Bergman interpolation on finite Riemann surfaces. Part II: Poincar\'e-Hyperbolic Case

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by  Dror Varolin

BERGMAN INTERPOLATION ON FINITE RIEMANN SURFACES.
PART II: POINCARE´-HYPERBOLIC CASE
DRORVAROLIN
5
1
0
2 ABSTRACT. WeformulatetheBergman-typeinterpolationproblemonfiniteopenRiemannsurfacescovered
bytheunit disk. Ourversion of theinterpolation problemgeneralizes Bergman-type interpolationproblems
c
previouslystudiedbySeip,Berntsson,OrtegaCerda`,andanumberofotherauthors. Wethenprovesufficient
e
D conditionsforasequencetobeinterpolating. Whenthecurvatureoftheweightinquestionisboundedinan
appropriatesense,weshowthatthesufficientconditionsarealmostnecessary,butnotquite.Theresultsextend

1 work of Ortega Cerda`, who resolved the case in which the boundary of the surface is pure 1-dimensional.
1 Our version of the interpolation problem effectively changes the geometry of the underlying space near the
0-dimensionalboundarycomponents,orpunctures,therebylinkinginacrucialwaywiththepreviousarticlein
] thistwo-partseries.
V
C
INTRODUCTION
.
h
t In this sequel to our paper [V-2015], we continue our investigation of interpolation in Bergman spaces
a
m over finite Riemann surfaces. Recall that for an open Riemann surface X with conformal metric ω and
weightfunction ψ,wedefinedtheso-called (generalized) Bergmanspace
[

2 H 2(X,e ψω):= g (X); g 2e ψω < + .
− −
v ∈ O | | ∞
5 (cid:26) ZX (cid:27)
In [V-2015] we defined another Hilbert space that measures the size of data along a closed discrete subset
2
2 Γ X. Inthepresent article,wechangethisHilbertspaceslightly. Todefinethisslightly modifiedHilbert
2 spa⊂ce,wefirstrecallthatthepointwiseinjectivity radiusofx X isthenumber
0 ∈
. ι (x) := sup r > 0; Dω(x)iscontractible ,
1 ω { r }
0 whereDω(x)denotes thesetofallpointswhoseω-geodesic distance toxislessthanr. Wedefine
5 r
1 ˆι (x) := min(ι (x),1),
ω ω
:
v
andlet
i
X
A (x) := ω
ω
ar ZDˆιω(x)(x)
betheω-areaofthediskD (x). Finally, givenacloseddiscretesubsetΓ X,weset
ˆιω(x) ⊂
ℓ2(Γ,e ψ):= f :Γ C; f(γ)2e ψ(γ)A (γ) < + .
− − ω
 → | | ∞
 Xγ∈Γ 
REMARK. When (X,ω) is asymptotically flat, which was the case in [V-2015], the areas Aω(γ) are uni-
formlyboundedaboveandbelowbyapositiveconstants,andthus,fromthepointofviewoftheinterpolation
problem,ourdefinitionshereareessentiallygeneralizationsofthoseof[V-2015]. Thereare,ofcourse,other
possible generalizations, butthisnaturaldefinitionallowsustoproveratherstrongresults.
⋄
WesaythatΓisaninterpolation sequence(fortheweightfunctionψandconformalmetricω)iftherestric-
tionmap
R : H 2(X,e ψω) ℓ2(Γ,e ψ)
Γ − −
→
1
is surjective. For a given triple (X,ω,ψ), the goal is to characterize interpolation sequences Γ in terms of
geometricproperties ofΓ,preferably expressedusingthemetricω andtheweightfunction ψ.
Inthepresentarticle, wefocusonfiniteopenRiemannsurfacesX whoseuniversalcoveristheunitdisk
(such Riemann surfaces were called Poincare´-hyperbolic in [V-2011]), and which have smooth bound-
ary consisting of compact, 1-dimensional boundary components (which we call border curves) and 0-
dimensionalboundarycomponents(whichwecallpunctures). EverysuchRiemannsurface(ormoregener-
ally, every Riemann surface covered bythe disk) possesses aunique metric ω ofcurvature 4, which we
P
−
callthePoincare´ metric(hencethename”Poincare´-hyperbolic”).
Themainresultofthepaperisthefollowingtheorem.
THEOREM 1. LetX beafiniteopenRiemannsurfacecoveredbythedisk,withitsPoincare´ metricωP. Let
ϕ C2(X) be a weight function satisfying the following conditions: there exist positive constants m and
∈
M suchthat
(⋆) eachpuncture p isanisolated boundary pointofanopensetP X thatisbiholomorphic tothe
j j
⊂
punctured disk,suchthat
4ω (ζ)+mωj(ζ) ∆ϕ(ζ) Mωj(ζ), ζ P ,
P c ≤ ≤ c ∈ j
whereωj isthecylindrical metricinP ,and
c j
(B) eachborder curveC istheouter boundary ofanopensetA that isbiholomorphic toanannulus,
j j
suchthat
mω (ζ) ∆ϕ(ζ) 2ω (ζ) Mω , ζ A .
P P P j
≤ − ≤ ∈
Let Γ X be a closed discrete subset. Then the restriction map R : H 2(X,e ϕω ) ℓ2(Γ,e ϕ) is
Γ − P −
⊂ →
surjective if
(i+) Γisuniformly separated, and
(ii+) theasymptotic upperdensityD+(Γ)ofΓisstrictlylessthan1.
ϕ
Conversely, ifR issurjective, then
Γ
(i-) Γisuniformly separated, and
(ii-) D+(Γ) 1,andifmoreoverX hasnoisolated boundary components thenD+(Γ) < 1.
ϕ ≤ ϕ
AfewremarksregardingtheprecisemeaningofsomeofthetermsinTheorem1areinorder.
(a) ThecylindricalmetricinC issomeconstantmultipleofthemetricω (z) = (2z 2) 1√ 1dz dz¯.
∗ c −
| | − ∧
InaRiemannsurfaceX coveredbythedisk,onehasspecialcoordinatesnearpunctures;coordinates
thatareadaptedtothehyperbolicgeometryofX inheritedfromthecover(cf. Section4.1). Inthese
coordinates, thecylindrical metricisgivenbythesameformula.
(b) Uniform separation ofaclosed discrete subset ismeasured withrespect tothegeodesic distance of
thecylindrical metricnearthepunctures, andofthePoincare´ metricnearthebordercurves.
(c) As in the prequel to this article, the asymptotic upper density D+(Γ) is the least upper bound of
ϕ
certain weighted densities of points of Γ in large geodesic disks (for the hyperbolic metric, except
near the puncture, in which case the disks are geodesic for the cylindrical metric), the least upper
bound being taken withrespect to the centers of these disks. Later inthe introduction wewillgive
a slightly imprecise version of the definition, and the precise definition will be given in Section 4,
afterthedefinitionofdensity hasbeengivenforthepunctured disk.
(d) Sinceϕissmooth,thereexistsasmooth,positive(1,1)-form ΘonX suchthat∆ϕ Θ.
≥ −
AscruciallyobservedbyOrtegaCerda` in[O-2008],sinceΓisclosedanddiscrete,defining(andcomput-
ing) the density reduces todoing sonear the boundary ofX. Infact, the density remains unchanged ifone
discards anyfinite subset of the sequence inquestion. Therefore itessentially suffices to define density for
thePoincare´ diskandthePoincare´ puncturedunitdisk. ThecaseofthePoincare´ diskhasbeenwell-studied,
but to the author’s knowledge the case of the Poincare´ punctured disk has not been directly considered in
interpolation problemsuntilnow.
2
If X has no punctures (and therefore at least one border curve), Theorem 1 is due to Ortega Cerda`
[O-2008]. Weshall discuss OrtegaCerda`’s theorem inmoredetail inSubsection 4.2. Inourprevious work
[SV-2008] with A. Schuster, we allowed punctures, but our results carried cumbersome hypotheses which
implied, in particular, that our sequences did not accumulate at the punctures. Essentially, punctures were
omitted in [O-2008], and were restrictive in [SV-2008], because the Green’s function was used to define
density,andGreen’sfunctionsdonotseeisolatedboundarypoints,asthelatterareirregularfortheDirichlet
problem. To some extent, the present article and its predecessor [V-2015] emerged from a desire to study
interpolation alongsequences thatcouldaccumulate onthepunctures.
REMARK. Thespecial casewhereX hasatleastonepuncture butnobordercurves canalmostbederived,
with some work, from the main result of [V-2015]. The proof there is very similar to the proof here, but
in the present article, we can weaken somewhat the requirements on our weight functions, because the
hyperbolicgeometryofthepunctures letsusmakeuseofatechniqueintroduced byDonellyandFefferman
[DF-1983],andfurtherdevelopedbyOhsawainanumberofarticles(Seealso[B-1996]). Thetechniqueof
Donnelly-Fefferman-Ohsawa willbepresented, initsmostgeneralform,inSubsection1.4.
⋄
Therearetwoimportant special casesofTheorem1, namelywhenX = Distheunitdisk, andX = D
∗
isthepunctureddisk. Thecaseoftheunitdisk,inwhichcondition(⋆)ofTheorem1isvacuous,wastreated
by a number of authors. The first results are found in the work of K. Seip in the unweighted (and also the
standardly weighted) Bergman disk [Seip-93]. Berndtsson and Ortega Cerda´ [BO-1995] were the first to
treat the weighted case, for which they proved sufficiency. (Berndtsson and Ortega Cerda` did not give an
explicit definition of asymptotic density in their paper, but it is effectively defined there.) To the author’s
knowledge, necessity seems never to have been completely written down in the case of the unit disk with
general weights, though in the work [OS-1998] of Ortega Cerda´ and Seip there is an essentially complete
sketchofhowtodoit. Wehavethereforedecidedtoprovidecompletedetailshere,wherewemostlyfollow
theideas ofOrtegaCerda´ andSeip, withonlyminormodifications that suitourowntaste; weconsider this
partoftheworktobeessentially known.
To the author’s knowledge, the case of the punctured disk has never been considered before the present
article. In fact, the punctured disk case contains nearly all the ideas needed to handle the general case.
Roughly speaking, a closed discrete subset of D can be written as a union of two sequences, the first of
∗
whichonlyaccumulates attheouter boundary, orborder, ofD ,andthesecond ofwhichaccumulates only
∗
atthepuncture. Thisdecompositionisnotunique,butthenotionsofuniformseparationandofupperdensity
arebothindependent ofthedecomposition.
Near the border, a sequence in D looks very much like a sequence in D, so its upper density can be
∗
defined, after asmallamountofcare, asthough thesequence isindeed asequence inD. However, nearthe
puncture, the geometry one must consider is determined by our definition of the ℓ2-spaces of functions on
the sequence. The rather natural definition we have chosen provides a geometry near the puncture that is
verymuchlikethecylindricalgeometryconsideredin[V-2015]. Weimportthedefinitionofdensitynearthe
puncturefromthecylindricalcase,thoughbecausecylindricalgeometryisessentiallyflatandourspacesare
negatively curved, formulating the definition ofdensity forsequences near thepuncture requires morecare
than was needed near the border. Finally, the density of the sequence Γ is defined as the maximum of the
densitynearthepunctureandthedensityneartheborder. Aswementionedearlier,thedensityofasequence
is unchanged if we throw away finitely many points of the sequence, and this is the reason why the upper
density isindependent ofthedecomposition ofthesequence intoborder-supported andpuncture-supported
subsequences.
Defining the density in the general case is now more clear. The sequence Γ is decomposed into a finite
part, and a union of ”tails”. Each tail, i.e., subsequence which accumulates near at most one boundary
component, hasanupperdensity, andthisupperdensityisliketheupperdensityinthediskiftheboundary
component is 1-dimensional, and like the upper density in the cylinder if the boundary component is 0-
dimensional. Theupperdensity isthenthemaximumofthefinitenumberofupperdensities thusobtained.
3
ToestablishthesufficiencyoftheconditionsofTheorem1forinterpolation, weactuallyproveastronger
result, whichwecallstrong sufficiency. Theresult weobtainisstronger inthesense thatwedonotrequire
the weight functions to be smooth, or to have Laplacian that is bounded from above. We do require the
strictly positive lower bound near the boundary, which was not always the case in [V-2015]. The reason,
vaguely speaking, is that there is at present no sharp L2 extension theorem in the setting of manifolds
that admit non-trivial functions with self-bounded gradient, i.e., in which one can apply the technique of
Donnelly-Fefferman-Ohsawa, discussed in Subsection 1.4. We hope to return to the sharp L2 extension
problem onanother occasion.
Thearticleisorganized asfollows.
In Section 1 we recall some background and establish notation that will be followed in the rest of the
article. In particular, we discuss metrics of constant negative curvature, and then recall the L2 extension
theorem, the Donnelly-Fefferman-Ohsawa Technique, some results on weights with bounded Laplacian in
theunitdisk,andthePoisson-Jensen Formula.
InSection2westateandproveTheorem1inthecaseoftheunitdisk. Aspreviouslymentioned,thisisour
owntakeonwhatisessentially workofBerndtsson, OrtegaCerda` andSeip. Butperhaps mostimportantly,
weprecisely formulatetheasymptoticupperdensityfortheunitdisk.
InSection 3, which is the longest section of the article, westate and prove Theorem 1 in the case of the
punctured disk. Inthissection, wedevelopthemostimportantpartsofthearticle: thecylindrical geometry
ofthepuncture, thedecomposition ofsequences intothosesupportedneartheborderandnearthepuncture,
andalltherelatedtechnicalmachinerythatisneededtotreatthetwotypesofsequences,andtogluetogether
dataobtained fromthesesubsequences intodatafortheentiresequence.
InSection4,webeginbyrecallingsomegeometryoftheendsofafiniteRiemannsurfacewithpunctures.
Wethenhaveallthetools weneedtocomplete theproof ofTheorem 1,butbefore doing sowediscuss the
special caseproved byOrtegaCerda`, whereX hasnopunctures. Wethenturnourattention totheproofof
Theorem 1. First, wedefine the asymptotic upper density. Thenweestablish necessity. Finally weprove a
strong sufficiency theorem as in the cases of the disk and the punctured disk, and show how it implies the
weakerformofsufficiencyrequired forthecompletion oftheproofofTheorem1.
The article ends with a short section that remarks on the equivalence of our interpolation problem with
theShapiro-Shields interpolation problem.
ACKNOWLEDGMENT. I am grateful to Henri Guenancia, Long Li, Jeff McNeal, Quim Ortega Cerda` and
AlexSchusterformanystimulatingconversationsbothpastandpresent,andwithoutwhichthisworkwould
nothavecometobe. Iamalsogratefultotheanonymousrefereeforveryusefulandinteresting remarks.
⋄
1. BACKGROUND
LetX beaRiemannsurface. Wewritedc = √ 1(∂¯ ∂),anddenoteby
−
2 −
∆ := ddc = √ 1∂∂¯
−
the Laplace operator (so normalized). Notethat our Laplacian sends functions to (1,1)-forms (orcurrents,
ifthefunctions areonlylocallyintegrable). Wedenote byφ thediskinvolution sending 0toz:
z
z ζ
φ (ζ):= − .
z
1 z¯ζ
−
Thefunction
(z,w) φ (w) = φ (z)
z w
7→ | | | |
iscalledthepseudohyperbolic distancebetweenz andwinD. Wedenoteby
D (z) := ζ D; φ (ζ) < r
r z
{ ∈ | | }
thepseudohyperbolic diskofradiusrandcenterz.
4
1.1. Complete metrics of constant negative curvature. Recall that if ω is a smooth conformal metric,
expressed inlocal coordinates z asω = e ψ(z)√ 1dz dz¯,then thecurvature R(ω)ofω isdefined asthe
− 2− ∧
(global) (1,1)-form
R(ω) = ∂∂¯ψ.
Wesaythecurvature isconstant (resp. positive, negative)ifthe(globally defined)function
√ 1R(ω) ∂2ψ
− = 2eψ :X R
ω ∂z∂z¯ →
(sometimes also called the curvature, or Gaussian curvature) is constant (resp. positive, negative). It is
well-known that every Riemann surface admits a complete conformal metric of constant curvature. This
curvatureispositiveifandonlyifX = P ,0ifandonlyifX iscoveredbythecomplexplane,andnegative
1
if and only if X is covered by the disk. Thus no open Riemann surface has a complete metric of constant
positive curvature, and an open Riemann surface X has a complete flat metric if and only if X = C or
X = C . Uptohomothety, inCthereisaunique flatmetric. InC thecompleteflatmetricisuniqueupto
∗ ∗
aconstant multiple. Inthehyperbolic case, things areevenbetter. IfX iscovered bythe unit disk, then X
hasauniquemetricofconstantcurvature 4,aswenowrecall.
−
1.1.1. Existenceanduniqueness ofthehyperbolic metric. Ontheunitdisk,onehasthePoincare´ metric
√ 1dz dz¯ √ 1 1
ω := − ∧ = − ∂∂¯log ,
P 2(1 z 2)2 2 1 z 2
−| | −| |
whichiscomplete andhasconstant negative curvature equalto 4. ThePoincare´ metrichastheadditional
featurethatAut(D) Isom(ω ). ItfollowsthatifX isaRiema−nnsurfacewithcoveringmapπ : D X,
P
then the group of dec⊂k transformations G Aut(D) consists of isometries of ω , and thus we can→push
π P
⊂
ω forward to X by π, obtaining a metric that we continue to denote by ω , and also call the Poincare´
P P
metric. Wenotethatthemetricω iscompleteonX. Explicitly,
P
dz(ξ)2
ω (dπ(z)ξ,dπ(z)ξ)= | | .
P 2(1 z 2)2
−| |
REMARK 1.1. SometimestheRiemannsurfaceX mayitselfbeanopensubsetofanotherRiemannsurface
Y that is covered by the unit disk. In this case, there may be unnecessary confusion in the notation ω .
P
Thus,whenweneedtospecify thesurfaceaswell,wemaywriteωX forthePoincare´ metricofX.
P ⋄
Finally, ω is the only metric of constant curvature 4. Indeed, if ω and ω are two complete metrics
P 1 2
ofconstant negativecurvature conX,wemayliftthe−mtotheunitdiskviaπ,andiftheyareequalonD,
thentheyareequalonX. Thus−wemightaswellassumeX = D. Write
√ 1
ω = eui − dz dz¯, i =1,2.
i
2 ∧
The remainder of the proof is due to Ahlfors [A-1938]. We want to show that u = u , and by symmetry
1 2
it suffices to show that u u . To establish the latter, let f : D (0) D;z z/r and write v =
1 2 r r
≤ → 7→
f u +logr2. Observethatthemetricev√ 1dz dz¯= f ω hasconstantnegativecurvature conD (0),
r∗ 2 2− ∧ r∗ 2 − r
and it is also complete there. Onthe other hand, ω , while having curvature con D (0), is ofcourse not
1 r
−
completethere.
LetE D (0)betheopensetofallpointswhereu > v. Seth := u v. Wehave
r 1 1
⊂ −
√ 1
∆h= c(eu1 ev) − dz dz¯.
− 2 ∧
Itfollows that hissubharmonic on E,and therefore cannot take its maximum inanyinterior point of E in
D (0). It must thus assume its maximum on the boundary of E. But at a boundary point of E that lies in
r
D (0),wemusthaveu = vbycontinuity, sothemaximumisnotachievedintheinteriorofD (0). Onthe
r 1 r
5
other hand, onthecircle∂D (0), h = because themetricf ω iscomplete inD (0)whilethemetric
r −∞ r∗ 2 r
ω isnot. ItfollowsthatE isempty,andtherefore u v. Lettingr 1,weseethatu u ,asdesired.
1 1 1 2
≤ → ≤
1.2. HyperbolicandpseudohyperbolicdistanceofD. Recallthattheω -geodesicdistancebetweentwo
P
pointsz,w Dis
∈
1 1+ φ (w)
z
dist (z,w) = log | |,
P
2 1 φ (w)
z
−| |
Indeed, sinceAut(D) Isom(ω ),foranappropriate θ Rwehave
P
⊂ ∈
dist (z,w) := dist (e√ 1θφ (z),e√ 1θφ (w)) = dist(0, φ (w)).
P P − z − z z
| |
Andsincethegeodesics emanatingfromtheoriginarerays,
r dt 1 1+r
dist(0,r) = = log .
1 t2 2 1 r
Z0 − −
Itfollowsthattheso-called pseudohyperbolic distance ρ(z,w) := φ (w) satisfies
z
| |
e2distP(z,w) 1
φ (w) = − = tanh(dist (z,w)).
| z | e2distP(z,w)+1 P
Inparticular, thehyperbolic distance ismonotonically increasing inthepseudohyperbolic distance, andthe
ratioofthetwodistances converges to1asthepairofpointscomestogether.
1.3. The L2 extension theorem. Since the work of Ohsawa and Takegoshi [OT-1987], there have been
manystatementsandproofs(aswellasapplications) oftheoremsonL2 extensionofholomorphicfunctions
andsections ofholomorphic linebundles. Wewillmakeuseofthefollowing version, proved bytheauthor
in[V-2008].
THEOREM 1.2. Let(X,ω) beaStein Ka¨hler manifold, andlet Z X beasmooth hypersurface. Assume
⊂
there exists a section T H0(X,L ) and a metric e λ for the line bundle L X associated to the
Z − Z
∈ →
smoothdivisor Z,suchthate λ isstillasingular Hermitianmetric,and
− Z
|
sup T 2e λ 1.
−
| | ≤
X
Let H X be a holomorphic line bundle with singular Hermitian metric e ψ such that e ψ is still a
− − Z
→ |
singular Hermitianmetric. Assumethat
√ 1(∂∂¯ψ+Ricci(ω)) √ 1∂∂¯λ
Z
− ≥ −
and
√ 1(∂∂¯ψ+Ricci(ω)) (1+δ)√ 1∂∂¯λ
Z
− ≥ −
forsomepositive constant δ 1. Thenforanysection f H0(Z,H)satisfying
≤ ∈
f 2e ψ
−
| | dA < +
dT 2e λ ω ∞
ZZ | |ω −
thereexistsasection F H0(X,H)suchthat
∈
24π f 2e ψ
F|Z = f and |F|2e−ψdVω ≤ δ d|T| 2e− λdAω.
ZX ZZ | |ω −
6
1.4. ThetheoremofDonnelly-Fefferman-Ohsawa. Let(X,ω)beaKa¨hlermanifold,L X aholomor-
→
phic line bundle with singular Hermitian metric e ϕ, and Ω X a pseudoconvex domain with smooth
−
boundary. Suppose there are positive functions τ and A on Ω⊂⊂with τ C2-smooth. One has the following
well-knownidentity. (See,forexample,[V-2008].)
THEOREM 1.3(Twistedbasicestimate). ForanyL-valued (0,1)-form β inthedomainof∂¯ψ∗ onΩonehas
theestimate
(τ +A)∂¯ β 2e ψdV + τ ∂¯ β 2e ψdV
| ψ∗ | − ω | ψ∗ | − ω
ZΩ ZΩ
∂τ ∂¯τ
τ(∂∂¯ψ+Ricci(ω)) ∂∂¯τ ∧ β,β e−ψdVω.
≥ − − A
ZΩ(cid:28)(cid:26) (cid:27) (cid:29)
Thetwistedbasicestimateisobtained fromtheBochner-Kodaira Identity
∂¯ β 2e ϕdV + ∂¯β 2e ϕdV = ∂∂¯ϕ+Ricci(ω) β,β e ϕdV
| ϕ∗ | − ω | | − ω − ω
ZΩ ZΩ ZΩ
(cid:10)(cid:8) (cid:9) (cid:11) dS
+ ¯β 2e ϕdV + ∂∂¯ρ β,β e ϕ
|∇ | − ω { } − ∂ρ2
ZΩ Z∂Ω | |
(cid:10) (cid:11)
by substituting e ϕ = τe ψ, using the non-negativity of the last two terms, and applying the Cauchy-
− −
SchwarzInequality. Ofcourse,theBochner-Kodaira Identityonlymakessenseforcontinuous forms,andit
is proved for smooth forms in the domain of ∂¯ . But since the latter are dense in the graph norm, after we
ϕ∗
usethepseudoconvexity ofΩ,itsufficestoprovethetwistedbasicestimateforsmoothforms.
If we take τ = e η and A = τ for some smooth function η and positive constant ν, then we have the
− ν
followingestimate: forallL-valued(0,1)-forms β inthedomainof∂¯ ,
ψ∗
1+ν
e η ∂¯ β 2e ψdV + e η ∂¯β 2e ψdV
ν − | ψ∗ | − ω − | | − ω
ZΩ ZΩ
(1) e η ∂∂¯ψ+Ricci(ω))+∂∂¯η (1+ν)∂η ∂¯η β,β e ψdV .
− − ω
≥ − ∧
ZΩ
Withtheestimate(1),wecann(cid:10)o(cid:8)wprovethefollowingtheorem, duetoOhsaw(cid:9)a,wh(cid:11)ichisananalogue for∂¯
ofatheoremprovedbyDonelly-Fefferman fortheexteriorderivatived.
THEOREM 1.4(Donnelly-Fefferman, Ohsawa). Let(X,ω)beaSteinKa¨hlermanifold,L X aholomor-
→
phiclinebundle withsingular Hermitianmetrice ψ,η W1,2(X)afunction, ν apositive number, andΘ
− ∈ ℓoc
anon-negative, almosteverywherepositive(1,1)-form suchthat
(2) √ 1(∂∂¯ψ+Ricci(ω)+∂∂¯η (1+ν)∂η ∂¯η) Θ.
− − ∧ ≥
ThenforanyL-valued ∂¯-closed(0,1)-form αsatisfying
eη α2e ψdV < +
| |Θ − ω ∞
ZX
thereexistsameasurable sectionuofLsuchthat
ν +1
∂¯u= α and eη u2e ψdV eη α2e ψdV .
| | − ω ≤ ν | |Θ − ω
ZX ZX
Proof. By standard approximation methods, we can replace X by a smoothly bounded pseudoconvex do-
mainΩ X,andassumethate ψ andη aresmoothfunctions. Withthesereductions, considerthelinear
−
⊂⊂
functional
L(∂¯ β) := β,α e ψdV
ψ∗ h i − ω
ZΩ
7
definedonthesubspace
Image(∂¯ ) := ∂¯ β ; β Domain(∂¯ ) .
ψ∗ { ψ∗ ∈ ψ∗ }
Since α Kernel(∂¯), it is orthogonal to the image of ∂¯ (the formal adjoint of ∂¯ acting on L-valued
∈ ψ∗
(0,2)-forms), andthusitsufficestorestrictL to∂¯ β forβ Kernel(∂¯). Butforsuchβ,wehave
ψ∗ ∈
L(∂¯ β) eη α2e ψdV e η Θ β,β e ψdV
ψ∗ ≤ | |Θ − ω − h{ } i − ω
(cid:18)ZΩ (cid:19)ZΩ
ν +1
eη α2e ψdV e η ∂¯ β 2e ψdV .
≤ ν | |Θ − ω − | ψ∗ | − ω
(cid:18)ZΩ (cid:19)ZΩ
Therefore L is continuous. Extending by 0 in Image(∂¯ ) and using the Riesz Representation Theorem,
ψ∗ ⊥
wefindasectionU ofLsuchthat
ν +1
e η(∂¯ β)U¯e ψdV = β,α e ψdV and e η U 2e ψdV eη α2e ψdV .
− ψ∗ − ω h i − ω − | | − ω ≤ ν | |Θ − ω
ZΩ ZΩ ZΩ ZΩ
Thefirstofthesesaysthat∂¯(e ηU) = α. Ifweletu= e ηU thenwehave
− −
ν +1
∂¯u= α and eη u2e ψdV eη α2e ψdV .
| | − ω ≤ ν | |Θ − ω
ZΩ ZΩ
Thiscompletes theproof. (cid:3)
In the case ofthe Poincare´ unit disk, wecan take η = log 1 to obtain the following result, which is
1 z2
stated(inaslightlydifferentbutequivalentform)andprovedin−[B| O| -1995],whereitisattributedtoOhsawa.
THEOREM 1.5. Letϕ :D [ , )beuppersemi-continuous andsatisfy
→ −∞ ∞
√ 1∂∂¯ϕ (2(1+ν)+c)ω
P
− ≥
forsomepositive numbersν andc. Thenforany(0,1)-form αonDsatisfying
α2 e ϕω < +
D| |ωP − P ∞
Z
thereexistsalocallyintegrable function usuchthat
ν +1
∂¯u= α and u2e ϕω α2 e ϕω
D| | − P ≤ cν D| |ωP − P
Z Z
InthecaseofthePoincare´ punctured disk(D ,ω ),weobtainthefollowingresult.
∗ P
THEOREM 1.6. Letϕ :D∗ [ , )beuppersemi-continuous andsatisfy
→ −∞ ∞
√ 1∂∂¯ϕ (2(1+ν)+c)ω
P
− ≥
forsomepositive numbersν andc. Thenforany(0,1)-form αonD satisfying
∗
α2 e ϕω < +
D∗| |ωP − P ∞
Z
thereexistsalocallyintegrable function usuchthat
ν +1
∂¯u= α and u2e ϕω α2 e ϕω
D∗| | − P ≤ cν D∗| |ωP − P
Z Z
Proof. InTheorem1.4letX = D ,L = ,η = loglog z 2 andψ = ϕ+η. Then
∗ −
O − | |
2ω = √ 1∂∂¯η = √ 1∂η ∂¯η,
P
− − ∧
andthus
∂∂¯ψ+Ricci(ω )+∂∂¯η (1+ν)∂η ∂¯η = ∂∂¯ϕ 2(1+ν)ω cω .
P P P
− ∧ − ≥
LettingΘ := cω completestheproof. (cid:3)
P
8
REMARK. NotethatHo¨rmander’sTheoremimpliestheseresultsifc 2,butnototherwise.
≥ ⋄
InSection4wewillextendextendtheorems1.5and1.6togeneralfiniteopenRiemannsurfacescovered
bytheunitdisk(cf. Theorem4.3).
1.5. WeightswithboundedLaplacianin(D,ω ). Werecallsomewell-knownmaterialthatwillbeused
P
intheproofofTheorem1.
Webegin with a result on a solution of Poisson’s Equation with locally uniform estimates. A proof can
befoundin[SV-2014].
LEMMA 1.7. Foreach0 < r < 1thereexists aconstant C = Cr > 0withthefollowing property. Forany
(1,1)-form θ C2(C)satisfying
∈
Mω θ Mω ,
P P
− ≤ ≤
thereexistsu C2(D (0))suchthat
r
∈
∆u= θ and sup(u + du ) CM.
| | | |ωP ≤
Dr(0)
Asacorollary, wehavefollowingresult, established in[BO-1995](seealso[SV-2014]).
LEMMA 1.8. Letϕ C2(D)satisfy
∈
Mω ∆ϕ Mω
P P
− ≤ ≤
for some positive constant M. Then for each r (0,1) there is a positive constant C such that for any
r
z Dthereisaholomorphic function F satisfyin∈g
∈
F(z) = 0, 2ReF(ζ) ϕ(ζ)+ϕ(z) C , and 2RedF(ζ) dϕ(ζ) C
| − | ≤ r | − |ωP ≤ r
forallζ D (z).
r
∈
Lemma1.8givesthefollowinggeneralizations ofBergman’sinequality. (See[OS-1998]foraproof.)
PROPOSITION 1.9. Letϕ C2(D)satisfy
∈
Mω ∆ϕ Mω .
P P
− ≤ ≤
Thenforeachr (0,1)thereexistsC = C (M)suchthatforallf H 2(D,e ϕω ),
r r − P
∈ ∈
(a)
f(z)2e ϕ(z) C f 2e ϕω ,
− r − P
| | ≤ | |
ZDr(z)
and
(b)
d(f 2e ϕ) (z) C f 2e ϕω .
| | | − |ωP ≤ r | | − P
ZDr(z)
COROLLARY 1.10. Let ϕ be a weight function as in Proposition 1.9. If Γ is a finite union of uniformly
pseudohyperbolically separated sequences then for each r (0,1) there exists a constant C = C (M)
r r
suchthatforallf H 2(D,e ϕω ), ∈
− P
∈
(a)
f(γ)2e ϕ(γ) C f 2e ϕω C˜ f 2e ϕω ,
− r − P r − P
γ Γ| | ≤ γ ΓZDr(γ)| | ≤ ZD| |
X∈ X∈
and
(b)
d(f 2e ϕ)(γ) C f 2e ϕω C˜ f 2e ϕω .
− r − P r − P
γ Γ| | | | ≤ γ ΓZDr(γ)| | ≤ ZD| |
X∈ X∈
9
1.6. Poisson-JensenFormula. IntheproofofnecessityofTheorem2.1,weshallmakeuseofthefollowing
weightedanalogueofthewell-knownPoisson-Jensen Formula,whichgivesweightedcountsofthenumber
ofzerosofaholomorphic functioninalargepseudohyperbolic disk. Toformulateit,wedenotetheGreen’s
function fortheunitdiskDwithpoleatz by
G (ζ) := log φ (ζ),
z z
| |
andthepseudohyperbolic diskofradiusrby
D (z) := ζ D; G(z,ζ) <logr .
r
{ ∈ }
THEOREM 1.11 (Poisson-Jensen Formula). Fix a weight function ψ C2(D). Let f (D), let z D,
∈ ∈ O ∈
andletr (0,1). Leta ,...,a bethe(possiblynotdistinct)zerosoff inD (z),andassumethatf(z) = 0,
1 N r
∈ 6
andthattherearenozerosoff ontheboundary ofthepseudohyperbolic diskD (z). Then
r
1 N r2 1 r2
log(f 2e ψ)dcG = log(f(z)2e ψ(z))+ log log ∆ψ.
π | | − z | | − φ (z)2 − 2π φ 2
Z∂Dr(z) j=1 | aj | ZDr(z) | z|
X
Proof. Recallthatdc = √ 1(∂¯ ∂),sothatddc = ∆. Let
2− −
f(ζ)2e ϕ(ζ)
−
K (ζ):= G (ζ) logr and H(ζ)= log | | .
z z −  N |φai(ζ)|2
j=1 r2
 
ByStokes’Theoremwehave Q
(3) HdcK K dcH = H∆K K ∆H.
z z z z
− −
Z∂Dro(z) ZDro(z)
Now,dcK = dcG ,K 0and∆K = πδ . Itfollowsthat
z z z ∂Do(z) z z
| r ≡
1 N r2
log f 2e ϕdcG = log f(z)2e ϕ(z)+ log +2 K dcG
π | | − z | | − φ (z)2 aj z
Z∂Dro(z) j=1 | ai | Z∂Dro(z) !
X
1 r
log ∆ϕ(ζ).
−π φ (ζ)
ZDro(z) | z |
ButsinceK 0,andapplication of(3)withH = K gives
z|∂Dro(z) ≡ aj
K dcG = K ∆K K ∆G = K (z) K (a ) = 0,
aj z aj z − z aj aj − z j
Z∂Dro(z) ZDro(z)
andthustheresultfollows. (cid:3)
2. INTERPOLATION IN (D,ωP)
Wewrite
A := ζ C; 1/2 < ζ < r .
r
{ ∈ | | }
Inthissection, weprovethefollowingspecialcaseofTheorem1.
THEOREM 2.1. Letϕ C2(D)beaweightfunction satisfying
∈
mω ∆ϕ 2ω Mω
P P P
≤ − ≤
forsomepositive constants mand M,and letΓ Dbeaclosed discrete subset. Then therestriction map
R :H 2(D,e ϕω ) ℓ2(Γ,e ϕ)issurjective⊂ifandonlyif
Γ − P −
→
(i) Γisuniformly separated withrespecttothegeodesic distance ofω ,and
P
10