Interpolating and sampling sequences in finite Riemann surfaces
arXiv:0704.0231v1  [math.CV]  2 Apr 2007
INTERPOLATING AND SAMPLING SEQUENCES IN FINITE
RIEMANN SURFACES
JOAQUIM ORTEGA-CERDÀ
Abstract. We provide a description of the interpolating and sampling sequences on a
space of holomorphic functions with a uniform growth restriction defined on finite Riemann
surfaces.
1. Introduction and statement of the results
Let S be an open finite Riemann surface endowed with the Poincaré (hyperbolic) met-
ric. We will study some properties of holomorphic functions in the Riemann surface with
uniform growth control. Namely we will deal with the Banach space Aφ(S) of holomorphic
functions in S such that ‖f‖ := supS |f |e
−φ <∞ where φ is a given subharmonic function
that controls the growth of the functions in the space.
The fact that φ is subharmonic is a natural assumption on the weight that limits the
growth since any other growth control given by a weight ψ, ‖f‖∗ = supS |f |e
−ψ can be
replaced by an equivalent subharmonic function because φ = sup‖f‖∗≤1 log |f | is a subhar-
monic function and Aψ(S) = Aφ(S) with equality of norms, supS |f |e
−ψ = supS |f |e
We have fixed a metric. It is then natural to restrict the possible weights φ, in a way
that the functions in Aφ oscillate in a controlled way when the points are nearby in the
Poincaré metric. This is achieved for instance by assuming that φ has bounded Laplacian
(the Laplace-Beltrami operator with respect to the hyperbolic measure). That is, if in a
local coordinate chart the Poincaré metric is of the form ds2 = e2ν(z)|dz|2, then we assume
that ∆φ = 4e−2ν(z) ∂
∂z∂z̄
satisfies C−1 ≤ ∆φ ≤ C. If we want to deal with other weights
then it is possible to introduce a natural metric associated to the weight as it is done in the
plane in [MMOC03]. In this work we will only consider the Poincaré metric and bounded
Laplacian since it already covers many interesting cases and it is technically simpler.
The problems that we will consider are the following:
(A) The description of the interpolating sequences for Aφ(S): i.e. the sequences Λ ⊂ S
such that it is always possible to find an f ∈ Aφ(S) such that f(λ) = vλ for all λ ∈ Λ
whenever the data {vλ}Λ, satisfies the compatibility condition supΛ |vλ|e
−φ(λ) < +∞
(B) The description of sampling sets for Aφ(S): i.e. the sets E ⊂ S such that there is
a constant C > 0 that satisfies
|f |e−φ ≤ C sup
|f |e−φ, ∀f ∈ Aφ(S).
Date: Working draft: July 20, 2021.
Supported by DGICYT grant MTM2005-08984-C02-02 and the CIRIT grant 2005SGR00611.
http://arxiv.org/abs/0704.0231v1
2 JOAQUIM ORTEGA-CERDÀ
In the solution of these problems the Poincaré distance and the potential theory in the
surface play a key role. This has already been observed by A. Schuster and D. Varolin in
[SV04], where they provide sufficient conditions for a sequence to be interpolating/sampling
for functions in a slightly different context where the weighted uniform control of the growth
of the functions is replaced by a weighted L2 control. Their condition basically coincides
with the description that we reach so our work can be considered as the counterpart of
their theorems, although we will give a different proof of their results as well. We will
rely on the well-known case of the disk and some simplifying properties of finite Riemann
surfaces. Their method of proof looks more promising if one wants to extend the result to
Riemann surfaces with more complicated topology.
When the surface is a disk, which will be our model situation, the corresponding problems
have been solved in [BOC95], [OCS98] and in a different way in [Sei98]. Of course, the
more basic problem of describing the interpolating sequences for bounded holomorphic
functions in finite Riemann surfaces (in our notation φ ≡ 0), has been known for a long
time, see [Sto65]).
We introduce now some definitions that will be needed to state our results. For any
point z ∈ S and any r > 0 we denote by D(z, r) the domain in the surface S that consits
of points at hyperbolic distance from z less than r. They are topological disks if the center
z is outside a big compact of S, or if r is small enough, as we will see in Section 2.
A sequence Λ of points in S is hyperbolically separated if there is an ε > 0 such that
the domains {D(λ, ε)}λ∈Λ are pairwise disjoint.
Let gr(z, w) be the Green function associated to the surface D(z, r) with pole at the
“center” z and g(z, w) = g∞(z, w) be the Green function associated to the surface S. We
define the densities
D+φ (Λ) := lim sup
1/2<d(z,λ)<r
gr(z, λ)
D(z,r)
gr(z, w)i∂∂̄φ(w)
D−φ (Λ) := lim inf
1/2<d(z,λ)<r
gr(z, λ)
D(z,r)
gr(z, w)i∂∂̄φ(w)
The main result is
Theorem 1. Let S be a finite Riemann surface and let φ be a subharmonic function with
bounded Laplacian.
(A) A sequence Λ ⊂ S is an interpolating sequence for Aφ(S) if and only if it is hyper-
bolically separated and D+φ (Λ) < 1.
(B) A set E ⊂ S is a sampling set for Aφ(S) if and only if it contains an hyperbolically
separated sequence Λ ⊂ E such that D−φ (Λ) > 1.
INTERPOLATING AND SAMPLING SEQUENCES IN FINITE RIEMANN SURFACES 3
In Section 2 we will prove some key properties of finite Riemann surfaces. In particular
we need to study the behavior of the hyperbolic metric as we approach the boundary of
the surface. We will also prove some weighted uniform estimates for the inhomogeneous
Cauchy-Riemann equation in the surface, Theorem 6, that has an interest by itself.
In the next section, we use the tools and Lemmas proved in Section 2 to reduce the
interpolating and sampling problem in S to a problem near the boundary that can be
reduced to the known case of the disk.
Finally in Section 4 we show how our results can be extended to other Banach spaces of
holomorphic functions where the uniform growth is replaced by weighted Lp spaces.
A final word on notation. By f . g we mean that there is a constant C independent of
the relevant variables such that f ≤ Cg and by f ≃ g we mean that f . g and g . f .
2. Basic properties of finite Riemann surfaces
We start by the definition and then we collect some properties of S that follow from the
restrictions that we are assuming on the topology of S.
Definition 2. A finite Riemann Surface is the interior of a smooth bordered compact
Riemann surface.
Our surface is an open Riemann surface and it is in fact an open subset of a compact
surface (the double, see [SS54]). See Figure 1 for a typical representation. Observe that
the genus is finite and the border of the surface consists of a finite number of smooth closed
Jordan curves. In most of what follows the particular case of a smooth finitely connected
open set in C has all the difficulties of the general case.
The following claim follows from instance from [Sch78, Prop 7.1-7.4]
Lemma 3. For any (0, 1)-form ω there is a solution u to the inhomogeneous Cauchy-
Riemann equation ∂̄u = ω. Moreover since S has an essential extension to a compact
Riemann surface if the data is a smooth form with compact support K in S then there is
a bounded linear solution u = T [w] with the bound |u| ≤ CK〈ω〉.
In this statement and in the following 〈ω〉 is the Poincaré length of the (0, 1)-form ω.
In the disk we have Blaschke factors that are very convenient to divide out zeros of
holomorphic functions without changing essentially the norm. The analogous functions
that provide us with the same property in the case of finite Riemann surfaces are given by
the next proposition:
Proposition 4. There is a constant C = C(S) > 0 such that for any point z ∈ S there is
a function hz ∈ H(S) with
| log |hz(w)| − g(z, w)| < C.
In particular hz(w) is a bounded holomorphic function that vanishes only on the point z
and for any ε > 0 K > |hz(w)| > C(ε) if d(z, w) > ε.
4 JOAQUIM ORTEGA-CERDÀ
Figure 1. A finite Riemann surface with three funnels
Proof. The obstruction for an harmonic function u to have an harmonic conjugate is that
for a set of generators {γi}
i=1 of the homology we have
∗du = 0, i = 1, . . . , m. If we
want u = log |f | for an f ∈ H(S), we just need that
∗du ∈ Z.
Being a finite Riemann surface there are {hj}
j=1 functions in the algebra of S without
zeros such that
∗d log |hj| = δij , see [Wer64, Lemma 1]. Thus the function
v(z) = u(z)−
log |hi(z)|
is the logarithm of an holomorphic function log |f | = v. Therefore there is a constant C such
that any harmonic function u in S admits an holomorphic function f with |u−log |f || < C.
Take a point z ∈ S and any holomorphic function kz ∈ H(S) that vanishes only on z. Then
g(z, w)− log |kz(w)| is harmonic in S and therefore there is a holomorphic function fz such
that |g(z, w)− kz(w)− log |fz|| < C. Thus we may define hz(w) = fz(w)kz(w) and it has
the estimate |g(z, w)− log |hz|| < C. The estimate |g(z, w)| > C(ε) when d(z, w) > ε holds
in finite Riemann surfaces, see for instance [Dil95, Theorem 5.5]. �
2.1. The hyperbolic metric in a finite Riemann surface. The open ends of the
Riemann surface can be parametrized as follows: The border of the Riemann surface S
is a finite union of smooth closed curves γ̃i, i = 1, . . . , n. Near each γ̃i there is a closed
geodesic γi that is homotopic to γ̃i. The subdomain of S bounded by γi and γ̃i is denoted
a “funnel” following the terminology of [DPRS87] and [Dil01].
We need to be more precise about the hyperbolic metric in the funnel. There are nice
coordinates in the funnel that provide good estimates. These are given by the collar
theorem. Let D be the universal holomorphic cover of S and let Tγ ∈ Aut(D) be the deck
transformation corresponding to the closed loop γ. Consider the surface Y = D/{T nγ }n∈Z.
This an annulus since π1(Y ) = Z. If we quotient it by the rest of the deck transformations
of the universal cover we get an holomorphic covering map πγ from Y → S which is
a local isometry (in Y and S we consider the Poincaré metric inherited from D). In fact
Y = {e−R < |z| < eR}, where R = π2/Length(γ), and πγ maps the unit circle isometrically
to γ. Moreover πγ is an isometric injection of the outer part of the annulus {1 < |z| < e
INTERPOLATING AND SAMPLING SEQUENCES IN FINITE RIEMANN SURFACES 5
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
�������������
|z|=eR
|z|=e −R
|z|=1
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
��������
Figure 2. Standard coordinates on the funnel
onto the funnel. These will be called the standard coordinates of the funnel. See [Dil01]
and [Bus92] for details.
The Poincare metric in the the funnel is explicit in the standard coordinates and it is
comparable to the hyperbolic metric on the disk in the coordinate disk |z| < eR when
restricted to |z| > 1.
We denote by Ai, i = 1, . . . , n the funnels of S bounded by γi and γ̃i.
2.2. The inhomogeneous Cauchy-Riemann equation on the surface. We want to
solve the inhomogeneous Cauchy-Riemann equation on S with weighted uniform estimates.
In order to get good estimates it is useful to find functions f ∈ H(S) with precise size
control, i.e., |f | ≃ eφ outside a neighborhood of the zero set of f . With this function we
can later modify an integral formula to get a bounded solution to the ∂̄-equation when the
data has compact support. The following Lemma provides such a function that in other
context has been termed a “multiplier”:
Lemma 5. Let S be a finite Riemann surface and let φ be a subharmonic function with
bounded Laplacian. Then there is a function f with hyperbolically separated zero set Σ such
that |f | ≃ eφ whenever d(z,Σ) > ε. Moreover if we fix any compact K in S it is possible
to find f with the above properties and without zeros in K.
Proof. In any of the funnels Ai we transfer the subharmonic weight φ to the standard
coordinate chart 1 < |z| < eRi . We define a weight φi on the disk |z| < e
Ri in such a
way that φi has bounded invariant Laplacian and moreover |φ − φi| < C on the region
1 < |z| < eRi . One way to do so is the following: we assume from the very beginning that
φ is smooth (this is no restriction since otherwise it can be approximated by a smooth
function). Define
(2) φi(z) = φ(z)χ(z) +Mi‖z‖
where χ is a cutoff function such that χ ≡ 1 in eRi/2 < |z| < eRi , χ ≡ 0 in |z| < 1 and
Mi is taken big enough such that φi is subharmonic and the invariant Laplacian of φi is
bounded above and below.
6 JOAQUIM ORTEGA-CERDÀ
We are under the hypothesis of the result from [Sei95] that states that there is an
holomorphic function in the disk fi with separated zero set Z(fi) (in the hyperbolic metric
of the disk) such that |fi| ≃ e
φi whenever d(z, Z(fi)) > ε. Since the hyperbolic metric of
the disk is comparable to the hyperbolic metric in the funnel, we have found a function
fi ∈ H(Ai) with separated zero set such that |fi(z)| ≃ e
φ(z) if d(z, Z(fi)) > ε. Moreover
dividing out fi by a finite Blaschke product we can assume that fi is zero free in any
prefixed compact of the disk.
We consider the “core” of S to be S \ Ãi, where Ãi are the outer part of the funnels
mapped by eSi < |z| < eRi . The values of the Si are taken so big as to make sure that the
compact K in the hypothesis of the Lemma is contained in the core of S. We adjust the
fi i = 1, . . . , n as mentioned before to make sure that they are zero free in the inner part
of the funnels 1 < |z| < eSi . We finally define f0 ≡ 1 in the core of S.
To patch the different fi together we will need to solve a Cousin II problem with bounds.
Our data is fi defined on the inner parts of the funnels mapped by 1 < |z| < e
Si. The data
are bounded above and below in the inner parts of the funnels (because φ is bounded above
and below in any compact of S and fi have no zeros there). We want to find functions
gi ∈ H(Ai) and g0 holomorphic on the core of S such that fi = g0/gi in the inner part of the
funnel. If moreover gi and g0 are bounded (above and below) then the function f defined
as figi in each of the funnels Ai and g0 on the core of S is holomorphic on S and has the
desired growth properties. To find the functions gi observe that since the intersection of the
funnel Ai with the core of S strictly separates the outer part of the funnel from the inner
part of the core we can reduce the Cousin II problem to solving a ∂̄-equation with bounded
estimates of the solution on S when the data is bounded and with compact support (the
support is in the inner part of the funnels). This can be achieved by Lemma 3. �
With this function we can then obtain the following result which is interesting by itself:
Theorem 6. Let S be a finite Riemann surface and let φ be a subharmonic function with
a bounded Laplacian. There is a constant C > 0 such that for any (0, 1)-form ω on S
there is a solution u to the inhomogeneous Cauchy-Riemann equation ∂̄u = ω in S with
the estimate
|u(z)|e−φ(z) ≤ C sup
〈ω(z)〉e−φ(z),
whenever the right hand is finite.
Recall that the notation 〈ω(z)〉 means the hyperbolic norm of ω at the point z.
Proof. Let wi be the form w restricted to the funnel Ai. We take a standard coordinate
chart and we may think of wi as a (0, 1)-form defined on the disk |z| < e
Ri and with
support in 1 < |z| < eRi. Consider as in the proof of Lemma 5 a subharmonic function φi
in the disk with bounded laplacian and such that |φ− φi| < C if 1 < |z| < e
By the results in [OC02, Thm 2] there is a solution ui to the problem ∂̄ui = wi in the
disk |z| < eRi with the estimate
|z|<eRi
|ui|e
−φi ≤ Ci sup
1<|z|<eRi
〈wi〉e
INTERPOLATING AND SAMPLING SEQUENCES IN FINITE RIEMANN SURFACES 7
Observe that the hyperbolic metric of the disk and of the surface S in the funnel are
equivalent. We consider ũi = uiχi, where χi is a cutoff function with support in 1 < |z| <
eRi and such that χi ≡ 1 if |z| > e
Ri/2. The function ũi is extended by 0 to the remaining
of S and it has the estimate supS |ũi|e
−φ ≤ Ci supS〈w〉e
−φ. Now ∂̄ũi coincides with w
on the outer part of the funnel Ai. Thus the (0, 1)-form wk = w −
i ∂̄ũi has compact
support in S and it satisfies supS〈wk〉e
−φ ≤ supS〈w〉e
−φ. The desired solution is then
ũi + v, where v is such that ∂̄v = wk. We must then solve ∂̄v = wk with weighted
uniform estimates but with the advantage that wk has compact support K.
Let T (ωk) be a solution operator for ∂ū = ωk. We take the operator T given by Lemma 3
the estimate supS |T [wk](z)| ≤ CK supK〈wk〉 holds. Take f with |f | ≃ e
φ and without zeros
in K as given in Lemma 5. Then we define R as
(3) R[ωk](z) = f(z)T [ωk/f ](z),
It solves ∂̄R[ωk] = ωk with the estimate
|R[ωk]|e
−φ ≤ CK sup
〈ωk〉e
The solution is thus v = R[wk]. �
3. The main results
Proposition 7. A separated sequence Λ ⊂ S is interpolating for Aφ(S) if and only if the
sequences Λi = Λ ∩Ai are interpolating in Aφ(Ai).
Proof. We only need to prove that we can pass from the local to the global interpolation
property. We split the proof in two steps
(1) From a funnel Ai to global S: We need to prove that there are finite sets Fi ⊂ Λi
such that ∪ni=1(Λi \ Fi) is interpolating globally.
(2) Filling up the remainder. We shall prove that by adding a finite number of points
to an interpolating sequence we still get an interpolating sequence. Thus Λ is
interpolating if (Λ1 \ F1) ∪ · · · ∪ (Λn \ Fn) is interpolating.
Let γ̃ be one of the closed curves on the boundary. Take a funnel A with outer end curve
in γ̃ and inner end curve in γ. The constant of interpolation in the funnel A is K > 0.
Take a cutoff function χε with support in the funnel such that 〈∂̄χε〉 < ε/(KC) (where C
is the constant in Theorem 6), the support is in a thick annulus of hyperbolic thickness
M = M(ε,K, C). We consider a smaller funnel where χε ≡ 1. The sequence Λ in this
smaller funnel has still at most interpolation constant K. We can interpolate arbitrary
values on Λ being small near the inner curve γ of A in the following way. Take some values
vλ with norm one. Take a function in the funnel f with norm at most K that solves the
interpolation problem. We are going to approximate it by a function in A that is small
near γ. Cut it off by χε and correct via the following inhomogeneous Cauchy-Riemann
equation:
8 JOAQUIM ORTEGA-CERDÀ
∂̄u = f∂̄χ
The function h = u − fχ is holomorphic. By using Theorem 6 on it is possible to solve
the equation with a solution u such that sup |u|e−φ ≤ ε. The function h does not solve
the problem directly but it almost does. We reiterate the procedure (interpolating the
error vλ − h(λ) and with a convergent series we get finally a function g such that h(λ) =
vλ, supA |h|e
−φ ≤ 2 and moreover in the inner half of the funnel that we denote by Ã,
supÃ |h|e
−φ ≤ ε.
Now it is easier to make it global. Take a new cutoff function χ with support in the
funnel A and that is one on the outer part of (i.e. A \ Ã. Then we need to solve
∂̄u = h∂̄χ,
with good global estimates in S. These are given by Theorem 6. We have solved the
interpolation problem when the sequence lies in the funnels. For the general situation
we only need to add a finite number of points. The existence of “Blaschke”-type factors
hλ(z) provided by Theorem 4 shows that Λ ∪ λ is interpolating if Λ is interpolating (it is
immediate to build functions in the space such that f |Λ ≡ 0 and f(λ) 6= 0). �
For the sampling part we need the following definition
Definition 8. Given the pair (S, φ) of a finite Riemann surface and a subharmonic function
defined on it, we associate to it the pairs: (Di, φi)i=1,...n of disks Di and subharmonic
functions φi defined on the disks as follow: If Ai = {1 < |z| < e
Ri}, i = 1, . . . , n are the
standard charts of the funnels of S we define Di = {|z| < e
Ri} and φi is any subharmonic
function in Di such that |φi − φ| < C in the region 1 < |z| < e
Ri, ∆φi = ∆φ in e
Ri/2 <
|z| < eRi and ∆φi ≃ 1 in |z| < e
Ri/2. They can be defined similarly as in (2), but to make
sure ∆φi = ∆φ we may take instead
φi(z) = φ(z)χ(z) +Miψ(z),
where ψ is any bounded subharmonic function in Di such that ∆ψ(z) = 1 if |z| < e
and 0 elsewhere.
The funnels Ai can be considered funnels of S and they are subdomains of Di too. We
will exploit this double nature in the following theorem
Theorem 9. Let S be a finite Riemann surface and let φ be a subharmonic function with
bounded Laplacian. A separated sequence Λ is sampling for Aφ(S) if and only if all the
sequences in the funnels Λi = Λi ∩ Ai ⊂ Di are sampling sequences for Aφi(Di), where
(Di, φi) are the associated pairs to S given by Definition 8.
Thus this Theorem and Proposition 7 show that the properties of sampling and inter-
polation only depend on the behavior of the sequence and the weight near the boundary
pieces.
To prove Theorem 9 we need some previous results
INTERPOLATING AND SAMPLING SEQUENCES IN FINITE RIEMANN SURFACES 9
Lemma 10. Let S be a finite Riemann surface and let φ be a subharmonic function with
bounded Laplacian. A sequence Λ ⊂ S is a uniqueness sequence for Aφ(S) if and only if
all the sequences in the funnels Λi = Λi ∩ Ai ⊂ Di are uniqueness sequences for Aφi(Di),
where (Di, φi) are the associated pairs to S given by Definition 8.
Proof. It is easier to deal by negation. Let Λ be contained in the zero set of a function
f ∈ Aφ(S). Therefore Λi is in the zero set of f ∈ Aφ(Ai). We divide by a finite number
of zeros Ei and we obtain a new function g ∈ Aφ(Ai) without zeros in 1 < |z| ≤ e
Ri/2 and
such that Λi \Ei ⊂ Z(g). Take the disk Di and consider the cover by two open sets |z| > 1
and |z| < eRi/2. On the first set we have the function g and on the second the function 1.
The quotient is bounded above and below in the intersection of the sets. This defines a
bounded Cousin II in the disk Di problem that can be solved with bounded data. We get
a new function h ∈ Aφi(Di) that vanishes in Z(g). We can now add the finite number of
zeros Ei without harm. The reciprocal implication follows with the same argument. �
The next result is inspired by a result of Beurling ([Beu89, pp. 351–365]) that relates
the property of sampling sequence to that of uniqueness for all weak limits of the sequence.
In the context of the Bernstein space (in the original work by Beurling) the space was
fixed (it was C, the space of functions was fixed, the Bernstein class, and he considered
translates and limits of it of the sampling sequence). Here we need to move and take limits
of the sequence (by zooming on appropriate portions of it) but we also need to change the
support space (portions of S near the funnel that look like the unit disk) and we will also
move the space of functions by changing the weights. We need some definitions:
Definition 11. We consider triplets (Dn, φn,Λn) where Dn are disks Dn = D(0, rn) ⊂ D,
φn are subharmonic functions defined in a neighborhood of Dn and Λn is a finite collection
of points in Dn. We say that (Dn, φn,Λn) converges weakly to (D, φ,Λ) (where D is the
unit disk, φ a subharmonic function in D and Λ a discrete sequence in D) if the following
conditions are fullfilled:
• The domains Dn tend to D, i.e.: rn → 1,
• The weights φn tend to the weight φ in the sense that ∆φn as measures converges
weakly to ∆φ.
• The sequences Λn converge weakly to Λ, i.e, the measure
δλ converges weakly
to the measure
λ∈Λ δλ.
Let us fix a point p ∈ S. If a sequence of points zn ∈ S goes to ∞, i.e. d(zn, p) → ∞,
from a point n0 on it will eventually belong to the union of the funnels A1 ∪ · · · ∪ An. If
we take the set of points Dn = {z ∈ S; d(z, zn) < d(zn, p)/2} then Dn is an hyperbolic
disk contained in the funnels if n is big enough. In each of the Dn we consider the function
φn = φ|Dn and Λn = Λ ∩ Dn. Thus for any sequence of points zn with d(p, zn) → ∞ we
build a triplet (Dn, φn,Λn) for n big enough.
Definition 12. Let W (S, φ,Λ) be the set of all triplets (D, φ∗,Λ∗) which are weak limits
of triplets (Dn, φn,Λn) associated to any sequence zn such that d(p, zn) → ∞.
The theorem of Beurling on our context is
10 JOAQUIM ORTEGA-CERDÀ
Theorem 13. Let S be Riemann surface of finite type and let φ be a subharmonic function
with bounded Laplacian. A separated sequence Λ is sampling for Aφ(S) if and only if
• The sequence Λ is a uniqueness set for Aφ(S)
• For any triplet (D, φ∗,Λ∗) ∈ W (S, φ,Λ), the sequence Λ∗ is a uniqueness set for
Aφ∗(D).
Proof. Let us prove that the uniqueness conditions imply that Λ is a sampling sequence. If it
were not, there would be a sequence of functions fn ∈ Aφ(S) such that supΛ |fn|e
−φ ≤ 1/n
and supS |fn|e
−φ = 1. Take a sequence of points zn with |fn(zn)|e
−φ ≥ 1/2. If zn are
bounded we can take a subsequence of points that we still denote zn convergent to z
∗ ∈ S
and by a normal family argument there is a partial of fn convergent to f ∈ Aφ, such that
f |Λ ≡ 0, f(z
∗) 6= 0 and this is not possible. Thus zn must be unbounded. Then we take
the triplets (Dn, φn,Λn) associated to zn and Dn → D because zn → ∞ and the hyperbolic
radius of Dn is d(zn, p)/2. Since φn has bounded Laplacian, the mass of ∆φn restricted to
any compact K in D is bounded, thus we can take a subsequence that converges weakly to
a positive measure µ in D which satisfies (1−|z|)2µ ≃ 1 because all the mesures ∆φn satisfy
this inequalities with uniform constants. Let φ be such that ∆φ∗ = µ. Since Λn = Dn ∩ Λ
are all separated with uniform bound, there is a weak limit Λ∗. The functions fn in the
disks can be modified by a factor egn in such a way that hn = fne
gn satisfies hn(0) = 1
and |hn| ≤ e
φn+Re(gn), if n big enough and supΛn |hn|e
−φn+Re(gn) ≤ 1/n. We can add an
harmonic function v to φ∗ in such a way that φn+Re(gn) → v+ φ
∗ uniformly on compact
sets. Thus hn has a partial convergent to h ∈ Aφ∗ , h(0) = 1 and h|Λ∗ ≡ 0 which was not
possible by assumption.
In the other direction, we assume that Λ is a sampling sequence forAφ(S), and (D, φ
∗,Λ∗) ∈
W (S, φ,Λ). We want to prove that any f ∈ A∗φ(D) that vanishes in Λ
∗ is identically 0.
Take a sequence of points zn that escapes to infinity and (Dn, φn,Λn) the associated triple
that converges weakly to (D, φ∗,Λ∗). As φn → φ
∗ and Λn → Λ
∗ uniformly on compact
sets we can take a sequence of radii sn such that d(Λ ∩ D(zn, sn),Λ
∗ ∩ D(0, sn)) < 1/n,
D(zn, sn) ⊂ D(zn, rn) and |φn − φ
∗| ≤ 1/n. If f vanishes in Λ∗ that means that f is very
small in D(zn, sn)∩Λ. Assume that f(0) = 1. Take a cutoff function χn such that χn ≡ 0
outside D(zn, sn), χ(zn) = 1, and 〈dχ〉 < εn. Define gn = fχn − un, where ∂̄u = f∂̄χn is
the solution estimates by Theorem 6. Clearly gn is small in all points of Σ and it has at
least norm 1. Thus we are contradicting the fact that Λ is sampling.
Observe that one particular instance of finite Riemann surface, where we can apply the
result are the disks Di associated to the funnels with the metric φi. The final piece for the
proof of Theorem 9 is then
Lemma 14. If S is a finite Riemann surface, φ a subharmonic function with bounded
Laplacian and Λ is a uniformly separated sequence, then all possible weak limits coincide
with the weak limits of the disks associated to the surface, i.e,
W (S, φ,Λ) =W (D1, φ1,Λ1) ∪ · · · ∪W (Dn, φn,Λn).
INTERPOLATING AND SAMPLING SEQUENCES IN FINITE RIEMANN SURFACES 11
Proof. The proof amounts to the observation that the metric in Di converges uniformly to
the metric in S as z → ∂Di, and in the definition of weak limits we only consider uniform
convergence over compacts. �
Theorem 9 follows now immediately from Theorem 13 and Lemmas 10 and 14. �
Now Theorem 9 and Theorem 7 show that the property of being a sampling/interpolating
sequence are determined by the behavior near the boundary, more precisely in the associ-
ated disks. In these disks there is a precise description of the interpolating and sampling
sequences (see [BOC95] and [OCS98]) that can be transported to the surface. If we rewrite
it we get the density conditions of Theorem 1, but the disks are not hyperbolic disks on
the surface, they correspond to hyperbolic disks in disks Di, but since the condition is
only relevant near the boundary, then the disks in both metrics look more an more sim-
ilar. Moreover the difference between the corresponding Green functions converge to 0
uniformly as we go to the boundary. Finally, as the sequence is uniformly discrete and the
Laplacian of the weight is bounded above and bellow, the small difference is absorbed by
the fact that the inequalities are strict and this proves Theorem 1. In fact it is possible to
replace in the definition of the density, (1) the Green function gr of D(z, r) by the Green
function g of S, because as before supw∈D(z,r) |gr(z, w)− g(z, w)| → 0 as z approaches the
boundary.
4. Some Lp-variants
We have considered up to now pointwise growth restrictions. It is possible to obtain from
our Theorem other results in different Banach spaces of holomorphic functions. Consider
for instance the weighted Bergman spaces
φ(S) = {f ∈ H(S);
|f |pe−φ dA < +∞},
where dA is s the hyperbolic area measure in S and p ∈ [1,∞). The natural problem in
this context is the following:
Definition 15. Let S be a finite Riemann surface, and let φ be a subharmonic function
with bounded Laplacian bigger than one, i.e., 1 + ε < ∆φ < M .
• A sequence Λ ⊂ S is interpolating for A
φ(S) if for any values vλ such that
pe−φ(λ) <∞
there is a function f ∈ A
φ(S) such that f(λ) = vλ.
The spaces A
φ can be empty if we only ask φ to be with positive bounded Laplacian. It is
then natural to require that the Laplacian is strictly bigger than one so that the Laplacian
plus the curvature of the metric in the manifold is strictly positive and there are functions
in the space (consider the case of the disk S = D for instance).
Let φ0 be a subharmonic function in S such that ∆φ0 = 1. The corresponding theorem
will be
12 JOAQUIM ORTEGA-CERDÀ
Theorem 16. Let S be a finite Riemann surface, and let φ be a subharmonic function
with bounded Laplacian strictly bigger than one. Let p ∈ [1,+∞) and Λ be a separated
sequence.
• The sequence Λ is interpolating for A
φ(S) if and only if D
(φ−φ0)
(Λ) < 1/p.
In the case of the unit disk dA(z) = (1 − |z|)−2 this description is well-known, see for
instance [Sei98, Thm 2,3].
Proof. The proof of the theorem is the same mutatis-mutandi as in the L∞ setting. The
basic tool that allows us to glue the pieces together is the next theorem which is the
generalization of Theorem 6 and it is proved in the same way:
Theorem 17. Let S be a finite Riemann surface, let φ be a subharmonic function with a
bounded Laplacian strictly bigger than one and let p ∈ [1,∞). There is a constant C =
C(p, S) > 0 such that for any (0, 1)-form ω on S there is a solution u to the inhomogeneous
Cauchy-Riemann equation ∂̄u = ω in S with the estimate
|u(z)|pe−φ(z)dA(z) ≤ C
〈ω(z)〉pe−φ(z)dA(z),
whenever the right hand is finite.
The proof of this result is again the same as in Theorem 6. We can separately solve the
C-R equation in each funnel using Theorem 2 from [OC02]. We glue them together with
a C-R equation with data that has compact support that can be solved with the operator
(3). �
References
[BOC95] Bo Berndtsson and Joaquim Ortega Cerdà, On interpolation and sampling in Hilbert spaces
of analytic functions, J. Reine Angew. Math. 464 (1995), 109–128. MR 96g:30070
[Beu89] Arne Beurling, The collected works of Arne Beurling. Vol. 1, Contemporary Mathematicians,
Birkhäuser Boston Inc., Boston, MA, 1989, Complex analysis, Edited by L. Carleson, P.
Malliavin, J. Neuberger and J. Wermer. MR 92k:01046a
[Bus92] Peter Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics,
vol. 106, Birkhäuser Boston Inc., Boston, MA, 1992. MR 93g:58149
[Dil95] Jeffrey Diller, A canonical ∂ problem for bordered Riemann surfaces, Indiana Univ. Math. J.
44 (1995), no. 3, 747–763. MR 96m:30063
[Dil01] , Green’s functions, electric networks, and the geometry of hyperbolic Riemann surfaces,
Illinois J. Math. 45 (2001), no. 2, 453–485. MR 2003j:30064
[DPRS87] Jozef Dodziuk, Thea Pignataro, Burton Randol, and Dennis Sullivan, Estimating small eigen-
values of Riemann surfaces, The legacy of Sonya Kovalevskaya (Cambridge, Mass., and
Amherst, Mass., 1985), Contemp. Math., vol. 64, Amer. Math. Soc., Providence, RI, 1987,
pp. 93–121. MR 88h:58119
[MMOC03] N. Marco, X. Massaneda, and J. Ortega-Cerdà, Interpolating and sampling sequences for entire
functions, Geom. Funct. Anal. 13 (2003), no. 4, 862–914. MR 2004j:30073
[OC02] Joaquim Ortega-Cerdà, Multipliers and weighted ∂-estimates, Rev. Mat. Iberoamericana 18
(2002), no. 2, 355–377. MR 2003j:32046
[OCS98] Joaquim Ortega-Cerdà and Kristian Seip, Beurling-type density theorems for weighted Lp
spaces of entire functions, J. Anal. Math. 75 (1998), 247–266. MR 2000k:46030
INTERPOLATING AND SAMPLING SEQUENCES IN FINITE RIEMANN SURFACES 13
[Sch78] Stephen Scheinberg, Uniform approximation by functions analytic on a Riemann surface, Ann.
of Math. (2) 108 (1978), no. 2, 257–298. MR 58 #17111
[Sei95] Kristian Seip, On Korenblum’s density condition for the zero sequences of A−α, J. Anal. Math.
67 (1995), 307–322. MR 97c:30044
[Sei98] , Developments from nonharmonic Fourier series, Proceedings of the International
Congress of Mathematicians, Vol. II (Berlin, 1998), no. Extra Vol. II, 1998, pp. 713–722
(electronic). MR 99h:42023
[SS54] Menahem Schiffer and Donald C. Spencer, Functionals of finite Riemann surfaces, Princeton
University Press, Princeton, N. J., 1954. MR 16,461g
[Sto65] E. L. Stout, Bounded holomorphic functions on finite Riemann surfaces, Trans. Amer. Math.
Soc. 120 (1965), 255–285. MR 32#1358
[SV04] A. Schuster and D. Varolin, Interpolation and Sampling for generalized Bergman spaces on
finite Riemann Surfaces., Preprint, 2004.
[Wer64] John Wermer, Analytic disks in maximal ideal spaces, Amer. J. Math. 86 (1964), 161–170.
MR 28 #5355
Dept. Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08071
Barcelona, Spain
E-mail address : jortega@ub.edu
