricisomorphism
= 1 ··· j
between Hp and the Hardy space Hp(D ). We will come back to the details of this relation
∞
in the next section, where we will show that it ensures an unambiguous definition of Hp in
thefullrange0 p . TheBohrliftisoffundamentalimportanceinoursubject,andwillin
< <∞
particularbewhatweneedinSection3andSection4toliftcoefficientestimatesinonecomplex
variabletoobtainresultsforHp.
Theadditivestructureoftheintegersplaysarolewheneverwerestrictattentiontotheprop-
ertiesof f(s)viewedasananalyticfunctioninahalf-planeorwhenweconsideranyproblemfor
whichtheorderofsummationmatters. Aparticularlyinterestingexampleisthatofthepartial
sumoperator
N
S f(s): a n s,
N n −
=
n 1
X=
viewedasanoperatoronHp.ByaclassicaltheoremofHelson[30],weknowthatitisuniformly
bounded on Hp when 1 p . In Section 5, we will give bounds that are essentially best
< <∞
possiblein therange 0 p 1 and an improvement by a factor 1/loglogN on the previously
< <
known bounds when p 1. We are however still far from knowing the precise asymptotics of
=
thenormofS whenitactsoneitherH1orH .
N ∞
We have found it interesting to relate our discussion and to apply part of our results to a
numbertheoreticproblemthatdealswiththeinterplaybetweentheadditiveandmultiplicative
structure of the integers. Thus in the final Section 6 we consider the computation of the so-
called pseudomoments of the Riemann zeta function ζ(s) which were studied by Conrey and
Gamburd [15] when p is an even integer. In our terminology, the pseudomomentsof ζ(s) are
pthpowersoftheHp normsoftheDirichletpolynomials
N
Z (s): n 1/2 s.
N − −
=
n 1
X=
Weobservethatifwewrite
1
(2) f (s): ,
N
= 1 p 1/2 s
pYj≤N − −j −
then Z S f . Hence Z canbeobtainedbyapplyingthepartialsumoperatortoaDirich-
N N N N
=
letserieswhosecoefficientsrepresenta completelymultiplicativefunction. Thiscomes asno
surpriseofcourse,buttheinterestingpointishowtoestimatethenormofS f . Wehavees-
N N
sentiallytwomethods,onerelyingonthemultiplicativeestimatesfromSection3andanother
relyingonanadditiveestimateofHelsonusedinSection5.Wewillshowthatourmultiplicative
estimatesimproveontheknownestimatesfrom[8]intherangep 2. Ingeneral,however,we
>
knowtherightorderofmagnitudeonlywhen p 1,therebeingahugegapbetweentheaddi-
>
tiveandmultiplicativeestimatesintherange0 p 1.Wearenotabletoremedythissituation,
< <
but we will shed lighton it by showing that the Nth partialsum of [f (s)]α for α 1 has Hp
N
>
HARDYSPACESOFDIRICHLETSERIESANDPSEUDOMOMENTSOFTHERIEMANNZETAFUNCTION 3
norm of an order of magnitudelarger than what is suggestedby our multiplicativeestimates,
providedthatp issufficientlysmall.
Ourstudyofthepseudomomentsofζ(s)andmoregenerallyζα(s)highlightsanotherimpor-
tantaspectofthespacesHp,namelyaprobabilisticinterpretationoftheBohrcorrespondence
and the use of probabilisticmethods. Our work on pseudomomentsin the range0 p 1 is
< <
inspired by the recent paper [26] and relies cruciallyon some delicate probabilisticestimates
duetoHarper[25].
To close this introduction, we note that there are many questions about Hp that are not
treated or only briefly mentioned in our paper. Our selection of topics has been governed by
what appear to be significant and doable problems for the whole range 0 p . We have
< < ∞
chosentobequitedetailedinthegroundworkinSection2,dealingwiththedefinitionofHp,
because the infinite-dimensional situation and the non-convexity of the Lp quasi-norms for
0 p 1requiresomeextracare. Inthatsection,wealsosummarizebrieflysomeknownfacts
< <
andeasyconsequences,suchasforinstancehowsomeresultsforH2canbetransferredtoHp
wheneitherp 2k orp 1/(2k)fork 2,3,... InSection3,whichdealswithupperandlower
= = =
weighted ℓ2 estimatesfor thecoefficients of functionsin Hp, we will record some functional
analytic consequences concerning respectively duality and local embeddings of Hp into ap-
propriate Bergman spaces when 0 p 2. For further information about known results and
< <
openproblems,werefertothemonograph[37]andtherecentpapers[12,43].
Notation. Wewillusethenotation f(x) g(x)ifthereissomeconstantC 0suchthat f(x)
≪ > | |≤
C g(x) for all (appropriate) x. If we have both f(x) g(x) and g(x) f(x), we will write
| | ≪ ≪
f(x) g(x). If
≍
f(x)
lim 1,
x g(x) =
→∞
thenwewrite f(x) g(x).Asabove,theincreasingsequenceofprimenumberswillbedenoted
∼
by(p ) ,andthesubscriptwillsometimesbedroppedwhentherecanbenoconfusion. The
j j 1
number≥ofprimefactorsinnwillbedenotedbyΩ(n)(countingmultiplicities).Wewillalsouse
thestandardnotations x max{n N:n x}and x min{n N:n x}.
⌊ ⌋= ∈ ≤ ⌈ ⌉= ∈ ≥
2. DEFINITIONS AND BASIC PROPERTIES OF THE HARDY SPACES Hp AND Hp(D∞)
2.1. Definition of Hp(D ). We use the standard notation T : {z : z 1} for the unit circle
∞
= | | =
which is theboundaryof theunitdisc D: {z : z 1} in thecomplexplane, and we equip T
= | |<
withnormalizedone-dimensionalLebesguemeasureµsothatµ(T) 1.Wewriteµ : µ µ
d
= = ×···×
fortheproductofd copiesofµ,whered maybelongtoN { }.
∪ ∞
Webeginbyrecallingthatforeveryp 0,theclassicalHardyspace Hp(D)(alsodenotedby
>
Hp(T))consistsofanalyticfunctions f :D Csuchthat
→
f p : sup f(rz) pdµ(z) .
k kHp(D) = | | <∞
0 r 1ZT
< <
ThisisaBanachspace(quasi-Banachincase0 p 1),andpolynomialsaredenseinHp(D),so
< <
itcouldaswellbedefinedastheclosureofallpolynomialsintheabovenorm(orquasi-norm).
Wereferto[19]orthefirstchaptersof[21]forthedefinitionandbasicpropertiesoftheHardy
spacesonD.
ForthefinitedimensionalpolydiscDd withd 2,thedefinitionofHardyspacescanbemade
≥
inasimilarmanner:Foreveryp 0,afunction f :Dd CbelongstoHp(Dd)whenitisanalytic
> →
4 ANDRIYBONDARENKO,OLEFREDRIKBREVIG,EEROSAKSMAN,KRISTIANSEIP,ANDJINGZHAO
separatelywithrespecttoeachofthevariablesz ,...,z and
1 d
f p : sup f(rz) pdµ (z) .
k kHp(Dd) = r 1ZTd| | d <∞
<
The standard source for these spaces is Rudin’s monograph [41]. As in the one-dimensional
case,foralmosteveryz inTd,theradialboundarylimit
f (z): lim f(rz)
∗
=r 1
→ −
exists,andwemaywrite
(3) f p f (z) pdµ (z).
k kHp(Dd)= Td| ∗ | d
Z
ThismeansthatHp(Dd)isasubspaceofLp(Td,µ ). Moreover,againasintheone-dimensional
d
case,forevery f inHp(Dd),wehavethat
(4) lim f f 0,
r 1 1k − rkHp(Dd)=
→ −
where f (z): f(rz). Thisimpliesthatthepolynomialsaredensein Hp(Dd),sothatthespace
r
=
couldequallywellbedefinedastheclosureofallpolynomialswithrespecttothenormonthe
boundarygivenby(3).
Both(3)and(4)aremostconvenientlyobtainedbyapplyingtheLp-boundednessofthera-
dialmaximalfunctionon Hp(Dd)forallp 0,aresultwhichcanbeobtainedbyconsideringa
>
dummyvariablew inDandcheckingfirstthat,given f inHp(Dd),thefunction
w f(wz ,...,wz )
1 d
7→
lies in Hp(Dd) for almost every (z ,...z ) Td. By Fubini’s theorem, the boundedness of the
1 d
∈
maximalfunctionthenreducestotheclassicalone-dimensionalestimate.
In order to define Hp(D ), some extra care is needed because functions in Hp(D ) will in
∞ ∞
generalnotbewelldefinedinthewholesetD . Tokeepthingssimple,wehenceforthconsider
∞
thesetD whichconsistsofelementsz (z ) D suchthatz 0onlyforfinitelymanyk.
∞fin = j j≥1∈ ∞ j 6=
Afunction f :D CisanalyticifitisanalyticateverypointzinD separatelywithrespectto
∞fin→ ∞fin
eachvariable.Obviouslyanyanalytic f :D CcanbewrittenbyaconvergentTaylorseries
∞fin→
f(z) c zκ, z D ,
= κ ∈ ∞fin
κ∈XN∞fin
andthecoefficientsc determine f uniquely.ThetruncationA f of f ontothefirstmvariables
κ m
A f (called“dermteAbschnitt”byBohr)isdefinedas
m
A f(z ,z ,...) f(z ,...,z ,0,0,...)
m 1 2 1 m
=
foreveryz inD . Byapplyingthefundamentalestimate g(0) g ,obtainedbyiterat-
∞fin | |≤k kHp(Dd)
ingthecased 1,wededucethat
=
(5) kAmfkHp(Dm)≤kAm′fkHp(Dm′)
wheneverm m.
′
≥
Definition. Letp 0. ThespaceHp(D )isthespaceofanalyticfunctionsonD obtainedby
> ∞ ∞fin
takingtheclosureofallpolynomialsinthenorm(quasi-normfor 0 p 1)
< <
f p : f(z) pdµ (z).
k kHp(D∞) = T | | ∞
Z∞
HARDYSPACESOFDIRICHLETSERIESANDPSEUDOMOMENTSOFTHERIEMANNZETAFUNCTION 5
FixacompactsetK inDd andembeditasthesubsetK ofD sothat
∞
K : z (z ,...,z ,0,0,...) D : (z ,...,z ) K .
1 d ∞ 1 d
= = ∈ e ⊂
For all polynomials g wee c©learly havesupz K |g(z)|≤CKkgkHp(D∞). It fªollows that any limitof
polynomials is analytic on D , whence H∈p(D ) is well defined. This also implies that every
∞fin ∞
element f in Hp(D ) has a well-defined Tayelor series f(z) c zκ and, in turn, this Taylor
∞ κ κ
=
seriesdetermines f uniquely. Namely, byrecalling(5), wehavethat A f isin Hp(Dm)forev-
m
P
ery m 1 and the A f are certainly determined by the Taylor series. Finally, by polynomial
m
≥
approximation,itfollowsthat
lim f Amf Hp(D ) 0.
m k − k ∞ =
→∞
Obviously, if a function f in Hp(D ) depends only on the variables z ,...z , then we have
∞ 1 d
kfkHp(D∞)=kfkHp(Dd).
ColeandGamelin[15]establishedanoptimalestimateforpointevaluationson Hp(D )by
∞
showingthat
1/p
1
∞
(6) |f(z)|≤Ãj 11−|zj|2! kfkHp(D∞).
Y=
ThustheelementsintheHardyspacescontinueanalyticallytothesetD ℓ2.
∞
∩
If f is anintegrablefunction(ora Borelmeasure)onT , thenwedenoteitsFouriercoeffi-
∞
cientsby
f(κ): f(z)z¯κdµ (z)
= T ∞
Z∞
for multi-indicesκ in Z . When p 1, it follows directly from the definitionof Hp(D ) that
∞fin b≥ ∞
it can be identified as the analytic subspace of Lp(T ), consisting of the elements in Lp(T )
∞ ∞
whose non-zero Fourier coefficients lie in the positive cone N (called the “narrow cone” by
∞fin
Helson[31]).
Thefollowingresultverifiesthat,alternatively, Hp(D )maybedefined intermsoftheuni-
∞
form boundedness of the Lp-norm of the sequence A f for m 1, and the functions A f
m m
≥
approximate f inthenormofHp(D ).
∞
Theorem2.1. Supposethat0 p andthat f isaformalinfinitedimensionalTaylorseries.
< <∞
Then f isinHp(D )ifandonlyif
∞
(7) sup Amf Hp(Dm) .
k k <∞
m 1
≥
Moreover,forevery f inHp(D∞),itholdsthat Amf f Hp(D ) 0asm .
k − k ∞ → →∞
Proofforthecasep 1. When p 1, thestatementsfollow from thefact that(A f) is ob-
m m 1
viously an Lp-mart≥ingale sequen>ce with respect to the natural sigma-algebras. It fo≥llows in
particular that there is an Lp limit function (still denoted by f) of the sequence A f on the
m
distinguishedboundaryT ,whichhastherightFourierseries,andthedensityofpolynomials
∞
follows immediatelyfrom thefinite-dimensionalapproximation. In thecase p 1, thisfactis
=
stated in [1, Cor. 3], and is derived as consequence of the infinite-dimensionalversion of the
brothers Riesz theorem on the absolute continuity of analytic measures, due to Helson and
Lowdenslager[32](asimplerproofoftheresultfrom[32]isalsocontainedin[1]). Theapproxi-
(cid:3)
mationpropertyofthe A f thenfollowseasily.
m
Thecase0 p 1requiresanewargumentandwillbepresentedinthenextsubsection.
< <
6 ANDRIYBONDARENKO,OLEFREDRIKBREVIG,EEROSAKSMAN,KRISTIANSEIP,ANDJINGZHAO
2.2. ProofofTheorem2.1for0 p 1. OuraimistoproveLemma2.3below,fromwhichthe
< <
claim will follow easily. In an effort to make the computations of this section more readable,
we temporarilyadopttheconventionthat f f , whereitshouldbeclear from the
k kLp(Td) =k kp
contextwhatd is. Westartwiththefollowingbasicestimate.
Lemma2.2. Let0 p 1. ThereisaconstantC suchthatall(analytic)polynomials f on
p
< < <∞
Tsatisfytheinequality
(8) f f(0) p C f p f(0) p f(0) p p2/2 f p f(0) p p/2 .
p p p − p
k − k ≤ k k −| | +| | k k −| |
³ ´
Proof. In this proof, we use repeatedly the elementary ineq¡uality a b p ¢ a p b p, which
| + | ≤| | +| |
is our replacement for the triangleinequality. We see in particular,by this inequalityand the
presenceoftheterm f p f(0) p insidethebracketsontheright-handside,that(8)istrivial
p
k k −| |
if, say, f p (3/2) f(0)2. We may therefore disregard this case and assume that f satisfies
p
k k ≥ | |
p
f(0) 1and f 1 εwithε 1/2.Ouraimistoshowthat,underthisassumption,
p
= k k = + <
(9) f 1 p C εp/2.
p p
k − k ≤
We begin by writing f UI, whereU is an outer function and I is an inner function, such
=
thatU(0) 0. Bysubharmonicityof U p,wehave1 U(0) (1 ε)1/p 1 c ε.Thismeans
p
> | | ≤| |≤ + ≤ +
thatI(0) (1 c ε) 1 1 c ε. Wewrite f 1 (U 1)I I 1andobtainconsequentlythat
p − p
≥ + ≥ − − = − + −
p p p
(10) f 1 U 1 I 1 .
p p p
k − k ≤k − k +k − k
In order to prove (9), it is therefore enough to show that each of the two summands on the
right-handsideof(10)isboundedbyaconstanttimesεp/2.
Webeginwiththesecondsummandontheright-handsideof(10)forwhichweclaimthat
(11) I 1 p C εp/2
k − kp ≤ p′
holds for some constantC . We write I u iv, where u and v are respectively the real and
p′ = +
imaginarypartofI.Since1 u 0,weseethat
− ≥
(12) 1 u (1 u(z))dm(z) 1 I(0) c ε.
1 p
k − k = − = − ≤
T
Z
UsingHölder’sinequality,wethereforefindthat
(13) 1 u p cPεp.
k − kp ≤ p
Inviewof(12)andusingthat I 1and(1 u2) 2(1 u),wealsogetthat
| |= − ≤ −
v p v p 1 u2 p/2 2 1 u p/2 (2c )p/2εp/2.
k kp ≤k k2 =k − k1 ≤ k − k1 ≤ p
Combiningthisinequalitywith(13),wegetthe¡desiredbo¢und(11).
Weturnnexttothefirstsummandontheright-handsideof(10)andtheclaimthat
(14) U 1 p C εp/2
k − kp ≤ p′′
holdsforsomeconstantC . Byorthogonality,wefindthat
p′′
Up/2 U(0)p/2 2 ε
k − k2≤
andhence
(15) Up/2 1 Up/2 U(0)p/2 (U(0)p/2 1)1/2 2ε1/2.
2 2
k − k ≤k − k + − ≤
HARDYSPACESOFDIRICHLETSERIESANDPSEUDOMOMENTSOFTHERIEMANNZETAFUNCTION 7
Since Up/2 1 U p/2 1 (p/2)log U andU(0) 1,thisimpliesthat
| − |≥|| | − |≥ | | ≥
+
(16) log U 2 log U log U(0) 8p 1ε1/2.
1 1 −
k | |k = k | |k − | |≤
+
Itfollowsthat
m z: log U(z) λ 8(pλ) 1ε1/2 and m z: argU(z) λ Cλ 1ε1/2,
− −
| | ||≥ ≤ | |≥ ≤
wherethe¡©latterinequalityisªt¢heclassicalweak-typeL1¡©estimateforthecªo¢njugationoperator.
WenowsplitTintothreesets
E : {z : U(z) 3/2} {z: U(z) 1/2},
1
= | |> ∪ | |<
E : z : 1/2 U(z) 3/2, argU(z) π/4 ,
2
= ≤| |≤ | |≥
E : T\(E E ).
3 © 1 2 ª
= ∪
Itisimmediatefrom(15)that
p
χ (U 1) ε.
k E1 − kp ≪
Sincem(E ) Cε1/2,wehavetriviallythat
2
≤
χ (U 1) p C(5/2)pε1/2.
k E2 − kp ≤
Finally,onE ,wehavethat Up/2 1 U 1 ,andsoitfollowsfrom(15)andHölder’sinequality
3
| − |≃| − |
that
χ (U 1) p εp/2.
k E3 − kp ≪
(cid:3)
Nowthedesiredinequality(14)followsbycombiningthelatterthreeestimates.
Onemaynoticethatthatinthelaststepoftheproofabovewecouldhaveused(16)andthe
fact that the conjugation operator is bounded from L1 to Lp. It seems that the exponent p/2
is the best we can get. It is also curious to note that with p 2/k and k 2 an integer, one
= ≥
couldavoidtheuseoftheweak-typeestimateforargU andgetaveryslickargumentbysimply
observingthatifg Up/2andω ,...,ω arethekthrootsofunity,thenbyHölder’sinequality,
1 k
=
k
U 1 g ω ,
p j 2
k − k ≤ k − k
j 1
Y=
andontherighthandsideoneL2-normisestimatedbyε1/2 andtheothersbyaconstantsince
we are assuming ε 1/2. Again one could raise the question if one can interpolate to get all
≤
exponents.
Lemma2.3. Supposethat0 p 1. Ifg isapolynomialonT ,then
∞
< <
p p p p p2/2 p p p/2
Am kg Amg p Cp Am kg p Amg p Amg p− Am kg p Amg p
k + − k ≤ k + k −k k +k k k + k −k k
³ ´
¡ ¢
holdsforarbitrarypositiveintegersm andk,whereC isasinLemma8.
p
Proof. We set h : A g and view h as a function on Tm Tk so that A g(w,w ) h(w,0).
m k m ′
Nowfixarbitrary=point+sw inTm andw inTk.Weapplythe×precedinglemmatothef=unction
′
f(z): h(w,zw ),
′
=
8 ANDRIYBONDARENKO,OLEFREDRIKBREVIG,EEROSAKSMAN,KRISTIANSEIP,ANDJINGZHAO
whichisananalyticfunctiononD. Thisyields
h(w,zw ) h(w,0) pdµ(z) C h(w,zw ) pdµ(z) h(w,0) p
′ p ′
| − | ≤ | | −| |
ZT µZT
p/2
h(w,0) p p2/2 h(w,zw ) pdµ(z) h(w,0) p .
− ′
+| | | | −| |
µZT ¶ ¶
The claim follows by integrating both sides with respect to (w,w ) over Tm k and applying
′ +
(cid:3)
Hölder’sinequalitytothelasttermontheright-handside.
ProofofTheorem2.1for0 p 1. If f isinHp(D ),thenclearly(7)holds.Toprovethereverse
∞
< <
implication, we start from a formal Taylor series f for which (7) holds. Then by assumption
A f is in Hp(D ), and we have that A (A f) A f whenever m m 1. Therefore the
m ∞ m m m ′
′ = ≥ ≥
quasi-norms Amf Hp(D )constituteanincreasingsequence,andhence(7)impliesthat
k k ∞
lim sup Am kf Hp(D ) Amf Hp(D ) 0.
m→∞k 1 k + k ∞ −k k ∞ =
≥ ¡ ¢
ByLemma2.3,wethereforefindthat(A f) isaCauchysequencein Hp(D ),whence f
m m 1 ∞
lim A f in Hp(D )sinceanelementin H≥ p(D )isuniquelydeterminedbythesequenc=e
m m ∞ ∞
→∞ (cid:3)
A f.
m
2.3. DefinitionofHp. ThesystematicstudyoftheHilbertspaceH2beganwiththepaper[29]
whichdefinedH2tobethecollectionofDirichletseries
f(s) ∞ a n s,
n −
=
n 1
X=
subject to the condition f 2 : a 2 1/2 . The space H2 consists of functions
k kH = ∞n 1| n| < ∞
analytic in the half-plane C : {s σ= it : σ 1/2}, since the Cauchy–Schwarz inequality
1/2 = ¡P= + ¢ >
shows thattheaboveDirichletseriesconverges absolutelyfor thosevalues of s. Bayart[5]ex-
tendedthedefinitiontoeveryp 0bydefiningHp astheclosureofallDirichletpolynomials
>
f(s): N a n s underthenorm(orquasi-normwhen0 p 1)
= n 1 n − < <
=
P 1 T 1/p
(17) f Hp : lim f(it) pdt .
k k =µT→∞2T Z−T | | ¶
Computingthelimitwhenp 2,weseethat(17)givesbacktheoriginaldefinitionofH2.How-
=
ever, atfirstsightitis notclear thattheabovedefinitionofHp is therightoneor thatiteven
yieldsspacesofconvergentDirichletseriesinanyrighthalf-plane.
TheclarificationofthesemattersisprovidedbytheBohrlift(1). ByBirkhoff’sergodictheo-
rem(orbyanelementaryargumentfoundin[44,Sec.3]),weobtaintheidentity
1/p
(18) f Hp Bf Hp(D ): Bf(z) pdµ (z) .
k k =k k ∞ =µZT∞| | ∞ ¶
SincetheHardyspacesontheinfinitedimensionaltorusHp(D )maybedefinedastheclosure
∞
ofanalyticpolynomialsintheLp-normonT ,itfollowsthattheBohrcorrespondencegivesan
∞
isomorphismbetweenthespaces Hp(D )andHp. Thislinearisomorphismisbothisometric
∞
and multiplicative,and thisresultsin a fruitfulinterplay: Manyquestionsin thetheoryof the
spacesHp canbebettertreatedbyconsideringtheisomorphicspaceHp(D ),andviceversa.
∞
An important example is the Cole-Gamelin estimate (6) which immediately implies that for
HARDYSPACESOFDIRICHLETSERIESANDPSEUDOMOMENTSOFTHERIEMANNZETAFUNCTION 9
everyp 0thespaceHp consistsofanalyticfunctionsinthehalf-planeC . Infact,weinfer
1/2
>
from(6)that
f(σ it) p ζ(2σ) f p
| + | ≤ k kHp
holds whenever σ 1/2, where ζ(s) is theRiemann zeta function. Moreover, since thecoeffi-
>
cientsofaconvergentDirichletseriesareunique,functionsinHp arecompletelydetermined
bytheirrestrictionstothehalf-planeC . ThismeansinparticularthatHp canbethoughtof
1/2
asaspaceofanalyticfunctionsinthishalf-plane.
To complete the picture, we mention that H is defined as the space of Dirichlet series
∞
f(s) a n s that represent bounded analytic functions in the half-planeσ 0. We en-
= ∞n 1 n − >
dowH =withthenorm
∞
P
f H : sup f(s), s σ it,
k k ∞ = | | = +
σ 0
>
andthentheBohrliftallowsustoassociateH withH (D ). Wereferto[37]forthisfactand
∞ ∞ ∞
furtherdetailsabouttheinterestingandrichfunctiontheoryofH .
∞
2.4. AprobabilisticinterpretationoftheBohrlift. Itisfrequentlyfruitfultothinkoftheprod-
uct measureµ on T as a probabilitymeasureand theinfinitelymanyvariables z as inde-
∞ k
∞
pendentidenticallydistributed(i.i.d.)randomvariables.FromtheviewpointoftheBohrcorre-
spondence,wethenassociatewiththesequenceofprimes(p ) asequenceofindependent
j j 1
≥
Steinhaus variables z(p ), which are random variables equidistributed on T. This sequence
j
definesarandommultiplicativefunctionz(n)onthepositiveintegersNbytherule
z(n) (z(p ))κ(n),
j
=
where we again use multi-index notation. Functions in Hp can then, via the Bohr lift, be
thoughtof as linear combinations of these random multiplicativefunctions. Indeed, we may
writetheBohrliftas
f(s) ∞ a n s F (z(p ) ∞ a z(n)
n − j n
= ←→ =
n 1 n 1
X= ¡ ¢ X=
p
andhenceexpress f asthepthmomentof F :
k kHp | |
f p EF p.
k kHp = | |
In thefinal section of this paper, we will make crucial use of this alternateviewpoint, and we
thenfinditnaturaltoswitchtothisprobabilisticterminology.
2.5. Summaryofknownresults. ThefunctiontheoryofthetwodistinguishedspacesH2 and
H is by now quite well developed; we refer again to [37, 43] for details. The results for the
∞
range1 p , p 2,arelesscomplete. Inthissection,wementionbrieflysomekeyresults
≤ <∞ 6=
thatextendtothewholerange0 p ,aswellassomefamiliardifficultiesthatariseinour
< <∞
attemptstomakesuchextensions.
We begin with the theorem on multipliers that was first established in [29] for p 2 and
=
extended to the range 1 p in [5]. We recall that a multiplier m for Hp is a function
≤ < ∞
suchthattheoperator f mf isboundedonHp,andthemultipliernormisthenormofthis
7→
operator. The theorem on multipliersasserts that the space of multipliersfor Hp is equal to
H ,andthisremainstruefor0 p 1,byexactlythesameproofasin[5]. Anotherresultthat
∞
< <
carriesoverwithoutanychange,istheLittlewood–Paleyformulaof[7,Sec.5]. Thelatterresult
wasalreadyusedin[12].
10 ANDRIYBONDARENKO,OLEFREDRIKBREVIG,EEROSAKSMAN,KRISTIANSEIP,ANDJINGZHAO
Forsomeresults,onlyapartialextensionfromthecasep 2isknowntohold.Awellknown
=
exampleis whether the Lp integral of a Dirichlet polynomial f(s) N a n s over any seg-
= n 1 n −
ment of fixed length on the vertical line Res 1/2 is bounded by a un=iversal constant times
p = P
f . Thisisknowntoholdfor p 2andthustriviallyfor p 2k fork apositiveinteger. As
k kHp = =
shown in [36], this embedding holds if and only if the following is true: The boundedly sup-
portedCarlesonmeasuresforHp satisfytheclassicalCarlesonconditioninC .
1/2
There is an interesting counterpart for p 2 to the trivial embedding for p 2k and k a
< =
positiveinteger 1. Thisisthefollowingstatementaboutinterpolatingsequences. IfS (s )is
j
> =
aboundedinterpolatingsequenceinC ,thenwecansolvetheinterpolationproblem f(s )
1/2 j
=
a inHp when
j
a p(2σ 1)
j j
| | − <∞
j
X
and p 2/k for k a positive integer. Indeed, choose any kth root a1/k and solve g(s ) a1/k
= j j = j
in H2. Then f gk solves our problem in Hp. We do not know if this result extends to any
=
p which is not of the form p 2/k. Comparing the two trivial cases, we observe that there
=
is an interesting “symmetry” between the embedding problem for Hp and the interpolation
problemforH4/p. Asimilarphenomenonwillbeexploredinthenextsection.
Beforeturningtothenexttwosectionswhichwilldealwithrespectivelyweightedℓ2andℓ
∞
boundsforthecoefficients,wewouldliketopointoutthattherearecertainlyotherinteresting
problemsofasimilarkind.Aninterestingexampleiswhethertheℓ1estimate
a
∞ n
| | C f H1
pnlogn ≤ k k
n 2
X=
holds when f(s) a n s. We refer to [13] for background on this problem and again to
= ∞n 1 n −
[43]forasurveyofoth=eropenproblems.
P
3. COEFFICIENT ESTIMATES: WEIGHTED ℓ2 BOUNDS
3.1. Contractive Hardy–Littlewood inequalities in the unit disc. We begin with some esti-
mates of the Hp(D) norms (or quasi-norms when 0 p 1) in terms of weighted ℓ2 norms
< <
of the coefficient sequence. Such inequalities were first studied systematically by Hardy and
Littlewood.
p
Forα 1,theweightedBergmanspaceA (D)isthespaceofanalyticfunctionsonDforwhich
α
>
1
f p : f(z) p (α 1) 1 z 2 α−2 dm(z) p ,
k kAα(D) =µZD − −| | π ¶ <∞
¯ ¯ ¡ ¢
wherem denotesLebesgueaream¯easu¯reonC. Weset
dm(z)
dmα(z): (α 1) 1 z 2 α−2 .
= − −| | π
¡ ¢
The Hardy space Hp(D) is the limit of the weighted Bergman spaces Ap(D) as α 1 , in the
α +
→
sensethat
kfkHp(D)=αlim1 kfkAαp(D).
→ +