Weak conditions for interpolation in holomorphic spaces

Publicacions Ma em`a iques, Vol. 44 (2000), 277–293
WEAK CONDITIONS FOR INTERPOLATION IN
HOLOMORPHIC SPACES
Alexande P. Schus e and K is ian Seip
Abs ac
An analogue o he no ion o uni o mly sepa a ed sequences, ex-
p essed in e ms o ex emal unc ions, yields a necessa y and suffi-
cien condi ion o in e pola ion in Lpspaces o holomo phic unc-
ions o Paley-Wiene - ype when 0 <p≤1, o Fock- ype when
0<p≤2, and o Be gman- ype when 0 <p<∞. Mo eo e , i a
uni o mly disc e e sequence has a ce ain uni o m non-uniqueness
p ope y wi h espec o any such Lpspace (0 <p<∞), hen i
is an in e pola ion sequence o ha space. The p oo s o hese
esul s a e based on an app oxima ion heo em o subha monic
unc ions, Beu ling’s esul s conce ning compac wise limi s o se-
quences, and he desc ip ion o in e pola ion sequences in e ms
o Beu ling- ype densi ies. De ails a e ca ied ou only o Fock
spaces, which ep esen he mos difficul case.
1. In oduc ion
This pape s udies some consequences o he ac ha in e pola ion
sequences o Be gman, Fock, and Paley-Wiene spaces a e in a ian un-
de ce ain na u al g oup ac ions and co esponding compac wise limi s.
We will show how and when his in a iance implies ha he defining
p ope y o in e pola ion sequences is equi alen o appa en ly weake
s a emen s abou sequences. Two such weak condi ions will be consid-
e ed: The fi s is he analogue o Ca leson’s condi ion o uni o m sepa-
a ion (c . [11]); he second is a condi ion o “uni o m non-uniqueness”,
in some sense dual o a condi ion o Beu ling used o desc ibe sampling
sequences.
Ou me hods apply o all o he spaces men ioned abo e, bu he
main ocus will be pu on Fock spaces, because hey ep esen he mos
difficul case. We begin by desc ibing ou esul s in ha se ing.
1991 Ma hema ics Subjec Classifica ion. 30H05, 46E15.
278 A. P. Schus e , K. Seip
Le dσ deno e Lebesgue a ea measu e on C. We define
 p
α,p =C
| (z)|pe−pα|z|2dσ(z)
o α>0 and p<∞, and  α,∞= supz| (z)|e−α|z|2. The Fock
space Fp
αconsis s o hose en i e unc ions such ha  α,p <∞.O
basic impo ance is ha he ansla ion ope a o Tζ,α (ζ∈C) defined
as
Tζ,α (z)= (z−ζ)eα(2ζz−|ζ|2)
ac s isome ically on Fp
α. This ac will some imes be e e ed o as he
ansla ion in a iance o Fp
α.
We say ha a sequence Γ = {γn}o dis inc poin s in Cis in e pola ing
o Fp
αi o e e y sequence {an}sa is ying

n
|an|pe−pα|γn|2<∞,
he e is an ∈Fp
αsuch ha (γn)=an o all n. Fo he case p=∞,
we equi e he exis ence o an ∈F∞
αwi h (γn)=anwhene e
sup
n
|an|e−α|γn|2<∞.
Fo a sequence Γ = {γn}o dis inc poin s, se Γk={γn}n:n=kand
le Fp
α(Γk) deno e he closed subspace o Fp
αconsis ing o unc ions an-
ishing on Γk. (We will keep his no a ion when Fp
αis eplaced by o he
spaces o holomo phic unc ions.) Ou fi s main heo em is he ollow-
ing:
Theo em 1. Le Γ={γn}be a sequence o dis inc poin s in C, and
suppose 0<p≤2. Then Γis in e pola ing o Fp
αi and only i he e is
aδ>0such ha
sup{| (γk)|: ∈Fp
α(Γk), α,p ≤1}≥δeα|γk|2 o all k.(1)
A no mal amily a gumen shows ha he sup emum on he le -
hand side o (1) is a ained, and hus i is in ac a maximum. A simple
example (see Sec ion 3) shows ha Theo em 1 ails o p>2.
We no e ha condi ion (1) is indeed “appa en ly weake ” han he
condi ion o being in e pola ing, because (1) says only ha we can sol e
he in e pola ion p oblems (γk)=1, (γn) = 0 o n=k, wi h con ol
o he no ms o he solu ions. The ac ha we can achie e such no m
con ol when Γ is in e pola ing ollows om he open mapping heo em
in a s anda d way. Thus he necessi y o (1) o Γ o be in e pola ing is
i ial, modulo he open mapping heo em.
In e pola ion in holomo phic spaces 279
I is ins uc i e o es a e and discuss condi on (1) in a mo e gene al
se ing. I Bis a no med o quasi-no med linea space o holomo phic
unc ions defined on some domain Ω, we say ha a sequence Γ o dis inc
poin s om Ω is a weak in e pola ion sequence o Bi he e exis s a δ>0
such ha
(2) sup{| (γk)|: ∈B(Γk), B≤1}
≥δsup{| (γk)|: ∈B, B≤1}
o all k. Thus (1) says ha Γ is a weak in e pola ion sequence o Fp
α.
The connec ion be ween in e pola ion and weak in e pola ion is pa
o Ca leson’s classical in e pola ion heo em (c . Chap e IX o [3]): A
sequence is in e pola ing o Hpi and only i i is a weak in e pola ion
sequence o Hp, whe e Hpdeno es he Ha dy space o he open uni
disk Do Cand 0 <p≤∞. (In his case, in e pola ing Blaschke
p oduc s a e ex emal unc ions o he le -hand side o (2).) In [11],
we showed ha he same s a emen is ue o he Be gman spaces Ap
β
o he uni disk (see Sec ion 4 o defini ion) when 0 <p<∞. (The
p oo echnique o [11] is diffe en om he one used he e and does no
ca y o e o he Fock and Paley-Wiene spaces.) In Sec ion 4, we show
ha o he Paley-Wiene spaces PWp
τ, his cha ac e iza ion holds only
when 0 <p≤1.
The p-dependence in hese esul s is qui e cu ious. I can be a -
ibu ed di ec ly o he unde lying geome y o ou spaces, espec i ely
o he line (Paley-Wiene ), he plane (Fock), and he disk (Be gman).
Howe e , in gene al, i is less clea why in some cases weak in e pola ion
implies in e pola ion (like o he Ha dy spaces), and in some i does no
(like o he Di ichle space; see Sec ion 5).
Theo em 1 will be ob ained as a consequence o a closely ela ed
esul , which we will now desc ibe. Fo his we need Beu ling’s no ion
o compac wise limi s. A sequence Qjo closed se s con e ges s ongly
o Q, deno ed Qj→Q,i [Q, Qj]→0; he e [Q, R] deno es he F ´eche
dis ance be ween wo closed se s Qand R, i.e., [Q, R] is he smalles
numbe such ha Q⊂{z:d(z,R)≤ }and R⊂{z:d(z,Q)≤ },
whe e d(·,·) deno es Euclidean dis ance in C.Qjcon e ges compac wise
o Q, deno ed QjQ, i o e e y compac se D,(Qj∩D)∪∂D →
(Q∩D)∪∂D. Fo a closed se Γ, we le W(Γ) deno e he collec ion o
se s Γsuch ha Γ+ajΓ o some sequence {aj}. I Γ is in e pola ing
o Fp
α, hen using he ansla ion in a iance o Fp
αand a no mal amily
a gumen , we see ha all sequences in W(Γ) a e also in e pola ing o
Fp
α.
280 A. P. Schus e , K. Seip
A sequence Γ is a uniqueness sequence o Fp
αi he only elemen
o Fp
α anishing on Γ is he ze o unc ion, and o he wise we say ha
Γisanon-uniqueness sequence o Fp
α. A sequence is called uni o mly
disc e e i he infimum o he Euclidean dis ances be ween dis inc poin s
is s ic ly posi i e. Wi h his e minology, ou second main heo em can
be s a ed as ollows:
Theo em 2. Le Γ={γn}be a sequence o dis inc poin s in Cand
suppose p<∞. Then Γis in e pola ing o Fp
αi and only i Γis
uni o mly disc e e and e e y Γ∈W(Γ) is a non-uniqueness sequence
o Fp
α.
This uni o m non-uniqueness condi on is known o be necessa y o a
sequence o be in e pola ing o Fp
α(see Sec ion 2), and so he p oblem
is o p o e ha his appa en ly a he weak condi ion implies ha a se-
quence is in e pola ing. Ou p oo o his implica ion is easily ans e ed
o Paley-Wiene and Be gman spaces, and in ac o all he weigh ed
spaces desc ibed in [15].
Theo em 2 appea s pa icula ly in e es ing when con as ing i wi h
he Hpcase: The analogue o Theo em 2 ails o Hp, as can be seen
by cons uc ing a sui able “uni o m” Blaschke sequence which does no
mee he Ca leson condi ion.
The nex wo sec ions con ain he p oo s o Theo em 2 and Theo em 1.
In Sec ion 4, we will summa ize ou findings in he Paley-Wiene and
Be gman spaces, wi h only indica ions o p oo s. Finally, in Sec ion 5,
he gene al ques ion abou he ela ionship be ween in e pola ion and
weak in e pola ion is discussed b iefly in he case ha Bis a Hilbe
space. In pa icula , we show ha weak in e pola ion does no imply
in e pola ion in he Di ichle space.
In wha ollows, we w i e gwhene e he e is a cons an Ksuch
ha ≤Kg, and ≃gi bo h gand g .
2. P oo o Theo em 2
We commen fi s on he necessi y o he wo condi ions o Theo em 2.
The easy p oo ha an in e pola ion sequence is uni o mly disc e e can
be ound in [9] (c . P oposi ion 3). The p oo o he ac ha he uni o m
non-uniqueness condi ion necessa ily holds can be ound in ha same
pape o p≥1 (c . Theo em 2). The case p≤2 ollows om Lemma 6.2
o [12] wi h L2 eplaced by Lp,p≤2. (C . also he p oo o Theo em 2
in Sec ion 3.)
We desc ibe nex he basic ing edien s o he p oo o he con e se
implica ion.
In e pola ion in holomo phic spaces 281
The sequence Γ = {γn}o dis inc poin s in Cis a sampling sequence
o Fp
αi
 α,p ≃{ (γn)e−α|γn|2}p
o ∈Fp
α. We will need he desc ip ion o sampling and in e pola ion
sequences in e ms o lowe and uppe uni o m densi ies. To his end,
fix Γ and deno e by n(z, ) he numbe o poin s o he sequence Γ in
D(z, ), which is he disk o cen e zand adius . The lowe uni o m
densi y o Γ is defined as
D−(Γ) = lim in
→∞ min
z∈C
n(z, )
π 2,
and he uppe uni o m densi y o Γis
D+(Γ) = lim sup
→∞
max
z∈C
n(z, )
π 2.
No e ha 0 ≤D−(Γ) ≤D+(Γ) <∞when Γ is uni o mly disc e e.
We shall need he ollowing heo em:
Theo em A. Suppose Γis uni o mly disc e e and 0<p≤∞. Then Γ
is sampling o Fp
αi and only i D−(Γ) >(2α)/π, and Γis in e pola ing
o Fp
αi and only i D+(Γ) <(2α)/π.
The esul s o p= 2 and p=∞a e p o ed in [12] and [16]; he
me hods o hose pape s ex end di ec ly o he case p≤2 and wi h
mino modifica ions o he case 2 <p<∞. A diffe en app oach o
he case p≥1 can be ound in [9]. (The “sampling pa ” o ha pape
con ains a p oo alid o all p.) No e also ha in ac Theo em 1 gi es
an in e es ing new p oo o he necessi y o he densi y condi ion o
in e pola ion when p≤2.
The second ing edien is a esul om [13], which is an analogue o
a heo em o Beu ling [2]. This heo em, which eflec s he ansla ion
in a iance o sampling sequences, is c ucial o p o ing he sampling pa
o Theo em A.
Theo em B. Suppose Γis uni o mly disc e e. Then Γis sampling o
F∞
αi and only i e e y Γ∈W(Γ) is a uniqueness sequence o F∞
α.
The hi d auxilia y esul is a special case o an in e es ing app oxi-
ma ion p inciple o Lyuba skii and Malinniko a [5]. I s a es ha any
subha monic unc ion can be app oxima ed by he loga i hm o he mod-
ulus o an en i e unc ion ou side a “small” excep ional se . We need he
ollowing special e sion o i , in essence p o ed ea lie by Lyuba skii
and Sodin [7] and s a ed as Theo em 3 in [5]:

282 A. P. Schus e , K. Seip
Theo em C. Suppose φis a subha monic unc ion in Csuch ha ∆φ≃
1, whe e ∆is he Laplacian ∂2
∂∂ . Then he e exis s an en i e unc ion G,
wi h uni o mly disc e e ze o sequence Γ, such ha
|G(z)|≃eφ(z)d(z,Λ).(3)
Suppose now ha Γ is uni o mly disc e e and W(Γ) con ains only
non-uniqueness sequences o Fp
α. Le us fi s explain in o mally he plan
o ou p oo . We will use Theo em C o cons uc ano he sequence Λ,
which “comple es” Γ in he sense ha he union o Γ and Λ has a uni o m
dis ibu ion jus like a la ice. We shall hen ans o m he p oblem
in such a way ha i is sol ed by applying Theo em B o he “dual”
sequence Λ and an app op ia e space F∞
α.
We u n o he de ails. Le Fbe some en i e unc ion anishing
p ecisely on Γ. We claim ha o any β>πD
+(Γ)/2 we can find a
uni o mly disc e e sequence Λ and an en i e unc ion G anishing on Λ
such ha
|F(z)G(z)|≃d(z,Γ)d(z,Λ)eβ|z|2.(4)
(Thus he ze os o FG a e dis ibu ed in he same egula way as he
poin s in a la ice o densi y (2β)/π.) This is done in he ollowing
ashion (c . [1, pp. 113–114]). Se
χ (z)=1/(π 2),|z|<
0,|z|≥ ,
and le
ν=
n
δγn
be he measu e consis ing o a poin mass a each poin o ou sequence Γ.
Choose so big ha 2β−πν ∗χ (z)≥), whe e )=β−πD+(Γ)/2 and
ν∗χ deno es he con olu ion o νand χ o e C. We se
=(ν−ν∗χ )∗E,
whe e E= log |z|. Ac ually, o be p ecise, his unc ion is defined as
(z) = lim →∞ (z), whe e co esponds o he fini e sequence con-
sis ing o hose γnwhich sa is y |γn|< . The unc ion is clea ly
well-defined, and he limi exis s because (z)= τ(z)i |z|+ < <τ,
by he mean alue p ope y o ha monic unc ions. We define
φ(z)=β|z|2+ (z)−log |F(z)|
In e pola ion in holomo phic spaces 283
and obse e ha )≤∆φ(z)≤β. I emains o apply Theo em C and
use he es ima e
| (z)−log d(z,Γ)|1,
which ollows om he defini ion o .
I is essen ial o no e ha Λ = {λn}is cons uc ed in such a way ha
we ob ain he iden i y
D+(Γ) = (2β)/π −D−(Λ).
Namely, he Riesz measu e ∆φ(z)dσ(z)o φis a omized by spli ing he
plane in o “cells”, each o mass 1 wi h espec o (2/π)∆φ(z)dσ(z), and
placing one λnin each “cell”; we e e o [5] o de ails.
To show ha Γ is an in e pola ion sequence o Fp
α, i he e o e suffices
o p o e ha D−(Λ) >2(β−α)/π. In iew o Theo em A, we need only
show ha Λ is a sampling sequence o F∞
β−αand by Theo em B, his
educes o showing ha W(Λ) con ains only uniqueness sequences o
F∞
β−α.
Suppose hen ha Λ∈W(Λ). Recall ha i is ob ained as a com-
pac wise limi o a sequence Λ + ak. Choose a subsequence o {ak},say
{akj}, such ha also Γ + akjΓ∈W(Γ). By a no mal amily a -
gumen applied o he unc ions Takj,β(FG), i is clea ha he e exis
unc ions ˜
Fand ˜
G anishing espec i ely on Γand Λsuch ha
|˜
F(z)˜
G(z)|≃d(z,Γ)d(z,Λ)eβ|z|2.(5)
This p o es ha Γ∪Λis a uniqueness sequence o Fp
β, because o he -
wise he e would ha e o exis an en i e unc ion such ha ˜
F˜
G∈Fp
β.
Howe e , using (5) and he subha monici y o | |pin small disks a ound
each poin in Γand Λ, we ob ain
 ˜
F˜
Gp
β,p ≃C
| (z)|pdσ(z)<∞,
which is a con adic ion unless ≡0. On he o he hand, since Γis
a non-uniqueness sequence o Fp
α, he e is a non i ial unc ion g∈Fp
α
which anishes on Γ.I Λ
is likewise a non-uniqueness sequence o
F∞
β−α, he e exis s a unc ion h∈F∞
β−α anishing on Λsuch ha gh ∈
Fp
β, which we ha e seen is impossible. So Λis a uniqueness sequence o
F∞
β−α, and he p oo o Theo em 2 is comple e.
284 A. P. Schus e , K. Seip
No e ha an applica ion o Theo em C wi h φ(z)=α|z|2shows ha
Theo em 2 ails when p=∞. Since he sequence Λ sa isfies (3), i is
a non-uniqueness sequence o F∞
α. A no mal amily a gumen shows
ha he same is ue o e e y membe o W(Λ). On he o he hand, i
is clea by cons uc ion ha D+(Λ) = (2α)/π, so ha by Theo em A,
Λ is no in e pola ing.
Theo em 2 does ha e an analogue o p=∞, howe e , i one conside s
ins ead he space F∞,0
α, which consis s o hose en i e unc ions sa is ying
| (z)|eα|z|2→0as|z|→∞. In his se ing, Γ is in e pola ing i he
p oblem (γn)=anis sol able whene e aneα|γn|2→0asn→∞. The
s a emen and p oo o he esul a e hen iden ical o ha o Theo em 2,
wi h Fp
α eplaced by F∞,0
α.
3. P oo o Theo em 1
Se e al ema ks a e in o de be o e we u n o he p oo o he su -
ficiency o he weak in e pola ion condi ion o Theo em 1. Fi s , no e
ha when 0 <p≤1, one can sol e he in e pola ion p oblem explici ly:
(z)=
k
ak
Gk(z)
Gk(γk),
whe e Gkis an ex emal unc ion o he le -hand side o (1). I is
clea ha his se ies con e ges locally uni o mly o a unc ion in Fp
αand
sol es he in e pola ion p oblem when he ak’s sa is y he compa ibili y
condi ion.
Second, since Fp
αis con inuously embedded in o F2
α o p≤2, and
since he in e pola ion sequences a e independen o p, i suffices o p o e
he heo em o he case p= 2, which is wha we will do.
Thi d, we no e ha he heo em ails when p>2. To show his, we
again appeal o Theo em C o find an en i e G, wi h uni o mly disc e e
ze o sequence Γ, such ha G(z)≃eα|z|2d(z,Γ). Then se ing Gk(z)=
G(z)/(z−γk), we ha e
sup
k
Gkα,p <∞
and
Gk(γk)eα|γk|2.
Howe e , Γ is no in e pola ing because D+(Γ) = (2α)/π, as ollows
om he cons uc ion o Γ.
In e pola ion in holomo phic spaces 285
To p o e Theo em 1, we will show ha Γ is uni o mly disc e e and
ha W(Γ) con ains only non-uniqueness sequences o Fp
α. The esul
will hen ollow om an applica ion o Theo em 2. Th oughou he
p oo , we will le Gkdeno e an ex emal unc ion o he le -hand side
o (1).
I ∈Fp
α anishes a some poin w, we find ha (z)/(z−w)∈Fp
α
wi h no m con olled by  α,p, and so
e−α|z|2| (z)||z−w| α,p.
We se =Gk,z=γkand w=γn, o see ha his inequali y becomes
1|γk−γn|, whence Γ is uni o mly disc e e.
I emains o p o e ha W(Γ) con ains only non-uniqueness se-
quences. To begin wi h, we no e ha he ansla ion in a iance o Fp
α
along wi h a no mal amily a gumen implies ha i Γ mee s (1), hen so
does e e y membe o W(Γ). Thus i suffices o demons a e ha e e y
sequence Γ sa is ying (1) is a non-uniqueness sequence o Fp
α.
Suppose hen, ha Γ is a uniqueness sequence. I D+(Γ) <(2α)/π,
hen Γ will be an in e pola ion sequence and hence a non-uniqueness
sequence, so we will assume ha D+(Γ) ≥(2α)/π and ob ain a con a-
dic ion.
Fix an a bi a y kand conside he unique unc ion gk∈Fp
αwi h
gk(γn)=eα|γk|2,n=k
0,n=k.
Se ing g(z)=(z−γk)gk(z), we obse e ha g(z)/(z−γn)∈Fp
α o
a bi a y n. Hence we mus ha e
C
g(z)
z−γn
p
e−pα|z|2dσ(z)|g(γn)|pe−pα|γn|2.(6)
By he subha monici y o |g(z)/(z−γn)|p, we also ha e
|g(γn)|pe−pα|γn|2)−2<|z−γn|<2
|g(z)|pe−pα|z|2dσ(z)
≤)−2D(γn,2)
|g(z)|pe−pα|z|2dσ(z),
(7)
independen ly o )>0.
Fo he a gumen s o be used nex , i is con enien o apply a heo em
o Landau [4], which implies ha we may w i e ins ead
D+(Γ) = lim sup
→∞
max
z∈C
m(z, )
2,
292 A. P. Schus e , K. Seip
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In e pola ion in holomo phic spaces 293
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Alexande P. Schus e :
Depa men o Ma hema ics
San F ancisco S a e Uni e si y
San F ancisco, Cali o nia 94132
U.S.A.
E-mail add ess:[email p o ec ed]
K is ian Seip:
Depa men o Ma hema ical Sciences
No wegian Uni e si y o Science and Technology
N-7491 T ondheim
No way
E-mail add ess:[email p o ec ed]
P ime a e si´o ebuda el 30 d’ab il de 1999,
da e a e si´o ebuda el 21 de se emb e de 1999.