Monomial convergence for holomorphic functions on $\ell\_r$

0.23 MB·

by  Frédéric Bayart

Monomial convergence for holomorphic
functions on ℓ
r
6 FrédéricBayart AndreasDefant† SunkeSchlüters‡
1 ∗
0
2

n
a
Abstract
J

9
LetF be either theset of allboundedholomorphic functionsor theset
2
ofallm-homogeneouspolynomialsontheunitballofℓr. Wegiveasys-
] tematicstudyofthesetsofallu ℓ forwhichthemonomialexpansion
A r
∂αf(0)uαofevery f F conve∈rges. Inspiredbyrecentresultsfromthe
F α α! ∈
. generaltheoryofDirichletseries,weestablishasourmaintool,indepen-
h P
t dentlyinteresting,upperestimatesfortheunconditionalbasisconstants
a
ofspacesofpolynomialsonℓ spannedbyfinitesetsofmonomials.
m r
[

1 1 Introduction
v
4
4 Let X beaBanachsequencespace(i.e.,ℓ X c suchthatthecanonicalse-
1 0
1 ⊂ ⊂
quences(e )forma1-unconditionalbasis)andR X aReinhardtdomain(i.e.,
8 k ⊂
0 a nonempty open set such that any complex sequence u belongs to R when-
.
1 ever thereexists z R with u z ; for instance, theopen unit ballBX of X).
∈ | |≤| |
0 Then each holomorphic (i.e., Fréchet differentiable) function f :R C has a
6 →
1 powerseriesexpansion α Nncα(n)zα oneveryfinitedimensionalsectionRn of
: ∈ 0 (n) (n 1)
v R, and for example fromPthe Cauchy formula we can see that cα cα+ for
rXi αn ∈NNan0n⊂dNal0nl+z1. RTh,us there is a unique family (cα(f))α∈N(0N) such t=hat, for all
a ∈ ∈ n
f(z) c zα.
α
=
α N(N)
∈X0
Thepowerseries c zαiscalledthemonomialexpansionof f,andc c (f)
α α α α
=
areitsmonomialcoefficients.
P
∗Laboratoire de Mathématiques Université Blaise Pascal Campus des Cézeaux, F-63177
AubièreCedex(France)
†InstitutfürMathematik.UniversitätOldenburg.D-26111Oldenburg(Germany)
‡InstitutfürMathematik.UniversitätOldenburg.D-26111Oldenburg(Germany)
1
Contrary to what happens on finite dimensional domains, the monomial
expansionof f does not necessarilyconvergeat every pointof R. Thisin [16]
motivatedtheintroductionofthefollowingdefinition: GivenasubsetF(R)of
H(R),thesetofallholomorphicfunctionsonR,wecall
monF(R) z R : c (f)zα forall f F(R)
α
= ∈ <∞ ∈
½ α N(N) ¾
∈X0 ¯ ¯
¯ ¯
thedomainofmonomialconvergencewithrespecttoF(R).
Bycontinuityofaholomorphicfunction,andsincetheequalityissatisfied
onR ,weknowthatforallz monF(R),
n
∈
f(z) c (f)zα.
α
=
α N(N)
∈X0
We are mostly interested in determining monF(R) when F(R) P(mℓ ) or
r
=
H (B )for1 r ; asusualwedenoteby H (B )theBanachspaceofall
bo∞undℓerdholom≤or≤ph∞icfunctions f :B C,and∞byPX(mX)itsclosedsubspace
X
→
ofallm-homogeneouspolynomialsP (i.e.,allrestrictionsofboundedm-linear
formsonXm totheirdiagonals).
Thecase r 1was solvedcompletelybyLempertin[20], andthecase r
= =
seems fairly well-understood through the results of [6] (for more on these
∞
resultsseetheintroductionsofthesections5.1 and5.2). However, for 1 r
< <
,despitetheresultsof[16],thedescriptionofmonP(mℓ )andmonH (B )
∞ r ∞ ℓr
remainsmysterious.Inthispaper,weimprovetheknowledgeonthesecases.
Ofcourse, for X ℓ thefact that each sequence in ℓ bydefinitionis ab-
1 1
=
solutely summable is a big advantage, and for X ℓ the crucial tool is the
= ∞
Bohnenblust-Hilleinequality(aninequalityform-linearformsonℓ )together
∞
withallitsrecentimprovments.ButforX ℓ withr 1, weneedalternative
r
= 6= ∞
techniques.
Theproblemistofindforeachu B anadditionalsummabilitycondition
∈ ℓr
which guarantiesfullcontrolofallsums c (f)uα , f H (B ).Thegen-
eralideaissimple. SplitthesetN(N) ofallmαultαiindicesα∈into∞auℓnrionoffinite
0 P ¯ ¯
sets Λ , and then each Λ into the disjoint¯union of¯ all its m-homogeneous
n n
partsΛ (i.e.,allα Λ withorder α m). Thechallengenowisasfollows:
n,m n
∈ | |=
Findacleverdecomposition
N(N) Λ , (1)
0 = m,n
m,n
[
whichallowsainasenseuniformcontroloverallpossiblepartialsums
c (f)uα , f H (B ), (2)
α Λn,m α ∈ ∞ ℓr
∈X ¯ ¯
¯ ¯
2
suchthatundertheadditionalsummabilitypropertyofu B weforallfunc-
∈ ℓr
tions f finallycanconcludethat
c (f)uα c (f)uα .
α α
≤ <∞
α N(N) n m α Λn,m
∈X0 ¯ ¯ X X ∈X ¯ ¯
¯ ¯ ¯ ¯
InordertostudydomainsmonF(R)ofmonomialconvergence,thedecompo-
sitionin(1)whichforourpurposesiscrucial,isinspiredbytheworkofKonya-
gin and Queffélec from [19] on Dirichletseries (see 11), and it is based on the
fundamental theorem of arithmetics. In order to handle (2), we study for ar-
bitraryfiniteindex sets Λ of multiindices upperboundsof theunconditional
basisconstantofthesubspaceinP(mℓ )spannedbyallmonomialszα,α Λ.
r
∈
Two tools of seemingly independent interest are established. The first one is
a fairly general upper estimate whenever all α Λ are m-homogeneous (i.e.,
∈
α m)(Theorem3.2).Thesecondoneleadstosuchestimatesforcertainsets
| |=
Λ ofnonhomogeneousα s,needed toapplytheabovetechniqueofKonyagin
′
andQueffélec(Theorem4.1and4.2).Finally,wepresentournewresultsonsets
of monomialconvergence for homogeneous polynomialsand bounded holo-
morphic functions on ℓ (for polynomials see part (3),(4) of Theorem 5.1 and
r
Theorem 5.3, and for holomorphicfunctions Theorem 5.5 with its corollaries
5.6and5.7).
2 Preliminaries
We usestandardnotationfrom Banach spacetheory. As usual,we denotethe
conjugateexponentof1 r byr ,i.e. 1 1 1.Givenm,n Nweconsider
≤ ≤∞ ′ r+r′ = ∈
thefollowingsetsofindices
M(m,n) j (j ,...,j );1 j ,...,j n {1,...,n}m
1 m 1 m
= = ≤ ≤ =
M(m) N© m ª
=
M NN
=
and
J(m,n) j M(m,n);1 j j n
1 m
= ∈ ≤ ≤···≤ ≤
J(m) © J(m,n) ª
=
n
[
J J(m).
=
m
[
Forindicesi,j M wedenoteby(i,j) (i ,i ,...,j ,j ,...)theconcatenationof
1 2 1 2
∈ =
i and j. An equivalence relationis defined in M(m)as follows: i j if thereis
∼
3
apermutationσsuchthati j forallk. Wewrite i forthecardinalityof
σ(k) k
= | |
the equivalence class [i]. Moreover, we note that for each i M(m) there is a
∈
uniquej J(m)suchthati j.
∈ ∼
Let us compare our index notation with the multi index notation usually
used in the context of polynomials. There is a one-to-one relation between
J(m)and
Λ(m) α N(N); α ∞ α m ;
=( ∈ 0 | |=i 1 i = )
X=
indeed, given j, one can define α by doing α {q j r}; conversely, for
r q
= | | = |
eachα,weconsiderjα (1,.α.1.,1,2,.α.2.,2,...,n,.α.n.,n,...).Inthesamewaywemay
=
identifyΛ(m,n) α Nn ; α m withJ(m,n). Notethat j m! forevery
= ∈ 0 | |= | α|= α!
α Λ(m).Takingthiscorrespondenceintoaccount,foreveryBanachsequence
∈ © ª
space X themonomialseriesexpansionofam-homogeneouspolynomialP
∈
P(mX)canbeexpressedindifferentways(wewritec c (P))
α α
=
c zα c z c z z .
α = j j= j1...jm j1··· jm
α Λ(m) j J(m) 1 j1 ... jm
∈X ∈X ≤ X≤ ≤
Given a Banach sequence space X and some index subset J J, we write
⊂
P(JX) for the closed subspace of all holomorphic functions f H (B ) for
X
whichc (f) 0forallj J \J. Clearly,P(mX) P(J(m)X). If J∈ J∞isfinite,
j
= ∈ = ⊂
then
P(JX) span z : j J ,
j
= ∈
wherez forj (j ,...,j )standsforthemnonomialoz :u u : u ... u . For
j = 1 ℓ j 7→ j = j1· · jℓ
J J(m),wecall
⊂
J∗ j J(m 1); k 1, (j,k) J
= ∈ − ∃ ≥ ∈
thereducedsetof J. © ª
3 Unconditionality
Given a compact groupG, theSidon constantof a finiteset C of characters γ
(in the dual group) is the best constant c 0, denoted by S(C), such that for
≥
everychoiceofscalarsc ,γ C,wehavethat
γ
∈
c c c γ .
γ γ
| |≤
γX∈C °γX∈C °∞
° °
AnimmediateconsequenceoftheCau°chy-Schw°arzinequalityisthat
1
1 S(C) C 2.
≤ ≤| |
ForthecirclegroupsG T,Tn andT differentvaluesarepossible:
∞
=
4
• Awell-knownresultofRudinshowsthatforthesetC {1,z,...,zn 1}of
−
=
charactersonG Twehave,uptoconstantsindependentofn,
=
S(C) pn. (3)
≍
• In[14]itwasprovedthatforeverym,n theSidonconstantofthemono-
mialsC {zα:α Λ(m,n)}onG Tn,uptothemthpowerCm ofsome
= ∈ =
absoluteconstantC,satisfies
S(C) Λ(m 1,n) 12 . (4)
≍| − |
• Incontrast,areformulationofaresultofAronandGlobevnik[2,Thm1.3]
showsthatforeverym theSidonconstantofthesparsesetC {zm : j
= j ∈
N}fulfills
S(C) 1. (5)
=
Let us transfer some of these results into terms of unconditional bases con-
stantsofspacespolynomialsonsequencespaces. RecallthataSchauderbasis
(x )of a Banach space X issaid to beunconditionalwhenever thereis a con-
n
stantc 0suchthat ε α x c α x foreveryx a x X and
k k k k k k k k k k
≥ k k≤ k k = ∈
all choices of (ε ) C with ε 1. In this case, the best constant c is de-
k k k
⊂P | | = P P
notedbyχ (x ) andcalledtheunconditionalbasisconstantof(x ). Ifsucha
n n
constantdoesn’texist,i.e. ifthebasisisnotunconditional,wesetχ (x ) .
¡ ¢ n =∞
GivenaBanachsequencespaceX andanindexset J J,suchthattheset
⊂ ¡ ¢
C {z : j J} of all monomialsassociated with J (counted in a suitableway)
j
= ∈
formsanbasisofP(JX),wewrite
χ P(JX) χ(C).
mon
=
¡ ¢
If we interpret each of these monomials z as a character on the group T ,
j ∞
then a straightforward calculation (using the distinguished maximum modu-
lusprinciple)provesthat
S(C) χ P(Jℓ ) .
mon
= ∞
¡ ¢
Asimplebutusefullemmashowsthatχ P(Jℓ ) isanupperboundofall
mon
χ P(JX) . ∞
mon ¡ ¢
¡ ¢
Lemma 3.1. Let X be a Banach sequence space and let J J, such that the
⊂
monomialsformabasisofP(JX). Then
χ P(JX) χ P(Jℓ ) .
mon mon
≤ ∞
¡ ¢ ¡ ¢
5
Proof. Assume χ P(Jℓ ) (otherwise there is nothing to show). For
mon
P P(JX) and a fixed u ∞B <de∞fineQ(w) P(wu) P(Jℓ ). Since B is a
∈ ¡ ∈ X¢ = ∈ ∞ X
Reinhardtdomain,wehave Q P . Itisnowsufficienttoobservethat
k k∞≤k k∞
c (P)u sup c (P)u w sup c (Q) w
j j j j j j j
| |= | || |= | || |
Xj∈J w∈Bℓ∞Xj∈J w∈Bℓ∞Xj∈J
χ P(Jℓ ) Q χ P(Jℓ ) P ,
mon mon
≤ ∞ k k∞≤ ∞ k k∞
¡ ¢ ¡ ¢
theconclusion.
Letusagainseesomeexamples: Given X,animmediateconsequenceof(3)is
thatforP(JX span{zj;0 j n 1}wehave,uptoauniversalconstant,
= 1 ≤ ≤ −
¢
χ P(JX) pn,
mon
≍
¡ ¢
andfrom(5)wemaydeducethatfor J (k, ,k);k N J(m)
= ··· ∈ ⊂
χ P(JX© ) 1. ª
mon
=
¡ ¢
Generalizing (4) is much more complicated. In the scale of all ℓ -spaces the
r
results from [4] (lower estimates)and [13, 14] (upper estimates)show thatfor
1 r
≤ ≤∞ χmon P(J(m,n)ℓr) J(m 1,n) 1−min1(r,2), (6)
≍ −
where meansthatth¡eleftandthe¢rig¯htsideequa¯luptothem-thpowerCm
≍ ¯ ¯
ofaconstantonlydependingonr (andneitheronm noronn).
Replacing the index set J(m,n) by an arbitrary finite subset J J(m,n)
⊂
thefollowingresultisastrongimprovementandourmaintoolwithinourlater
studyofsetsofmonomialconvergence.
Theorem3.2. Given1 r and m 1, thereis aconstantC(m,r) 1 such
≤ ≤∞ ≥ ≥
thatforeveryn 1,everyP P(J(m,n)ℓ ),every J J(m,n),andeveryu ℓ
r r
≥ ∈ ⊂ ∈
wehave
cj(P) |uj|≤C(m,r)|J∗|1−min1(r,2)kukmr kPk∞, (7)
j J
X∈ ¯ ¯
¯ ¯
where
eme(m 1)/r if1 r 2
−
C(m,r) ≤ ≤
≤(em2(m−1)/2 if2 r .
≤ ≤∞
Inparticular,foreveryfinite J J(m)
⊂
χmon P(Jℓr) C(m,r) J∗ 1−min1(r,2). (8)
≤ | |
¡ ¢
6
Theproofisgiveninthefollowingtwosubsections;itisdifferentforr 2and
≤
for r 2. The case r of (6) is given in [14], and it uses the hypercontrac-
≥ =∞
tiveBohnenblust-Hilleinequality. Thegeneralcase1 r from[13]needs
≤ ≤∞
sophisticatedtoolsfromlocalBanachspacetheory(asGordon-Lewisandpro-
jection constants). Theseargumentsin fact only work for thewholeindex set
J(m,n), and they seem to fail in full generality for subsets J of J(m,n). We
hereprovideatricky,butquiteelementary,argumentwhichworksforarbitrary
J;moreover,wepointoutthatevenforthespecialcase J J(m,n)weobtain
=
betterconstantsC(m,r)for(8)thanin[13].
From [15] we know that for each infinite dimensional Banach sequence
spaceX,theBanachspaceP(mX)neverhasanunconditionalbasis.Inpartic-
ular,theunconditionalbasisconstantχ P(mX) ofallmonomials(zj)
mon j J(m)
∈
isnotfinite.ButletusnotethatincontrasttothisthereareX suchthatforeach
¡ ¢
m
supχ P(J(m,n)X)
mon
<∞
n
¡ ¢
(thiscanbeeasilyshownforX ℓ ,butfollowing[15]thereareevenexamples
1
=
ofthistypedifferentfromℓ ).
1
3.1 Thecaser 2
≤
Weneedseverallemmas. ThefirstoneisaCauchyestimateandcanbefound
in[7,p. 323].Forthesakeofcompletenessweincludeastreamlinedargument.
Lemma3.3. Let1 r andα Nn with α m. ThenforeachP P mℓn
≤ ≤∞ ∈ 0 | |= ∈ r
wehave
mm 1 ¡ ¢
r
c (P) P .
| α |≤ αα k k∞
³ ´
Inparticular,foreachj J(m,n)wehavethat
∈
m 1
cj(P) e r j r P .
| |≤ | | k k∞
Proof. Defineu =m−1/r(α11/r,...,α1n/r)∈Bℓnr. ThenbytheCauchyintegralfor-
mulaforeachP P mℓn
∈ r
¡ ¢ 1 P(z)
c (P) ... dz.
α = (2πi)n zαz ...z
Z|zn|=un Z|z1|=u1 1 n
Henceweobtain
1 mm 1
r
c (P) P P ,
| α |≤ uα k k∞= αα k k∞
| | ³ ´
theconclusion. Forthesecondinequalitynotefirstthat mm r1 em/r m! 1/r,
αα ≤ α!
andrecallthatifweassociatetojthemultiindexα,then³m! ´j . ³ ´
α! =| |
7
Corollary3.4. ConsiderthelinearoperatorQ L ℓn,P(m 1ℓn) definedby
∈ r − r
n ¡ ¢
Q(z,w) b z w ,
(j,k) j k
=j J(m 1,n)Ãk 1 !
∈ X− X=
wherez,w ℓn. Thenforanyj J(m 1,n),
∈ r ∈ −
n 1/r′
b(j,k) r′ emr−1 j 1/r Q .
à | | ! ≤ | | k k∞
k 1
X=
Proof. Let us fix w ∈ Bℓnr. Then Q(·,w) ∈ P(m−1ℓnr). Thus, by the preceding
lemmaforanyj J(m 1,n),
∈ −
n
b(j,k)wk emr−1 j 1/r sup Q(z,w) emr−1 j 1/r Q .
¯¯kX=1 ¯¯≤ | | z∈Bℓnr | |≤ | | k k∞
¯ ¯
Wenowta¯¯kethesupr¯¯emumoverallpossiblew ∈Bℓnr.
Lemma3.5. LetP P(mℓn). Thenforanyj J(m 1,n)
∈ r ∈ −
n 1/r′
c(j,k)(P)r′ me1+mr−1 j 1/r P .
à | | ! ≤ | | k k∞
k jm 1
=X−
Proof. Let A:ℓn ... ℓn Cbethesymmetricm-linearformassociatedtoP,
r × × r →
A(z(1),...,z(m)) a (A)z(1) z(m);
= i i1 ··· im
i M(m,n)
∈ X
inparticular,foreachj J(m,n)wehavea (A) cj(P).Forz,w ℓn definethe
∈ j = j ∈ r
linearoperator ||
Q(z,w)=A(z,...,z,w)∈L ℓnr,P(m−1ℓnr) ;
thenasimplecalculationproves ¡ ¢
Q(z,w) a (A)z z w
= i i1··· im 1 im
i M(m,n) −
∈ X
n
a (A)z z w
= (i,k) i1··· im 1 k
i M(m 1,n)k 1 −
∈ X− X=
n
a (A)z z w
= (i,k) i1··· im 1 k
j J(m 1,n)i [j]k 1 −
∈ X− X∈ X=
n
a (A)z z w
= j J(m 1,n)k 1Ãi [j] (i,k) i1··· im−1! k
∈ X− X= X∈
n
a (A)j z z w .
= (j,k) | | j1··· jm 1 k
j J(m 1,n)k 1 −
∈ X− X= ¡ ¢
8
Nownotethatforeveryj J(m 1,n)wehave (j,k) m j ,andhencebythe
∈ − | |≤ | |
precedingcorollaryforsuchj
1/r′ 1/r′
c (P) r′ a (A)(j,k) r′
(j,k) (j,k)
à ! =à | | !
k:jm 1 k k:jm 1 k
X− ≤ ¯ ¯ X− ≤ ¯ ¯
¯ ¯ n ¯ 1/r¯′
m a(j,k)(A)j r′ memr−1 j 1/r Q .
≤ Ã | | ! ≤ | | k k∞
k 1
X= ¯ ¯
¯ ¯
Finally,byHarris’polarizationformulaweknowthat Q e P ,andhence
k k∞≤ k k∞
weobtainthedesiredconclusion.
Nowwearereadytogivethe
ProofofTheorem3.2for1 r 2. Take P P(J(m,n)ℓ ), J J(m,n) and u
r
≤ ≤ ∈ ⊂ ∈
ℓ . Then,byLemma3.5,foranyj J ,
r ∗
∈
1/r′ n 1/r′
c(j,k)(P)r′ c(j,k)(P)r′ me1+mr−1 j 1/r P .
à | | ! ≤à | | ! ≤ | | k k∞
k:(j,k) J k jm 1
X∈ =X−
NowbyHölder’sinequality(twotimes)andthemultinomialformulawehave
c (P) u c u u
j j (j,k) j k
| || | = | || || |
j J j J k:(j,k) J
X∈ X∈ ∗ X∈
1/r′ 1/r
u c r′ u r
j (j,k) k
≤ j J | | Ãk:(j,k) J| | ! Ã k | | !
X∈ ∗ X∈ X
me1+mr−1 j 1/r uj u r P
≤ | | | |k k k k∞
j J
X∈ ∗
1/r 1/r′
me1+mr−1 j uj r 1 u r P
≤ Ãj J | || | ! Ãj J ! k k k k∞
X∈ ∗ X∈ ∗
1/r 1/r′
me1+mr−1 j uj r 1 u r P
≤ Ãj∈J(Xm−1,n)| || | ! ÃjX∈J∗ ! k k k k∞
= me1+mr−1|J∗|1−1rkukmr kPk∞.
In orderto deduce(8), notethatfor everyfinite J J(m)thereis n such that
⊂
J J(m,n). Then every P P(Jℓ ) can be considered as a polynomial in
r
⊂ ∈
P(J(m,n)ℓ )withequalnorm,whichimpliestheconclusion.
r
9
3.2 Thecaser 2
≥
Note first that the simple argument from the proof of Lemma 3.1 shows that
we onlyhaveto deal with thecase r . For r we need anotherlemma
=∞ =∞
whichsubstitutestheargument(byCauchy’sestimates)fromLemma3.3. Itis
animprovementofParseval’sidentity,anditsproofcanbefoundin[6,Lemma
2.5].
Lemma3.6. LetP P(J(m,n)ℓ ). Then
∈ ∞
1/2
n
 c(j,k)(P)2 em2m2−1 P .
| | ≤ k k∞
k 1 j J(m 1,n)
X= ∈ X− 
 jm−1≤k 
Wearenowreadyforthe
ProofofTheorem3.2forr . LetP P(J(m,n)ℓ ). Then,foranyu B ,by
ℓ
=∞ ∈ ∞ ∈ ∞
theCauchy-SchwarzinequalityandtheprecedingLemma3.6wehave
n
c (P) u  c 
j j (j,k)
| || | ≤ | |
Xj∈J kX=1 jX∈J∗ 
(j,k) J
 ∈ 
  1/2
n
 c(j,k) 2 {j J∗:(j,k) J} 1/2
≤ | | ∈ ∈
kX=1 jX∈J∗  ¯ ¯
(j,k) J
 ∈  ¯ ¯
  1/2
n
 c(j,k) 2 J∗ 1/2
≤ | | | |
k 1 j J(m 1,n)
X= ∈ X− 
em2m2−j1m−J1∗≤k1/2 P . 
≤ | | k k∞
For the second statement, see again the argument from the proof in the case
1 r 2. ThisfinallycompletestheproofofTheorem3.2.
≤ ≤
Remark3.7. It isnaturaltoaskfor lower boundsofχ P(Jℓ ) using J or
mon r
| |
J . For the whole set of m-homogeneous polynomials, this has been done
| ∗| ¡ ¢
in [10] for r 2 and in [4] for 1 r 2. Using the Kahane-Salem-Zygmund
≥ ≤ ≤
inequality,wecangivesuchalowerboundatleastforthecaser . Indeed,
=∞
10