interpolation and sampling on riemann surfaces

Interpolation and Sampling on Riemann Surfaces

Alexander P. Schuster and Dror Varolin

Introduction

The purpose of this paper is to employ methods from several complex variables, an-alytic geometry and potential theory on Riemann surfaces to study the problem ofwhether a discrete sequence in a Riemann surfaceX is an interpolation sequence or asampling sequence for certain Hilbert spaces of holomorphic functions onX.

The origin of our paper is the work of Berndtsson, Ortega-Cerda, Seip and Wallsten,among others. The work of these authors reveals the importance of the notion of densityof a sequence; in order to characterize the interpolation and sampling property fora discrete sequence, one should define upper and lower densities of the sequence inquestion. These densities should be such that a sequence is interpolating when theupper density, properly normalized, is less than 1, and sampling when the lower densityis greater than 1.

In this paper, we define a large collection of upper and lower densitiesD+f (Γ) and

D−f (Γ) of a discrete sequenceΓ. The densities are parameterized by a family of locallyintegrable functionsf on the positive real line. Our main results, Theorems 1.1 and 1.2,can be stated informally as follows.

Main Theorem (informally stated): LetX be a Riemann surface satisfying certainadditional hypotheses, and letΓ ⊂ X be a uniformly separated discrete subset. ThenΓ is an interpolating sequence ifD+

f (Γ) < 1 and a sampling sequence ifD−f (Γ) > 1.

The additional hypotheses we require from our Riemann surfaces are satisfied, forexample, on bordered or compact Riemann surfaces with a finite number of pointsremoved. See section 4. It is possible that these hypotheses hold for all open Riemannsurfaces, but we’ve not yet been able to verify this statement. The precise statement ofour results appears in section 1.

Our results are extensions to more general Riemann surfaces of the celebrated re-sults, in the complex plane and the unit disk, of Seip and Wallsten [Seip-93, SW-92],and of their generalizations in [BO-95]. In fact, the proofs of our main theorems arebased on the techniques of [BO-95], but the novelty in our work lies elsewhere, namelyin the use of potential theory, in the introduction of a large collection of densities, andin our case-independent proofs.

Indeed, we make clear the role of potential theory in the definition of separation ofsequences, and therefore in the spaceb2(Γ, g, ϕ) of values of holomorphic functionslying in certain generalized Bergman spaces onX. To elaborate somewhat, one mustuse disksDσ(γ), defined via a certain canonical potential on the surface, to uniformly

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separate the points of the sequences under consideration. Correspondingly, one mustintroduce a weightAg(Dσ(γ)) in the Hilbert norm onb2(X, g,Γ), equal to the metricarea of these potential-theoretic disks. (Again, the notation is explained in section 1.)The observation that separation is potential theoretic rather than metric is obscured inthe cases of the plane and the disk. In these cases a certain metric, called the fun-damental metric, obtained from the potential theory ofX agrees with the Euclideanmetric and the Poincare metric respectively. This accident ceases to occur in generalon Riemann surfaces that are not simply connected. For example, the twice puncturedplane is negatively curved but is not hyperbolic; every bounded subharmonic functionon it is constant.

Next, we have introduced an arbitrary locally integrable function in our definitionof the density. This implies that, in general, our density conditions furnish only suffi-cient conditions for interpolation and sampling. Nevertheless there are two significantadvantages to our approach. On the one hand, with the flexibility afforded by the pres-ence of an arbitrary functionf , the densities can be tailored to produce rather preciseresults of a different nature. Our main examples areShannon-typeresults on samplingand interpolation: we give rather sharp estimates on theNyquist radiusfor interpolationand sampling. (The Nyquist radius is a natural analogue of the Nyquist rate of sam-pling, discussed by Shannon in his famous theory of information.) We also give someexamples of simple summation formulas that allow one to test more easily whether asequence is interpolating or sampling.

On the other hand, the introduction of an arbitrary locally integrable functionf inthe definition of the densities paves the way to formalizing the problem of finding thosedensities that furnish not only sufficient, but also necessary conditions for a sequenceto be interpolating or sampling. We do not address the question of necessity in thispaper, hoping to return to it at a later date.

Finally, the methods of this paper unify known results. Our proofs of Theorems1.1 and 1.2 apply to both the plane and for the disk. Previously these two results haddistinct proofs.

Since the paper is rather geometric, we include some background material in partfor purposes of self containment. That said, the paper is not self contained, but wehave tried to make it easier for the reader not to follow endless reference lists. Weinclude a brief discussion of the needed potential theory and also a discussion of theBochner-Kodaira technique with two weights, following Siu [Siu-82, Siu-02]. In thecase of Riemann surfaces, much of the technique greatly simplifies, thus allowing usto give complete proofs that are rather short.

The paper is organized as follows. In section 2 we recall some background material.More precisely, we discuss the notion of extremal fundamental solution and some basicpotential theory. We also present a variant, due to Ohsawa, of Hormander’s theoremon solutions of the∂-equation withL2 estimates. In section 3 we discuss interpolationand sampling theorems for holomorphic sections of line bundles on compact Riemannsurfaces. Section 4 is devoted to showing that our hypotheses on the surfaces we con-sider are all satisfied if the surface in question is a finite surface. The main techniqueis to use compactness to reduce the problem to the boundaries, where we can com-pare the fundamental geometry with either the Euclidean geometry (parabolic case) orPoincare geometry (hyperbolic case). In section 5 we discuss our main theorems in

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some cases where explicit formulas can be given for our densities. More specifically,we look at how our theorems specialize in the case of the plane and the disk. We com-pare our results with those of [BO-95], but also obtain some new results that may beof independent interest. In section 6 we define the notions of Nyquist interpolation andsampling radii, and calculate these in some classical cases. In section 7 we give theproofs of our main theorems.

1 Statements of the main results

Let X be an open Riemann surface and letg be a conformal metric onX, expressedlocally asg = e−2ψ|dz|2. The area associated tog is denoteddAg. Every openRiemann surfaceX supports a fundamental solution to Laplace’s equation having cer-tain extremalproperties. IfX supports a bounded subharmonic function, then it has aGreen’s function. OtherwiseX supports a so-called Evans kernel. (See section 2.) Ineither case, we denote byE : X × X → R an extremal fundamental solution onX,and set

ρz(ζ) := eE(z,ζ), Dε(z) := {ζ ∈ X ; ρz(ζ) < ε} and Sε(z) = ∂Dε(z).

Let Γ ⊂ X be a discrete set. We will always assume thatΓ is uniformly separatedandsparse. These terms are explained in definition 2.3, as is a precise notion of theminimum separationσ = σ(Γ) between any two points ofΓ. Finally, we equipX witha weight functionϕ.

To the data(X,Γ, g, ϕ) we assign the generalized Bergman space

B2 = B2(X, g, ϕ) :={h ∈ O(X) ; ||h||2 :=

∫X

|h|2e−2ϕdAg < +∞},

whereO is the sheaf of holomorphic functions, and the associatedlittle space

b2 = b2(Γ, g, ϕ) :=

(sγ)γ∈Γ ;∑γ∈Γ

|sγ |2e−2ϕ(γ)Ag(Dσ(γ)) < +∞

,

whereAg(B) =∫BdAg.

Definition A uniformly separated sequenceΓ ⊂ X is said to be

(1) interpolating if for each(sγ) ∈ b2 there existsh ∈ B2 such thath(γ) = sγ , and

(2) sampling if there exists a constantA > 1 such that for anyh ∈ B2,

1A||h||2 ≤

∑γ∈Γ

|h(γ)|2e−2ϕ(γ)Ag(Dσ(γ)) ≤ A||h||2. (1)

The extremal fundamental solution defines a metrice−2ν for the Riemann surface,which we call thefundamental metric, as follows.

e−2ν(z)|dz|2 = limζ→z

|∂ρz(ζ)|2.

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We associate to our metricg = e−2ψ the two functions

uψ := ψ − ν and τψ := e2ν(∆ψ + 2∆uψ − 2|∂uψ|2

).

Remark. Here and in the rest of the paper, we employ the convention

∆f =∂2f

∂z∂z,

which differs from the usual Laplacian by a factor of1/4.

We defineRE := 1 if E is the Green’s function, andRE = +∞ if E is anEvans kernel. For each locally integrable functionf : [0, RE) → [0,∞) and eachr ∈ (0, RE), let cr := 2π

∫ r0tf(t)dt and

ξr(z, ζ) =1crf(ρz(ζ))e2ν(ζ)|dρz(ζ)|21Dr(z)(ζ), (2)

where1A is the characteristic function of a setA. We associate to every uniformlyseparated sequenceΓ upper and lower densities, defined by

D+f (Γ) := lim sup

r→REsupz∈X

∑γ∈Γ

π2 ξr(γ, z)

e2ν(z)∆ϕ(z) + τψ(z), (3)

and

D−f (Γ) := lim infr→RE

infz∈X

∑γ∈Γ

π2 ξr(γ, z)

e2ν(z)∆ϕ(z). (4)

In the following theorems, we consider Riemann surfaces satisfying certain technicalhypotheses that will be defined later.

Theorem 1.1. LetX be a fundamentally finite and harmonically homogeneous openRiemann surface with conformal metricg = e−2ψ and letϕ be a weight function onXsuch that, for somec > 1, 1

c ≤ e2ν∆ϕ ≤ c,

e2ν∆(ϕ) + τψ ≥1c

and e2ν |∂uψ|2 ≤ c. (5)

Then every uniformly separated, sparse sequenceΓ satisfyingD+f (Γ) < 1 is an inter-

polation sequence.

Theorem 1.2. Let (X, g) be a fundamentally finite and harmonically homogeneousopen Cauchy-Green surface andΓ ⊂ X a uniformly separated and sparse sequence.Supposeϕ is a weight function onX such that, for someC > 1, 1

C ≤ e2ν∆ϕ ≤ C.Assume also that the metricg = e−2ψ is bounded above by the fundamental metrice−2ν (i.e.,uψ ≥ 0) and moreover satisfies the differential inequality

2e2ν |∂uψ|2 ≤ e2ν∆uψ. (6)

ThenΓ is a sampling sequence wheneverD−f (Γ) > 1.

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Several remarks are in order. First, the definitions of fundamentally finite, harmonicallyhomogeneous and Cauchy-Green surface are given in section 4. In that section we alsoshow that every finite Riemann surface, i.e., bordered or compact surfaces with a finitenumber of points removed, is fundamentally finite, harmonically homogeneous andCauchy-Green.

The Cauchy-Green hypothesis is essentially the hypothesis that certain uniformCauchy estimates hold for possibly large domains on the surface in question. As thereader can infer from section 4.3, the hypothesis is rather weak. It is likely to befulfilled on any Riemann surface, but we are unable to show this.

We don’t know if every open Riemann surface is fundamentally finite and harmon-ically homogeneous. Theorem 1.1 will be proved in section 7.2 and Theorem 1.2 insection 7.3.

Remark. The expert in the study of interpolation and sampling sequences will observe,and perhaps find surprising, that the upper and lower densities are given by formulasthat differ by more than just the process of takings suprema versus infema. In partic-ular, the denominator in the upper density differs from the denominator in the lowerdensity. However, such phenomena should not be surprising. We illustrate this pointin section 3 by discussing interpolation and sampling properties for finite sequences.In this situation, the generalized Bergman space is the (finite dimensional) space ofholomorphic sections of a line bundleL on a compact Riemann surface of genusg. Inthat setting, the upper and lower densities

D+(Γ) :=#Γ

deg(L) + 2− 2gand D−(Γ) :=

#Γdeg(L)

differ when the genus is not1.

2 Background material

In this section we discuss the basic tools needed in the statements and proofs of Theo-rems 1.1 and 1.2.

2.1 Extremal fundamental solutions

Let δζ denote the Dirac mass atζ. Recall the following definition.

Definition 2.1. The Green’s function on a Riemann surfaceX is the functionG :X ×X → [−∞, 0) with the following properties.

a. For eachζ ∈ X, ∆zG(z, ζ) = π2 δζ(z).

b. If H : X ×X → [−∞, 0) is another function with propertya, thenH(z, w) ≤G(z, w) wheneverz 6= w.

It can easily be deduced that the Green’s function is symmetric.

Recall that a Riemann surface is said to behyperbolicif it admits a bounded sub-harmonic function,elliptic if it is compact, andparabolicotherwise. It is well known

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that a Riemann surface has a Green’s function if and only if it is hyperbolic. Property(b) guarantees that the Green’s function is unique.

On the other hand, a Riemann surface admits an Evans kernel if and only if itis parabolic (see page 352 of [NS-70]). Moreover, after prescribing (with somewhatlimited possibility) the logarithmic singularity at infinity, the Evans kernel is unique upto an additive constant.

Definition 2.2. An Evans kernel on a Riemann surfaceX is a symmetric functionS : X ×X → [−∞,+∞) with the following properties.

a. For eachζ ∈ X, ∆zS(z, ζ) = π2 δζ(z).

b. For eachr ∈ R andp ∈ X, the level set{ζ ∈ X ; S(ζ, p) = r} is compact andnon-empty.

As mentioned in section 1, we denote byE either the Green’s function of a hyperbolicsurface or the Evans kernel of a parabolic surface, and set

RE :={

1 X is hyperbolic+∞ X is parabolic

In particular,ρz(ζ) < RE for anyz, ζ ∈ X.

For a measurable subsetA ⊂ X, let

Dr(A) = {w ∈ X ; w ∈ Dr(a) for some a ∈ A}.

We define two separation conditions on a sequenceΓ.

Definition 2.3. LetΓ ⊂ X be a discrete set.

1. The separation constant ofΓ is the number

σ(Γ) := sup{r ; Dr(γ) ∩Dr(γ′) = ∅}.

and say thatΓ is uniformly separated ifσ(Γ) > 0.

2. We sayΓ is sparse if there is a positive constantNr,ε, depending only on0 <r, ε < RE , such that the number of points ofΓ lying in the setDr(Dε(z)) is atmostNr,ε for all z ∈ X.

2.2 Green’s formula

Let X be any open Riemann surface and letY ⊂⊂ X be an open connected subsetwhose boundary consists of finitely many smooth Jordan curves. It is well known thatthe Green’s functionGY for Y exists, is unique, is continuous up to the boundary andvanishes on the boundary. Moreover, the exterior derivatived(GY (ζ, ·)) is continuouson the boundary ofY .

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Remark. One can construct the Green’s functionGY from the extremal fundamentalsolutionE of X as follows. SinceY has smooth boundary, the Dirichlet Problem ofharmonic extension from the boundary can be solved onY . We then take

GY (ζ, z) := E(ζ, z)− hζ(z),

wherehζ is the harmonic function inY that agrees withE(ζ, ·) on the boundary ofY .

We writeHr,ζ(z) := GDr(ζ)(ζ, z).

In fact, the functionHr,ζ has a particularly simple form in terms of the extremal fun-damental solutionE:

Hr,ζ(z) = E(z, ζ)− log r, z ∈ Dr(ζ). (7)

Moreover, in this case we don’t need to assume thatr is a regular value ofρζ .Recall that on a Riemann surface with a conformal metric, the Hodge star operator

simplifies somewhat when expressed in analytic coordinatesz = x+√−1y: if f is a

real-valued function,α = α1dx+α2dy is a real 1-form andϕdx∧ dy is a real 2-form,then one has

∗f = fdAg = e−2ψfdx ∧ dy∗α = −α2dx+ α1dy

∗(ϕdx ∧ dy) = e2ψϕ.

Using this, we have4∆ = d ∗ d (recall that∆ = ∂∂ = 14 (∂2

x + ∂2y) in our convention),

and Green’s formula can be written

4∫D

f∆h− h∆f =∫∂D

f ∗ dh− h ∗ df. (8)

PuttingD = Dr(z) andh = Hr,z and using the definition of Green’s function, weobtain the following lemma.

Lemma 2.4. Let r < RE andζ ∈ X. Then

2πf(z) =∫Sr(z)

f ∗ dEz +∫Dr(z)

Hr,z∆f. (9)

In particular, if f is subharmonic, then

f(z) ≤ 12π

∫Sr(z)

f ∗ dEz (10)

with equality whenf is harmonic.

Recalling the definition (2) ofξr, we observe that, for any functionF ,∫X

F (w)ξr(z, w)e−2ν(w)√−1dw ∧ dw =

1cr

∫Dr(z)

Ff(ρz)dρz ∧ ∗dρz

=1cr

∫ r

0

tf(t)

(∫St(z)

F ∗ dEz

)dt.

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Thus, in view of (10) of Lemma 2.4, ifh is subharmonic then

h(z) ≤∫X

ξr(z, w)h(w)e−2ν(w)√−1dw ∧ dw (11)

with equality ifh is harmonic.

2.3 Bochner-Kodaira technique with two weights

In our proof of the interpolation theorem, we require a theorem for solving∂ withcertainL2 estimates. Such a theorem has been stated by Ohsawa [O-94] in a verygeneral situation, but there seem to be counterexamples at this level of generality (see[Siu-02]). However, in the case of Riemann surfaces there is a short proof of Ohsawa’stheorem. Since it is not easily accessible in the literature, we shall give a proof hereusing methods adapted from [Siu-02].

Let X be a Riemann surface with conformal metricg = e−2ψ and letV → Xbe a holomorphic line bundle with Hermitian metrich = e−2ξ that is allowed to besingular, i.e.,ξ may be inL1

`oc. One has the Bochner-Kodaira identity

||∂∗β||2 = ||∇β||2 + (2e2ψ∆(ξ + ψ)β, β), (12)

where∇(fdz) := (fz + 2ψzf)dz⊗2.

Indeed, straight-forward calculations show that the formal adjoints∂∗

of ∂ and∇∗ of∇ are given by

∂∗(hdz) = −e2ψ

(∂h

∂z− 2

∂ξ

∂zh

)and∇∗(hdz⊗2) = −e2ψ

(∂h

∂z− 2

∂ξ

∂zh

). (13)

Using these, another calculation shows that∂∂∗β −∇∗∇β = 2e2ψ∆(ξ + ψ)β, which

gives (12).We shall now make a simple but far-reaching modification of the identity (12).

To this end, lete−2ξ ande−2(ξ−u) be two metrics of the same line bundle. (Thusuis a globally defined function.) We assume moreover thate2(ψ−u)|∂u|2 is uniformlybounded.

Formula (13) implies that

∂∗ξ−uβ = ∂

∗ξβ − 2e2ψ∂u ∧ β. (14)

Substituting into the Bochner-Kodaira identity (12), we obtain

||∂∗ξ−uβ||2ξ = ||∇β||2ξ (15)

+(2e2ψ

{∆(ξ + ψ)− 2|∂u|2

}β, β

)ξ

−2Re(∂∗ξβ, 2e

2ψ∂u ∧ β)ξ.

Identity (15) is sometimes called the Bochner-Kodaira identity with two weights. TheCauchy-Schwarz inequality then shows that for anyε > 0 we have

(1 + ε−1)||∂∗ξ−uβ||2ξ ≥(2e2ψ

{∆(ξ + ψ)− 2(1 + ε)|∂u|2

}β, β

)ξ. (16)

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LettingTf := ∂(e−uf), we can rewrite (16) as

||T ∗β||2ξ−u ≥ Cε

(2e2(ψ−u)

{∆(ξ + ψ)− 2(1 + ε)|∂u|2

}β, β

)ξ−u

(17)

Suppose now that, for someδ > 0, one has the estimate

e2(ψ−u)(∆(ξ + ψ)− 2|∂u|2

)≥ δ.

Sincee2(ψ−u)|∂u|2 is bounded, we may chooseε > 0 sufficiently small in (17) toobtain

||T ∗β||2ξ−u ≥ C||β||ξ−u. (18)

A standard Hilbert space argument yields a functionf such that

Tf = α

with the estimate ∫X

|f |2e2ue−2ξdAg ≤ C||α||ξ−u. (19)

Finally, choosingu = uψ = ψ − ν, ϕ := ξ − 2u andU = e−uf gives the following.

Theorem 2.5. [O-94] Suppose that for someδ > 0,

e2ν∆ϕ+ τψ ≥ δ and e2ν |∂uψ|2 <1δ.

Then there exists a constantC = Cδ such that for anyα satisfying∫X

√−1α ∧ αe−2uψe−2ϕ < +∞

the equation∂U = α has a solution satisfying∫X

|U |2e−2ϕdAg ≤ C

∫X

α ∧ αe−2uψe−2ϕ.

3 Compact Riemann surfaces

Let X be a compact Riemann surface and letV → X be a holomorphic line bundle.We denote byVx the fiber ofV overx ∈ X. ThenΓ is interpolating if and only if theevaluation map

H0(X,L) 3 s 7→∑γ∈Γ

s(γ) ∈⊕γ∈Γ

Vγ (20)

is surjective, and sampling if and only if (20) is injective.

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Let Λ be the line bundle corresponding to the effective divisorΓ. One can under-stand the situation completely using the short exact sequence of sheaves

0 → OX(L⊗ Λ∗) → OX(L) →⊕γ∈Γ

Vγ → 0,

whereVγ(U) = Vγ if γ ∈ U andVγ(U) = 0 if γ 6∈ U . Passing to the long exactsequence, we have that

0 → H0(X,L⊗Λ∗) i0−→H0(X,L) eΓ−→⊕γ∈Γ

Vγδ0−→H1(X,L⊗Λ∗) i1−→H1(X,L) → ...

We see thate is injective if and only if Image(i0) = 0 and surjective if and only ifi1 isinjective, i.e., Image(δ0) = 0. We then have the following proposition.

Proposition 3.1. LetX be a compact Riemann surface of genusg, Γ ⊂ X a finitesubset andL→ X a holomorphic line bundle.

1. If #Γ < deg(L) + 2− 2g, thenΓ is interpolating.

2. If #Γ > deg(L), thenΓ is sampling.

Proof. To establish 1, note that by Serre duality,h1(X,L⊗Λ∗) = h0(X,KX⊗Λ⊗L∗),and the latter vanishes if

#Γ + 2g − 2− deg(L) = deg(KX ⊗ Λ⊗ L∗) < 0.

Similarly, if deg(L)−#Γ = deg(L⊗ Λ∗) < 0, thenh0(X,L⊗ Λ∗) = 0.

Part 1 of Proposition 3.1 can also be proved using Theorem 2.5. Because it is similarto the proof of our main interpolation theorems, we sketch this method here.

Analytic proof of Proposition 3.1.1.Let∑vγ ∈

⊕Vγ . First, observe that there is a

smooth sectionη of L such thatη(γ) = vγ for all γ ∈ Γ. In fact, by the usual cutoffmethod, we can takeη supported nearΓ and holomorphic in a neighborhood ofΓ.

Fix a conformal metrice−2ψ on X. Let τ be the canonical section ofΛ corre-sponding to the divisorΓ. By the degree hypothesis, there is a metrice−2ϕ for the linebundleL ⊗ Λ∗ such that the curvature

√−1∂∂(ϕ + ψ) of L ⊗ Λ∗ ⊗ K∗

X is strictlypositive onX. Thene−2ϕ/|τ |2 is a singular metric forL such that the curvature cur-rent ofe−2(ϕ+ψ)/|τ |2 is still strictly positive onX. Moreover, sinceη is holomorphicin a neighborhood ofΓ, we have

∫X|∂η|2|τ |−2e−2ϕ < +∞. By Theorem 2.5 (with

uψ ≡ 0; c.f. Proposition 4.10) there is a sectionu of L such that∂u = ∂η and∫X|u|2|τ |−2e−2(ϕ+ψ) < +∞. But sinceτ vanishes onΓ, so doesu. Thusσ = η − u

is holomorphic and solves the interpolation problem.

Remark. We note that ife−2ϕ is a metric for a holomorphic line bundleL, then wehave

deg(L) =1

2π√−1

∫X

∆ϕ,

showing the resemblance between Proposition 3.1 and our main theorems.

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4 Finite Riemann surfaces

We have made assumptions on the Riemann surfaces to which our main theorems apply.In this section, we give the definitions of those assumptions, and show that they allhold whenX is either a bordered Riemann surface with a finite number of puncturesor a compact Riemann surface with a finite number of punctures. Bordered Riemannsurfaces are hyperbolic, while compact punctured Riemann surfaces are parabolic. Weshall refer to all of these surfaces asfinite Riemann surfaces.

Although it will not matter to us, we note that on a bordered Riemann surface theGreen’s function vanishes on the boundary andρz is an exhaustion for everyz. Afterwe remove a finite number of points, the Green’s function for the resulting surfaceagrees with the green’s function of the unpunctured surface away from the punctures.Thusρz ceases to be an exhaustion.

4.1 Fundamental finiteness

Definition 4.1. We say that an open Riemann surfaceX with extremal fundamentalsolutionE is fundamentally finite if for each sufficiently smallσ ∈ (0, RE) there is aconstantC = Cσ > 0 such that for allz ∈ X andζ ∈ Dσ(z), the following estimateholds:

1C≤ e2ν(ζ)|∂ρz(ζ)|2 ≤ C. (21)

In this paragraph we prove the following result.

Theorem 4.2. Every finite Riemann surface is fundamentally finite.

Before getting to the proof, we make a few observations. Suppose given an ex-tremal fundamental solutionE on our Riemann surfaceX. By definition of funda-mental solution of∆, if z is a local coordinate onU ⊂ X then there exists a functionhU (ζ, η), harmonic in each variable separately, such thathU (ζ, η) = hU (η, ζ) and

E(p, q) = log |z(p)− z(q)|+ hU (z(p), z(q)).

The dependence ofhU onz is determined by the fact thatE is globally defined.For simplicity of exposition, we abusively write

E(z, ζ) = log |ζ − z|+ h(z, ζ).

We then have thatρz(ζ) = |z − ζ|eh(z,ζ) and, differentiating, we obtain

∂ρz(ζ) =ζ − z

|ζ − z|eh(z,ζ)

(12

+ (z − ζ)∂ζh(z, ζ)).

It follows that

e−2ν(ζ) = e2h(ζ,ζ), and

e2ν(ζ) |∂ρz(ζ)|2 = e2(h(z,ζ)−h(ζ,ζ))∣∣∣∣1 + 2(ζ − z)

∂h(z, ζ)∂ζ

∣∣∣∣2 (22)

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In particular, we point out that the right hand side of (22) is a well defined, since this isthe case for the left hand side.

Proof of theorem 4.2.We shall break up the proof into the hyperbolic and paraboliccase.

The case of bordered Riemann surfaces.We realizeX as an open subset of its doubleY . (For the definition of the double,

see [SS-54].) SinceX = X ∪ ∂X is compact, it suffices to bound the right hand sideof (22) in a setU ∩X, whereU is a coordinate chart inY . For coordinate charts whoseclosure lies in the interiorX, it is clear that this can be done. Indeed, ifU ⊂⊂ Xandz, ζ ∈ U , thenh is a smooth function that is harmonic in each variable separately,andρz(ζ) � |ζ − z| uniformly onU . Thus by takingσ sufficiently small, we obtainthe estimate (21) for allz ∈ U andζ ∈ Dσ(z). We thus restrict our attention to theboundary.

LetU ⊂ Y be a coordinate neighborhood of a boundary pointx ∈ ∂X. By takingU sufficiently small, we may assume thatU is the unit disk in the plane, thatU ∩ Xlies in the upper half plane and that∂X lies on the real line. It follows that the Green’sfunction is given by

E(z, ζ) = log |z − ζ| − log |z − ζ|+ F (z, ζ),

whereF (z, ζ) is smooth and harmonic in each variable on a large open set containingthe closure ofU . Indeed, the Green’s function for the upper half plane islog |z − ζ| −log |z−ζ|. The regularity ofF then follows from the construction of Green’s functionson finite Riemann surfaces using harmonic differentials on the double. (See [SS-54],§4.2.) It follows that inU ,

2∂h(z, ζ)∂ζ

= − 1z − ζ

+ 2∂F (z, ζ)∂ζ

and ρz(ζ) ≥ C|z − ζ||z − ζ|

.

Thus ∣∣∣∣2(ζ − z)∂h(z, ζ)∂ζ

∣∣∣∣ ≤ |z − ζ||z − ζ|

+ 2|z − ζ|∣∣∣∣∂F (z, ζ)

∂ζ

∣∣∣∣≤ C

|z − ζ||z − ζ|

≤ C ′ρz(ζ),

where the constantC ′ depends only on the neighborhoodU . The proof in the hyper-bolic case is thus complete.

The case of compact Riemann surfaces with punctures.Let E be the Evans kernel ofX. Fix p ∈ X and chooser so large that the setX − Dr(z) is a union of punctureddisksU1, ..., UN . We may think of eachUj as sitting inC, with the puncture at theorigin.

SinceDr(z) ⊂⊂ X, eachx ∈ Dr(z) has a neighborhoodU for which the ex-pression (22) is bounded above and below by positive constants, depending only onU ,

12

wheneverρz(ζ) < σ for some sufficiently smallσ again depending only onU . In-deed, in any such neighborhood the functionh is very regular, andρz(ζ) is uniformlycomparable to|z − ζ|.

Forz, ζ ∈ Uj , the Evans kernel has the form

E(z, ζ) = log |z − ζ| − λj log |ζ|+ F (z, ζ), (23)

whereλj > 0 with λ1 + ... + λN = 1, andF (z, ζ) is smooth across the origin(again see [NS-70]). Indeed, using the method of constructing harmonic differentialswith prescribed singularities (see [SS-54]§2.7) we can construct a function with theright singularities, defined everywhere onX. Such a function clearly can be writtenin the form (23) near the puncture. Thus by the uniqueness of the Evans kernel withprescribed singularities at the punctures, this function must differ fromE by a constant.

It follows that inU ,

2∂h(z, ζ)∂ζ

= −λjζ

+∂F (z, ζ)∂ζ

and

ρz(ζ) ≥ C|z − ζ||ζ|

.

Thus ∣∣∣∣2(ζ − z)∂h(z, ζ)∂ζ

∣∣∣∣ ≤ λj|z − ζ||ζ|

+ 2|z − ζ|∣∣∣∣∂F (z, ζ)

∂ζ

∣∣∣∣≤ C

|z − ζ||ζ|

≤ C ′ρz(ζ),

where again the constantC ′ depends only on the neighborhoodU . The proof of Theo-rem 4.2 is thus complete.

4.2 Harmonic homogeneity

Definition 4.3. X be a open Riemann surface with extremal fundamental solutionE.We say that say thatX is harmonically homogeneous if there is a constantC such that,for sufficiently smallσ > 0 and allz ∈ X,

supw∈Dσ(z)

exp

(4π

∫D2σ(z)

−G(w, ζ)e−2ν(ζ)

)(24)

≤ C infw∈Dσ(z)

exp

(4π

∫D2σ(z)

−G(w, ζ)e−2ν(ζ)

)< +∞,

whereG is the Green’s function for the domainD2σ(z).

13

Lemma 4.4. Let σ > 0 be a fixed, sufficiently small constant. IfX is fundamentallyfinite and harmonically homogeneous, andϕ is a weight for whiche2ν∆ϕ is boundedabove and below by positive constants, then there is a constantC = Cσ such that, forall z ∈ X and allw ∈ Dσ(z),

exp

(4π

∫D2σ(z)

−G(w, ζ)∆ϕ(ζ)

)≤ C (25)

Proof. By fundamental finiteness, harmonic homogeneity and the boundedness of∆ϕ,it suffices to prove the result when∆ϕ(ζ) = |dρz(ζ)|2 andw = z. In this case, it iseasy to show that the integral is equal to8σ2.

Proposition 4.5. Every finite Riemann surface is harmonically homogeneous.

Sketch of proof.Once again we can use compactness of the finite surfaces. The finite-ness of the integrals in question is easy, since Green’s functions have only a logarithmicsingularity, and are thus locally integrable. Thus we restrict ourselves to estimatingnear the boundary.

The local analysis used in the proof of Theorem 4.2 shows that, near the boundary,the disksDσ(z) are simply connected and that the metrice−2ν is equivalent to thePoincare metric of the disk in the hyperbolic case, and the metric|z|−2|dz|2 in theparabolic case.

The hyperbolic case follows from the fact that the Green’s functionG(w, ζ) iscomparable to the Green’s function of the disk. To see the parabolic case, it is easierto work with the complement of the unit disk rather than the punctured disk. Then themetric e−2ν is comparable to the Euclidean metric, the Green’s functionG(w, ζ) iscomparable to the Green’s function of the plane, and the necessary estimate follows asin the Euclidean case. This completes the proof.

4.3 Cauchy estimates

An ingredient in the proof of Theorem 1.2 is a version of the Cauchy estimates onrather large domains in a Riemann surface. In order to obtain such estimates, we needthe surface to support an integral kernel with certain properties.

Definition 4.6. A Riemann surfaceX with a conformal metricg is said to be Cauchy-Green, or simply CG, if for eachx ∈ X there exists a functionKx : X × X → R, called a Cauchy-Green kernel atx, such that the following conditions hold. Letσ ∈ [0, RE).

1. In the sense of distributions,∆zKx(z, ζ) = π

2 δz(ζ) for all z, ζ ∈ Dσ(x).

2. For everyε < σ/4 there exists a constantCε,σ such that for anyx ∈ X thefollowing estimates hold:

supz∈Dε(x)

∫Vσ(x)

e2ψ∣∣∣∣∂ρx∂ζ ∂Kx(z, ζ)

∂ζ

∣∣∣∣2 ≤ Cε,σ (26)

supz∈Dε(x)

|∂ρx(z)|−2

∫Vσ(x)

e2ψ|∂ρx|2∣∣∣∣∂2Kx(z, ζ)

∂z∂ζ

∣∣∣∣2 ≤ Cε,σ (27)

14

HereVσ(x) := Dσ(x)−Dσ/2(x).

We have the following lemma.

Lemma 4.7. Let (X, g) be a CG surface. Then there exists a constantCε,σ such thatfor anyx ∈ X the following Cauchy estimates hold.

supDε(x)

|h|2 ≤ Cε,σ

∫Dσ(x)

|h|2dAg, (28)

and

supDε(x)

|∂ρx|−2|h′|2 ≤ Cε,σ

∫Dσ(x)

|h|2dAg. (29)

Proof. Let f ∈ C∞0 (Dσ(x)) and writeKxz (ζ) = Kx(z, ζ). Applying formula (8) with

h(ζ) = Kxz (ζ), we obtain

π

2f(z) =

∫Dσ(x)

Kxz d∂f =

∫Dσ(x)

∂f ∧ ∂Kxz .

Now let ε < σ/4 and letχ ∈ C∞0 ([0, 3σ/4)) be such that

χ|[0, σ/2] ≡ 1 and sup |χ′| ≤ 5σ.

If h ∈ O(Dσ(x)), then withz ∈ Dε(x) we have

h(z) =∫Dσ(x)

hχ′(ρx)∂ρx ∧ ∂Kxz . (30)

An application of the Cauchy-Schwarz inequality and the estimate (26) gives the in-equality (28), while differentiation of (30) followed by an application of the Cauchy-Schwarz inequality and the estimate (27) gives the inequality (29).

Remark. If we allowed the constantsCε,σ to depend onx as well, Lemma 4.7 wouldfollow without(26)and(27).

Proposition 4.8. Every finite Riemann surfaceX with any conformal metric is aCauchy-Green surface.

Sketch of proof.In the case of a bordered Riemann surface with a finite number ofpunctures, one can find a Cauchy-Green kernel that does not depend on the pointx.This is done as follows. LetY be the double ofX, and fix any smooth distance functiononY . We letXε be the set of allx ∈ Y that are a distance less thanε fromX. For εsufficiently small,Xε −X is a finite collection of annuli whose inner boundaries formthe boundary ofX. We may take for our Cauchy-Green kernel the Green’s function ofXε. We leave it to the reader to check that the relevant estimates hold.

15

In the case of anN -punctured compact Riemann surface, one decomposesX as

X = Xcore ∪N⋃j=1

Uj ,

whereXcore is a bordered Riemann surface, and eachUj is a neighborhood of a punc-ture biholomorphic to the punctured disk. Each surface in the union has a Cauchy-Green kernel by the construction in the bordered Riemann surface case, and thus weare done.

4.4 Sparseness and separation

In the complex plane, the triangle inequality allows one to easily show that a uniformlyseparated sequence is sparse. In the unit disk, it is also the case that a uniformlyseparated sequence is sparse. Here one can use a hyperbolic triangle inequality (See[DS-03]).

In both of these situations, the stated inequality allows one to estimate the diameterof a setDε(Dr(a)) in terms ofε andr, uniformly for all a.

Such an estimate can always be found if it is allowed to depend on the base pointa. In any finite Riemann surfaceX we can, as in the proofs of fundamental finiteness,harmonic homogeneity and globally uniform Cauchy estimates, take advantage of thecompactness in the picture. In particular, we have uniform estimates if we have themin neighborhoods of the boundary. But on the boundary, the potential theory ofX iseither like that (near the boundary) in the upper half plane or (near infinity) in the plane,where we know from triangle inequalities that the needed estimates hold. We thus havethe following proposition.

Proposition 4.9. In a finite Riemann surfaceX every uniformly separated sequence issparse.

4.5 Existence of metrics satisfying (6)

LetX be an open Riemann surface. Observe that a functionu onX satisfies (6) if andonly if

∆(−e−2u) ≥ 0.

Since the function−e−2u is bounded above, we immediately obtain the followingproposition.

Proposition 4.10. LetX be a compact or parabolic Riemann surface. Then any func-tion satisfying the inequality(6) is constant.

In particular, we may assume that whenX is parabolic, the metricg in Theorem 1.2 isjust the fundamental metric.

Let us turn now to the hyperbolic case. Leto ∈ X be any point. Then

ρo∆ρo = ρ2o∆E(o, ·) + ρ2

o|∂E(o, ·)|2 = ρ2o|∂E(o, ·)|2 = |∂ρo|2,

16

and thus

∆ρ2o = 4|∂ρo|2 = |dρo|2.

We let

u = −12

log(1− ρ2o).

The reader versed in Several Complex Variables will recognize this function as thenegative log-distance-to-the-boundary. Calculating, we have

∆u− 2|∂u|2 =(

∆ρ2o

2(1− ρ2o)

+|∂ρ2

0|2

2(1− ρ2o)2

)− 2

|∂ρ2o|2

4(1− ρ20)2

=|dρ0|2

2(1− ρ20)≥ 0.

Moreover, observe that

|∂u|2 =ρ2o|∂ρo|2

(1− ρ2o)2

.

Proposition 4.11. Let X be a hyperbolic Riemann surface. Then there is always afunctionu ≥ 0 satisfying the differential inequality(6). Moreover, ifX is a finitehyperbolic surface, then one can chooseu such thate2ν |∂u|2 is uniformly bounded.

Sketch of proof.It remains only to verify the last assertion. By compactness, it sufficesto prove the desired estimate in a neighborhood of the form{z ∈ C ; |z| < 1, Imz >0} in the upper half plane. As above, one can takeν = − log Imz the Poincare poten-tial. Moreover, one can show that

1− ρo(z) = Imz + higher order terms.

The proposition now follows.

5 Explicit examples

5.1 The Euclidean plane

In this paragraph, we consider the case of the complex planeX = C. The plane is themain example of a parabolic Riemann surface. The Evans kernel inC is unique, and isgiven by

E(z, ζ) = log |z − ζ|.Thusρz(ζ) = |z− ζ| and the disksDσ(z) are simply the Euclidean disks|z− ζ| < σ.A simple calculation shows that

|dρz(ζ)|2 = 4|∂ρz(ζ)|2 = 1,

and thus the Green metric is just a multiple of the Euclidean metric. It follows that theweighted Bergman space in this case is

BF2 ={h ∈ O(C) ; ||h||2ϕ :=

∫C|h|2e−2ϕdm < +∞

},

17

wheredm is Lebesgue measure in the plane, and the space of sequences is

bf2 =

(sγ) ⊂ C ; ||(sγ)||2ϕ :=∑γ∈Γ

|sγ |2e−2ϕ(γ) < +∞

.

These spaces are sometimes calledgeneralized Bargmann-Fock spaces. The classicalBargmann-Fock space is the caseϕ(z) = |z|2/2.

The upper and lower densities are given by

D+f (Γ) = lim sup

r→∞supz∈C

∑Γ∩Dr(z)

f(|z − γ|)4∆ϕ

∫ r0tf(t)dt

and

D−f (Γ) = lim infr→∞

infz∈C

∑Γ∩Dr(z)

f(|z − γ|)4∆ϕ

∫ r0tf(t)dt

wheref is any non-negative, locally integrable function on[0,+∞).It is interesting to consider various different possibilities for the functionf appear-

ing in the definition of the densities.

(a) If f = 1 we obtain the densities

D+#(Γ) = lim sup

r→∞supz∈C

#(Γ ∩Dr(z))2r2∆ϕ

D−#(Γ) = lim infr→∞

infz∈C

#(Γ ∩Dr(z))2r2∆ϕ

which are the upper and lower densities in [BO-95].

(b) Letf(t) = e−t. Then

D+f (Γ) = lim sup

r→∞supz∈C

∑Γ∩Dr(z)

e−|z−γ|

4∆ϕ

D−f (Γ) = lim infr→∞

infz∈C

∑Γ∩Dr(z)

e−|z−γ|

4∆ϕ.

In particular we see that, for the classical Bargmann-Fock space,Γ is an interpo-lating sequence whenever

supz∈C

∑γ∈Γ

e−|γ−z| < 2

and a sampling sequence whenever

infz∈C

∑γ∈Γ

e−|γ−z| > 2.

18

Integration by parts shows that∫ r

0

#(Γ ∩Ds(z))ds

es= −e−r#(Γ ∩Dr(z)) +

∑γ∈Γ∩Dr(z)

e−|z−γ|.

A geometric covering argument shows that sinceΓ is uniformly separated, thereis a constantC such that, for allr > 0, #(Γ ∩Dr(z)) ≤ Cr2. Therefore

D+f (Γ) = sup

z∈C

∫ ∞

0

#(Γ ∩Ds(z))4∆ϕ(z)

ds

es

D−f (Γ) = infz∈C

∫ ∞

0

#(Γ ∩Ds(z))4∆ϕ(z)

ds

es

In particular, in the classical Bargmann-Fock spaceΓ is interpolating if

supz∈C

∫ ∞

0

#(Γ ∩Ds(z))ds

es< 2

and sampling if

infz∈C

∫ ∞

0

#(Γ ∩Ds(z))ds

es> 2

5.2 The disk

In this paragraph we consider the case of the unit diskX = D. The disk is the mainexample of a regular hyperbolic Riemann surface. The Green’s function for the disk is

E(z, ζ) = log |φz(ζ)|, where φz(ζ) =z − ζ

1− zζ

is the standard involution. Thusρz(ζ) = |φz(ζ)| and the disksDσ(z) are the well-known pseudo-hyperbolic disks; they are geometrically Euclidean disks, but their Eu-clidean centers and radii are different.

Standard calculations show that

|dρz(ζ)|2 =∣∣∣∣ 1− |z|2

(1− zζ)2

∣∣∣∣2 =(1− ρz(ζ)2)2

(1− |ζ|2)2,

so we haveν(ζ) = log 2 + log(1− |ζ|2), and it is clear that (21) holds.As suggested by the proof of Proposition 4.11, we take

uψ(z) = −12

log(1− |z|2),

and thus we have

e2ν(z)(∆uψ(z)− 2|∂uψ(z)|2) =12(1− |z|2) ≥ 0 and τψ(z) =

12(1− |z|2)

.

19

We also haveAg(Dσ(γ)) = Cσ(1− |γ|2).

Thus our Hilbert spaces are

B2ϕ :=

{h ∈ O(D) ;

∫D|h|2e−2ϕ dm

(1− |z|2)< +∞

}and

b2ϕ :=

(sγ) ;∑γ∈Γ

|sγ |2e−2ϕ(γ)(1− |γ|2) < +∞

.

The densities are given by

D+f (Γ) = lim sup

r→1supz∈D

∑ρz(γ)<r

f(ρz(γ))(1− ρz(γ)2)2

4((1− |z|2)2∆ϕ(z) + 1

2 (1− |z|2)) ∫ r

0tf(t)dt

,

and

D−f (Γ) = lim infr→1

infz∈D

∑ρz(γ)<r

f(ρz(γ))(1− ρz(γ)2)2

4(1− |z|2)2∆ϕ(z)∫ r0tf(t)dt

.

(a) Let

D+] (Γ) = lim sup

r→1supz∈D

∑12<ρz(γ)<r

log 1ρz(γ)

log 11−r (1− |z|2)2∆ϕ(z)

and

D−] (Γ) = lim supr→1

supz∈D

∑12<ρz(γ)<r

log 1ρz(γ)

log 11−r (1− |z|2)2∆ϕ(z)

It is shown in [BO-95] that ifΓ is uniformly separated, then, for the Hilbertspaces(B2

ϕ, b2ϕ), Γ is interpolating ifD+

] (Γ) < 1 and sampling ifD−] (Γ) > 1.The converse is proved in [OS-98].

To compare our theorems with those of [BO-95], we let

f(t) =− log t

(1− t2)21h

12 , 1

”.Theorems 1.1 and 1.2 then yield the same results as [BO-95]. In fact, Theorem1.1 states thatΓ is interpolating if

lim supr→1

supz∈D

∑12<ρz(γ)<r

log 1ρz(γ)

/log 1

1−r((1− |z|2)2∆ϕ(z) + 1−|z|2

2

) < 1.

Interestingly, on the surface this result seems as though it might be stronger thanthat of [BO-95].

20

(b) For the sake of illustration, we consider from here on only the weight

ϕ = −12

log(1− |z|2).

This choice corresponds to the classical (unweighted) Bergman space

B2C :=

{h ∈ O(D) ;

∫D|h|2dm < +∞

}and the corresponding space of sequences

b2C :=

(sγ) ;∑γ∈Γ

|sγ |2(1− |γ|2)2 < +∞

.

(i) Letting f = 1, we see thatΓ is

interpolating if supz∈D

∑(1− ρz(γ)2)2 < 1

andsampling if inf

z∈D

∑(1− ρz(γ)2)2 > 1.

(ii) If we definef(t) = (1− t2)−2, we see thatΓ is

interpolating if lim supr→1

supz∈D

#(Γ ∩Dr(z))Ahyp(Dr(z))

< 1

and

sampling if lim infr→1

infz∈D

#(Γ ∩Dr(z))Ahyp(Dr(z))

> 1,

where

Ahyp(Dr(z)) =12π

∫Dr(z)

dm(z)(1− |z|2)2

denotes hyperbolic area ofDr(z).

6 Nyquist radii

Let X be an open Riemann surface with a conformal metricg and a weightϕ. Wedefine

IR(X, g, ϕ) := {a > 0 ; #(Γ ∩Da(z)) ≤ 1 for all z ∈ X ⇒ Γ is interpolating}

and

SR(X, g, ϕ) := {a > 0 ; #(Γ ∩Da(z)) ≥ 1 for all z ∈ X ⇒ Γ is sampling} .

21

Definition 6.1. The Nyquist interpolation radius is the number

NI(X, g, ϕ) := inf IR(X, g, ϕ),

and the Nyquist sampling radius is the number

NS(X, g, ϕ) := supSR(X, g, ϕ).

Remark. If (X, g, ϕ) has the property that no interpolation sequence is also a sam-pling sequence, then we must have

NS(X, g, ϕ) ≤ NI(X, g, ϕ).

More generally,NI(X, g, ϕ)−NS(X, g, ϕ) should be an interesting number.

Let us consider first the classical Bargmann-Fock spaces in the entire plane

BF2α :=

{f ∈ O(C) ;

∫C|f(z)|2e−α|z|

2dm < +∞

},

whereα > 0. For these spaces, the Nyquist radii can be computed exactly.

Theorem 6.2.

NI(C, |dz|2, 12α|z|

2) = NS(C, |dz|2, 12α|z|

2) =1√α

Proof. Fix a > 1√α

and take, in the definition of the upper density, the characteristicfunctionf(t) = 1[0,a]. Then the upper density becomes

D+f (Γ) = sup

z∈C

#(Γ ∩Da(z))αa2

.

If now #(Γ ∩Da(z)) ≤ 1, then

D+f (Γ) ≤ 1

αa2< 1,

so that, by Theorem 1.1,a ∈ IR(C, |dz|2, 12α|z|

2). In particular,

NI(C, |dz|2, 12α|z|

2) ≤ 1√α.

Next fix a < 1√α

and take the same characteristic functionf(t) = 1[0,a] in thedefinition of the lower density. We obtain

D−f (Γ) = infz∈C

#(Γ ∩Da(z))αa2

.

If now #(Γ ∩Da(z)) ≥ 1, then

D−f (Γ) ≥ 1αa2

> 1,

22

so that by Theorem 1.2a ∈ SR(C, |dz|2, 12α|z|

2). In particular,

NS(C, |dz|2, 12α|z|

2) ≥ 1√α.

It is an easy consequence of the results of Seip et. al. (see also [DS-03]) that thereare no sequences in the Bargmann-Fock space that are both sampling and interpolating.We conclude by the remark made above that

NS(C, |dz|2, 12α|z|

2) ≤ NI(C, |dz|2, 12α|z|

2).

This completes the proof.

Next we consider the classical Bergman spaces on the unit diskD:

B2α :=

{f ∈ O(D) ;

∫D|f(z)|2(1− |z|2)α−1dm < +∞

},

whereα > 0.

Theorem 6.3. Letgo = (1− |z|2)−1|dz|2 andϕo = −α2 log(1− |z|2). Then

NI(D, go, ϕo) = NS(D, go, ϕo) =1√α+ 1

Proof. A direct calculation shows

e2ν∆ϕo + τψo =α

2+

1− |z|2

2and e2ν∆ϕo =

α

2.

Now, for a > 0, choose the functionf(t) = (1 − t2)−21[0,a] in the definitions of theupper and lower densities. We thus obtain

D+f (Γ) := sup

z∈D

#(Γ ∩Da(z))αa2(1− a2)−1

and D−f (Γ) := infz∈D

#(Γ ∩Da(z))αa2(1− a2)−1

.

From here on, the proof proceeds exactly as that of Theorem 6.2.

7 Proofs of Theorems 1.1 and 1.2

7.1 Functions and singular metrics

In this paragraph we define certain functions that play important roles in the proofs ofTheorems 1.1 and 1.2.

7.1.1 A local construction of a holomorphic function

In the proofs of Theorems 1.1 and 1.2 we will need, for eachγ ∈ Γ, a holomorphicfunction defined in a neighborhood ofγ and satisfying certain global estimates. Forreasons that will become clear later, the size of this neighborhood cannot be takentoo small. As a consequence, we must overcome certain difficulties presented by thetopology of the neighborhood.

23

Lemma 7.1. LetX be a harmonically homogeneous open Riemann surface. Assumee2ν∆ϕ is bounded above and below by positive constants. LetΓ be a uniformly sepa-rated sequence,γ ∈ Γ, andσ = σ(Γ). There exists a constantC = CΓ > 0 and, foreachγ ∈ Γ, a holomorphic functionFγ ∈ O(Dσ(γ)) such thatFγ(γ) = 0 and for allz ∈ Dσ(γ),

1Ce−2ϕ(γ) ≤

∣∣e−2ϕ+2Fγ∣∣ ≤ Ce−2ϕ(γ). (31)

Proof. LetG be the Green’s function for the domainD2σ(γ). Consider the function

Tγ(z) :=2π

∫D2σ(γ)

−G(z, ζ)∆ϕ(ζ).

By Green’s formula, we have that

Tγ(z) = −ϕ(z) +12π

∫S2σ(γ)

ϕ(ζ) ∗ dζG(z, ζ).

We claim that the harmonic function

hγ :=12π

∫S2σ(γ)

ϕ(ζ) ∗ dζG(z, ζ)

has a harmonic conjugate, i.e., it is the real part of a holomorphic function. Indeed, ifC is a Jordan curve inDr(γ), then∫

C∗dhγ(z) =

12π

∫S2σ(γ)

ϕ(ζ) ∗ dζ(∫

C∗dzG(z, ζ)

). (32)

Now, sinceS2σ(γ)∩C = ∅, the functionz 7→ G(z, ζ) is harmonic and thus∗dzG(z, ζ)is a closed form. It follows that the term in the parentheses on the right hand side of(32) depends only on the homology class[C] ∈ H1(X,Z). SinceH1(X,Z) is discreteand∗dzG(z, ζ) is continuous inζ, we see that the right hand side of (32) vanishes, asclaimed.

Let

Hγ := hγ +√−1∫ z

γ

∗dhγ

be the holomorphic function whose real part ishγ , and letFγ := Hγ − Hγ(γ). Wehave

|2ϕ(γ)− 2ϕ(z) + 2ReFγ(z)| = 2 |Tγ(γ)− Tγ(z)| ≤ 2|Tγ(γ)|+ 2|Tγ(z)|.

Taking exponentials and applying Lemma 4.4 completes the proof.

7.1.2 A function with poles alongΓ

Forz, ζ ∈ X andr < RE , let

I(ζ, z) =∫X

ξr(ζ, w)E(w, z)e−2ν(w)√−1dw ∧ dw

=1cr

∫ r

a

tf(t)

(∫St(ζ)

E(w, z) ∗ dEζ(w)

)dt.

24

SinceE is a fundamental solution to∆,

e2ν(z)∆zI(ζ, z) =π

2

∫X

ξr(ζ, w)δz(w) =π

2ξr(ζ, z).

Next it follows from (11) that, sinceE(·, z) is subharmonic,E(ζ, z) ≤ I(ζ, z) and,sinceE(·, z) is harmonic in the region{w ∈ X : ρζ(w) > r}, E(ζ, z) = I(ζ, z) ifρz(ζ) > r. Moreover, in view of (7), an application of (9) shows that

12π

∫St(ζ)

E(w, z) ∗ dEζ(w) = E(z, ζ)− 1Dt(z)(ζ) (E(z, ζ)− log t) .

We see that

I(ζ, z) =2πcr

(log(ρz(ζ))

∫ ρz(ζ)

0

tf(t)dt+∫ r

ρz(ζ)

tf(t) log tdt

)if ρz(ζ) < r. Note that ∣∣∣∣∣ 1

cr

∫ r

ρz(ζ)

tf(t) log(t)dt

∣∣∣∣∣ ≤ Dr,

whereDr depends only onr. We then have

|I(ζ, z)| ≤ Krρz(ζ) |log(ρz(ζ))|+Dr

for all z, ζ ∈ X satisfyingρz(ζ) < r. Since the expression on the right hand side isbounded by a constant that depends only onr, we have

|I(ζ, z)| ≤ Cr (33)

wheneverρz(ζ) < r.Let Γ be a discrete sequence. We define the function

vr(z) =∑γ∈Γ

(E(γ, z)− I(γ, z)) .

By the preceding remarks,vr(z) ≤ 0 and

vr(z) =∑

γ∈Γ∩Dr(z)

(E(γ, z)− I(γ, z)) .

Moreover,

e2ν∆vr =π

2

∑γ∈Γ

(e2νδγ − ξr(γ, ·)). (34)

Writing

XΓ,ε :={z ∈ X ; min

γ∈Γργ(z) > ε

},

we have the following lemma.

25

Lemma 7.2. Let Γ be a sparse, uniformly separated sequence and letε ≤ σ(Γ).The functionvr is uniformly bounded onXΓ,ε. Moreover,vr satisfies the followingestimate: ifγ ∈ Γ andργ(z) < σ(Γ), then

|vr(z)− log ργ(z)| ≤ Cr,ε. (35)

Proof. Let z ∈ XΓ,ε. SinceΓ is sparse, there are at mostN = Nr,0 members ofΓ, sayγ1, . . . , γN , lying inDr(z), and so

|vr(z)| ≤N∑j=1

(|E(γj , z)|+ |I(γj , z)|) ≤N∑j=1

(|log(ρz(γj))|+ Cr) .

Note that the numberN does not depend onz. Sinceε < ρz(γj) < r, the terminvolving the logarithm has a bound that depends only onε andr. We thus see thatvris uniformly bounded onXΓ,ε.

Let γ ∈ Γ. SinceΓ is sparse, there are at mostN = Nr,ε elements ofΓ that lie inDr(Dε(γ)). We writeΓ∩Dr(Dε(γ)) = {γ1, . . . , γN}, whereγ1 = γ. Again,N doesnot depend onz. Then

|vr(z)− log ρz(γ)| ≤

N∑j=2

|E(γj , z)|+N∑j=1

|I(γj , z)|

+ |E(γ, z)− log ρz(γ)|.

The first sum is bounded becauseσ(Γ) < ρz(γj) < r for j = 2, . . . , N . The secondsum is bounded by (33), and the third term vanishes. This completes the proof of thelemma.

7.1.3 A function with bumps alongΓ

In this paragraph, we shall use area forms associated to the points ofΓ. We define

dAE,γ(ζ) := dργ(ζ) ∧ ∗dργ(ζ).

Let

AE,γ(D) :=∫D

dAE,γ .

Given a distributionf , we consider its regularization

1AE,γ(Dε(z))

∫Dε(z)

fdAE,γ

using the area elementdAE,γ , whereγ ∈ Γ.Observe that

AE,γ(Dε(γ)) =∫Dε(γ)

dργ ∧ ∗dργ =∫Dε(γ)

ργdργ ∧ ∗dE(γ, ·)

=∫ ε

0

t

(∫St(γ)

∗dE(γ, ·)

)dt = 2π

∫ ε

0

t dt = πε2. (36)

26

Consider the function

vr,ε(z) = t∑γ∈Γ

1πε2

∫Dε(γ)

(E(ζ, z)− I(ζ, z)) dAE,γ(ζ)

where0 << t < 1.

Lemma 7.3. The functionvr,ε has the following properties.

1.

e2ν(z)∆vr,ε(z) = t∑γ∈Γ

12ε2

e2ν(z)|dργ(z)|21Dε(z)

−t∑γ∈Γ

12ε2

∫Dε(γ)

ξr(·, z)dAE,γ .

In particular,

limε→0

e2ν∆vr,ε =π

2t∑γ∈Γ

(e2νδγ − ξr(γ, ·)

)in the sense of distributions.

2. There exists a positive constantCr,ε such that

z ∈ X ⇒ −Cr,ε ≤ vr,ε(z) ≤ 0 (37)

and for anyγ ∈ Γ,

ργ(z) < ε ⇒

∣∣∣∣∣ vr,ε(z)− t

πε2

∫Dε(γ)

E(ζ, z)dAE,γ(ζ)

∣∣∣∣∣ ≤ Cr,ε (38)

Proof. 1. The formula for the Laplacian is a straightforward calculation, and the limitis a standard consequence of regularization of currents.

2. SinceE(ζ, z) = I(ζ, z) wheneverρz(ζ) > r, we have, in view of formula (36),

vr,ε(z) =∑

γ∈Dε(Dr(z))

t

πε2

∫Dε(γ)

(E(ζ, z)− I(ζ, z)) dAE,γ(ζ).

Chooseγ ∈ Γ. SinceΓ is sparse, there existγ1, ..., γN ∈ Γ − {γ} such that for allz ∈ Dε(γ)

vr,ε(z) =t

πε2

∫Dε(γ)

(E(ζ, z)− I(ζ, z)) dAE,γ

+N∑j=1

t

πε2

∫Dε(γj)

(E(ζ, z)− I(ζ, z)) dAE,γj

27

Moreover,N is independent ofγ, and depends only onr andε. It follows that∣∣∣∣∣vr,ε(z)− t

πε2

∫Dε(γ)

E(·, z)dAE,γ

∣∣∣∣∣≤ t

πε2

∫Dε(γ)

|I(·, z)|dAE,γ +N∑j=1

t

πε2

∫Dε(γj)

(|E(·, z)|+ |I(·, z)|) dAE,γj .

We have estimates forI(ζ, z) as in the proof of Lemma 7.2, and since, by uniformseparation,ρz(ζ) > σ for any ζ ∈ Dε(γj), we can estimate the right hand side by aconstant that depends only onr. This proves (38), and (37) follows from (38), Lemma7.2 and the fact thatvr ≤ 0.

Finally, we shall have use for the following lemma.

Lemma 7.4. For anyz ∈ Dε(γ),

1AE,γ(Dε(γ))

∫Dε(γ)

E(z, ζ)dAE,γ(ζ) ≤ log1ε

+12. (39)

Proof. Observe that ifz ∈ Dε(γ) andt ∈ (0, ε], then∫St(γ)

∗dζE(z, ζ) =∫Dt(γ)

dζ ∗ dζE(z, ζ) = 2π1Dt(γ)(z) ≤ 2π.

Applying Green’s formula (8) withf = E(z, ·) andh = E(γ, ·), we obtain∫St(γ)

E(z, ζ) ∗ dζE(γ, ζ) =∫St(γ)

E(γ, ζ) ∗ dζE(z, ζ).

We thus have

−∫Dε(γ)

log ρzdργ ∧ ∗dργ = −∫ ε

0

t

(∫St(γ)

E(z, ζ) ∗ dζE(γ, ζ)

)dt

= −∫ ε

0

t

(∫St(γ)

E(γ, ζ) ∗ dζE(z, ζ)

)dt

= −∫ ε

0

t log t

(∫St(γ)

∗dζE(z, ζ)

)dt

≤ −2π∫ ε

0

t log tdt = πε2(

12− log ε

).

The lemma now follows from (36).

7.2 Proof of Theorem 1.1

Let (sγ) ∈ b2(Γ, g, ϕ). We begin by constructing a smooth functionη ∈ L2(X, g, ϕ)that interpolates(sγ). To this end, letχ ∈ C∞0 ([0, σ)) satisfy

0 ≤ χ ≤ 1, χ|[0, σ/2] ≡ 1 and |χ′| ≤ 3σ.

28

We defineη(z) :=

∑γ∈Γ

χ ◦ ργ(z)sγeFγ(z),

whereFγ is as in lemma 7.1. Observe thatη(γ) = sγ for all γ ∈ Γ, and that∫X

|η|2e−2ϕdAg =∑γ∈Γ

|sγ |2∫Dσ(γ)

|χ ◦ ργ |2∣∣e2Fγ−2ϕ

∣∣ dAg≤ C

∑γ∈Γ

|sγ |2e−2ϕ(γ)Ag(Dσ(γ)) < +∞.

Next we wish to correctη by adding to it a functionU that lies inL2(X, g, ϕ)and vanishes alongΓ. The standard approach is to solve the equation∂U = ∂η withsingular weights, using Ohsawa’s∂ Theorem 2.5. The singular weight we will use isthe weight

ϕ := ϕ+ vr,

and one computes that

∂η =∑γ∈Γ

χ′(ργ)∂ργsγeFγ . (40)

SinceD+f (Γ) < 1, there existr < RE andδ > 0 such that

e2ν∆ϕ+ τψ = e2ν∆ϕ+ τψ + e2ν∆vr

≥(e2ν∆ϕ+ τψ

)1−∑γ∈Γ

π2 ξr(·, γ)

e2ν∆ϕ+ τψ

> δ(e2ν∆ϕ+ τψ),

where the first inequality follows from (34). It follows from hypothesis (5) in Theorem1.1 that

e2ν∆ϕ+ τψ ≥ Cδ > 0.

Next, (40) and Lemma 7.2 imply thatϕ is comparable toϕ on the support of∂η, whichlies inVσ(γ) := Dσ(γ)−Dσ

2(γ). We then have the estimate∫

X

|∂η|2e−2uψe−2ϕ ≤ C

σ2

∑γ∈Γ

|sγ |2e−2ϕ(γ)

∫Vσ(γ)

|∂ργ |2e−2uψ

≤ C

σ2

∑γ∈Γ

|sγ |2e−2ϕ(γ)

∫Dσ(γ)

|∂ργ |2e−2uψ

≤ C ′∑γ∈Γ

|sγ |2e−2ϕ(γ)

∫Dσ(γ)

e−2(ν+uψ)

< +∞,

where the first inequality follows from Lemma 7.1 and the last inequality follows from(21). Applying Theorem 2.5, we obtain a functionU ∈ L2(X, g, ϕ) ⊂ L2(X, g, ϕ)

29

such that∂U = ∂η. Moreover, sincee−2ϕ ∼ 1|z−γ|2 for z sufficiently close toγ, we

see thatU(γ) = 0 for all γ ∈ Γ. Thus the function

f := η − U ∈ B2(X, g, ϕ)

interpolates(sγ), and the proof of Theorem 1.1 is complete.

7.3 Proof of Theorem 1.2

Let ϕ := ϕ + vr,ε. The main idea behind the proof of Theorem 1.2 is the followingsampling type lemma.

Lemma 7.5. Suppose the metrice−2ψ satisfies the differential inequalities(6). Foreachh ∈ B2(X, g, ϕ), ∫

X

|h|2e−2ϕe2ν∆ϕdAg ≥ 0. (41)

Proof. Consider the functionS = |h|2e−2ϕ. Then

∆SS

= ∆ logS +1S2|∂S|2 =

1S2|∂S|2 + ∆ log |h|2 − 2∆ϕ

and thuse2ν∆S ≥ −2S

(e2ν∆ϕ

)≥ −2S

(e2ν∆ϕ

).

We claim that ∫X

e2ν∆S dAg ≤ 0.

To prove the claim, letz0 ∈ X. Takeλ ∈ C∞0 ([0, 1/2]) such thatλ(t) ≡ 1 for0 ≤ t ≤ 1/4, and put

χa(r) := λ(r2(1− a)).

Then∫X

e−2(ψ−ν)∆S =∫X

e−2uψ∆S

= lima↗1

∫X

e−2uψχa ◦ ρz0∆S

= lima↗1

∫X

S∆(e−2uψ · (χa ◦ ρz0)

)= lim

a↗1

∫X

S(∆(e−2uψ )χa ◦ ρz0 + ∂(e−2uψ ) ∧ ∂(χa ◦ ρz0)

+∂(e−2uψ ) ∧ ∂(χa ◦ ρz0) + e−2uψ∆(χa ◦ ρz0))

= lima↗1

∫X

S(∆(e−2uψ )χa ◦ ρz0 + e−2uψ∆(χa ◦ ρz0)

),

30

where the third equality follows from Stokes’ Theorem. Now,

lima↗1

∫X

Se−2uψ∆(χa ◦ ρz0)

= lima↗1

∫X

Se−2uψ(χ′′a(ρz0)|∂ρz0 |2 + χ′a(ρz0)∆ρz0

)= lim

a↗1

∫X

Se−2uψ

(χ′′a(ρz0) +

χ′a(ρz0)ρz0

)|∂ρz0 |2

= lima↗1

∫X

|h|2e−2ϕ

(χ′′a(ρz0) +

χ′a(ρz0)ρz0

)e2ν |∂ρz0 |2dAg = 0,

where the last equality follows from (21) and the definition ofχa.It follows that ∫

X

e−2(ψ−ν)∆S =∫X

S∆e−2uψ .

Since

∆e−2uψ = 2e−2uψ (2|∂uψ|2 −∆uψ) = 2e−2ψe2ν(2|∂uψ|2 −∆uψ

),

the lemma now follows from (6).

Conclusion of the proof of Theorem 1.2.Let h ∈ B2(X, g, ϕ). By Lemma 7.3 wecalculate that

e2ν(z)∆ϕ(z) = e2ν(z)∆ϕ(z) + e2ν(z)∆vr,ε(z)

= e2ν(z)∆ϕ(z)

1− t∑γ∈Γ

12ε2

∫Dε(γ)

ξr(ζ, z)e2ν(z)∆ϕ(z)

dAE,γ(ζ)

+t∑γ∈Γ

e2ψ1ε2e2ψ(z)|∂ργ(z)|2

e2ν(z)∆ϕ(z)1Dε(γ)(z)

.

Applying the hypothesesD−f (Γ) > 1 and (21), we see therefore that, fort sufficientlyclose to1, there existr, δ, C > 0 such that

e2ν∆ϕ ≤ −te2ν∆ϕ

δ − C∑γ∈Γ

e2ψ2ε2

1Dε(γ)

. (42)

31

from (36). We thus obtain∫X

|h|2e−2ϕdAg

≤∑γ∈Γ

(C1

ε2+2t|h(γ)|2e−2ϕ(γ)Ag(Dσ(γ)) + C2ε

2−2t

∫Dσ(γ)

|h|2e−2ϕdAg

)

≤∑γ∈Γ

(C1

ε2+2t|h(γ)|2e−2ϕ(γ)Ag(Dσ(γ))

)+ C2ε

2−2t

∫X

|h|2e−2ϕdAg.

By taking ε sufficiently small, we obtain the left hand side of (1). For the right handside of (1), we argue as follows.∑

γ∈Γ

|h(γ)|2e−2ϕ(γ)Ag(Dσ(γ)) =∑γ∈Γ

|h(γ)e−Fγ(γ)|2e−2ϕ(γ)Ag(Dσ(γ))

≤ Cσ2∑γ∈Γ

e−2ϕ(γ)

∫Dσ(γ)

|he−Fγ |2dAg

≤ C ′∑γ∈Γ

∫Dσ(γ)

|h|2e−2ϕdAg

≤ C ′′∫X

|h|2e−2ϕdAg,

where the first inequality follows from (28), the second from Lemma 7.1 and the thirdfrom definition 2.3. The proof of Theorem 1.2 is thus complete.

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interpolation and sampling on riemann surfaces

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