Interpolating and sampling sequences for entire functions

We characterise interpolating and sampling sequences for the spaces of entire functions f such that f e^{-phi} belongs to L^p(C), p>=1 (and some related weighted classes), where phi is a subharmonic weight whose Laplacian is a doubling measure. The results are expressed in terms of some densities adapted to the metric induced by the Laplacian of phi. They generalise previous results by Seip for the case phi(z)=|z|^2, and by Berndtsson&Ortega-Cerd\`a and Ortega-Cerd\`a&Seip for the case when the Laplacian of phi is bounded above and below.


INTRODUCTION
In this paper we provide Beurling-type density conditions for sampling and interpolation in certain generalised Fock spaces. We consider a rather general situation, with only mild regularity conditions on the possible growth. Let φ be a (nonharmonic) subharmonic function whose Laplacian ∆φ is a doubling measure (see definition and properties in Section 2.1), and let ω denote a flat weight, that is, a positive measurable function with slow growth (see details in Section 2.2). The spaces we deal with are parametrised by an index p ∈ [1, ∞], as follows: The function ρ −2 is a regularised version of ∆φ, as described in [Chr91]. More precisely, if µ = ∆φ and z ∈ C, then ρ φ (z) (or simply ρ(z) if no confusion can arise) denotes the positive radius such that µ(D(z, ρ(z)) = 1. Such a radius exists because doubling measures have no mass on circles.
Two particular families of spaces seem of special interest. The first one are the usual weighted L p -spaces of entire functions, obtained with ω = ρ 2/p . The second case arises when ω = 1; then the spaces F p φ,ω coincide with {f ∈ H(C) : C |f | p e −pφ ∆φ < ∞}.
Since functions f in the spaces F p φ,ω are determined by the growth of |f |, their restriction to a sequence should be described as well in terms of growth.
Let Λ ⊂ C be a sequence and let v = {v λ } λ∈Λ be an associated sequence of values. Definition 1. A sequence Λ is an interpolating sequence for F p φ,ω , 1 ≤ p < ∞ (denoted Λ ∈ Int F p φ,ω ), if for every sequence of values v such that v p An application of the open mapping theorem shows that when Λ ∈ Int F p φ,ω there is M > 0 such that for any v ∈ ℓ p φ,ω (Λ), there exists f ∈ F p φ,ω with f |Λ = v and ||f || F p φ,ω ≤ M||v|| ℓ p φ,ω (Λ) .
The least possible M in (1) is called the interpolating constant of Λ and is denoted by M p φ,ω (Λ), or M(Λ) if no confusion is possible.
Also, Λ ∈ Samp F ∞ φ,ω if there exists C > 0 such that for every f ∈ F ∞ φ,ω The least constant C verifying these inequalities is called the sampling constant of Λ and is denoted L p φ,ω (Λ), or simply L(Λ). The definitions of interpolating and sampling sequences in the spaces defined by L ∞ norms reflect the maximal growth for functions in the space, and are natural. The definition for p < ∞ can be motivated in the following way. Consider for instance the case p = 2. The estimates of the normalised Bergman kernel k φ,ω (λ, z) in F 2 φ,ω (see Lemma 20) show that k φ,ω (λ, ·), f ≃ f (λ)ω(λ)e −φ(λ) for all f ∈ F 2 φ,ω . Thus Λ ∈ Samp F 2 φ,ω if and only if that is, if and only if {k φ,ω (λ, ·)} λ∈Λ is a frame in F 2 φ,ω . Similarly, Λ ∈ Int F 2 φ,ω if and only if {k φ,ω (λ, ·)} λ∈Λ is a Riesz basis in its closed linear span in F 2 φ,ω . These are the standard problems of interpolation and sampling in Hilbert spaces of functions with reproducing kernels [SS61]. For p = 2 the previous definitions give the appropriate notions of interpolation and sampling as well, in view of the pointwise growth of functions in the spaces (see Lemma 18 and Remark 5).
Our description of interpolating and sampling sequences is expressed in terms of certain Beurling-type densities adapted to the metric induced by ∆φ, or more precisely, by its regularisation ρ −2 (z)dz ⊗ dz. Before introducing the densities we need the notion of ρ-separation.
This is equivalent to saying that the points in Λ are separated by a fixed distance in the metric above (Lemma 4).
The main theorems are the following. Let Ω φ denote the class of flat weights.
In particular, there are no sequences which are simultaneously sampling and interpolating (it should be mentioned that this is not obtained as a corollary of the theorems; it is actually an important ingredient of the proofs).
These results generalise previous work, beginning with the papers by Seip and Seip-Wallstén [Sei92], [SW92]. They described the interpolating and sampling sequences for the classical Fock space in terms of the so-called Nyquist densities. In the notation above this corresponds to φ(z) = |z| 2 and ω ≡ 1. This was extended in [LS94], [BOC95] and [OCS98] to the case of entire functions f such that f e −φ ∈ L p (C), where φ is subharmonic with bounded Laplacian ε < ∆φ < M. The description was given again in terms of some Nyquist type densities. In these cases the function ρ is bounded above and below, hence the metric ρ −2 (z)dz ⊗ dz is equivalent to the Euclidean metric. In particular, ρ(z) can be replaced by the constant 1 in the definition of the uniform densities.
There are also some partial results in several complex variables. The classical Fock space has been studied in [MT00] and the weighted scenario in [Lin01]. In this context there exist necessary or sufficient density conditions, which do not completely characterise the sampling or interpolating sequences.
Interpolation problems for other spaces of functions related to these weights have been considered by Squires and Berenstein and Li (see for instance [Squ83], [BL95] and the references therein).
The results mentioned above relied on the remarkable work by Beurling [Beu89] and on Hörmander's weighted L 2 -estimates for the∂ equation [Hör94]. In our proofs we first extend Beurling's tools to the context of certain spaces which are non-invariant under translations. We need as well a Hörmander type theorem giving precise estimates for the∂ equation in Banach norms other than L 2 .
The plan of the paper is the following: In Section 2 we study the properties of doubling measures and introduce the flat weights. Recall that the only assumption on our subharmonic weight φ is that the measure ∆φ is doubling. We will need a regularisation of φ and the construction of a multiplier associated to φ (that is, an entire function f such that |f | approximates e φ ), very much in the spirit of [LM01] and [OC99].
In Section 3 we state and prove some basic properties of functions in F p φ,ω . The main result in this section is the following Hörmander type theorem.

Theorem C. Let φ be a subharmonic function such that ∆φ is a doubling measure. For any
We also include the estimates of the Bergman kernel that justify the notion of interpolating and sampling sequences we have considered. Finally , we study the invariance of our spaces under some appropriate scaled translations. This leads to the notion of weak limit and the corresponding analysis analogous to Beurling's. Section 4 is devoted to some preliminary (but important) properties of interpolating and sampling sequences, including their behaviour under weak limits. The main results in this section are some inclusion relations between various spaces of interpolating and sampling sequences, and the fact that there are no sequences which are simultaneously interpolating and sampling for the same space of functions F p φ,ω . In Section 5 we prove the sufficiency part of Theorem A. We use again an approach similar to that of Beurling.
Section 6 includes the proof of the necessity part of Theorem A. For this we need once more Beurling's analysis, plus the non-existence of sampling and interpolating sequences. We use some theorems that relate the densities of sampling and interpolating sequences, following the ideas by Ramanathan and Steger [RS95].
Section 8 is devoted to the proof of the necessity part of Theorem B. We use Ramanathan and Steger's theorem plus an original argument that shows that the density inequality is actually strict.
Finally, in Section 7 we deal with the sufficiency part of Theorem B. In the course of the proof, whose main tool is the multiplier, we need to express the density in terms of rectangles instead of disks. The usual argument of Landau [Lan67] does not work, in view of the inhomogeneity of our measures. Theorem 42 takes care of this. A final word on notation: C denotes a finite constant that may change in value from one occurrence to the next.The expression f g means that there is a constant C independent of the relevant variables such that f ≤ Cg, and f ≃ g means that f g and g f .

SUBHARMONIC FUNCTIONS WITH DOUBLING LAPLACIAN
In this chapter we recap some results on doubling measures and subharmonic functions φ whose Laplacian ∆φ is doubling. We start with regularity and integrability conditions on doubling measures. Next we show that φ can be regularised, in the sense that there exists ψ subharmonic and regular for which the interpolation and sampling problems for F p φ,ω and F p ψ,ω are equivalent. The final part is dedicated to the construction of the multiplier. A useful application of this is the existence of holomorphic "peak functions" with controlled growth.
for all z ∈ C and r > 0. We denote by C µ the minimum constant C for which the inequality holds.
Recall that when φ is subharmonic ∆φ is a nonnegative Borel measure, finite on compact sets.
For convenience we write D r (z) = D(z, rρ(z)) and D(z) = D 1 (z). We will write D r φ (z) when we need to stress that the radius depends on φ.
Henceforth dm denotes the Lebesgue measure in C. We also use the measure dσ = dm/ρ 2 , which should be thought of as a doubling regularisation of ∆φ (see Theorem 14).
2.1. Doubling measures. Throughout this section we assume that µ is a positive doubling measure non-identically zero. We begin with a result of Christ [Chr91, Lemma 2.1].
Lemma 1. Let µ be a doubling measure in C. There exists γ > 0 such that for any disks D, D ′ of respective radius r(D) > r(D ′ ) with D ∩ D ′ = ∅: In particular, the support of µ has positive Hausdorff dimension.
As a consequence of Lemma 1 and (5) we have Corollary 3. For every r > 1 there exists γ > 0 such that if ζ ∈ B(z, r) then It will be convenient to express some of the results in terms of the distance d φ induced by the metric ρ −2 (z)dz ⊗ dz.
The lower inequalities are contained in [Chr91, Lemma 3.1] and its proof.
The upper estimate in case (a) is immediate from Corollary 3. In case (b) take γ(t) = z + t(ζ − z) and use Lemma 2; then From now on, given z ∈ C and r > 0, we denote Doubling measures satisfy certain integrability conditions.
Lemma 5. Let µ be a doubling measure. There exist C > 0 and m ∈ N depending on C µ such that for any r > 0 (b) According to Lemma 4 it is enough to consider the integral on |z − ζ| ≥ rρ(z). Applying Fubini's theorem we see that Let x 0 = log 2 C µ , and for a given t denote k(t) = inf{k ∈ N : 1/rt ≤ 2 k }. Then This and Lemma 4(b) show that the result holds for m big enough.
Remark 2. It is clear from the proof that There is a discrete version of the previous Lemma.
Lemma 6. Let Λ be a ρ-separated sequence. There exists m ∈ N such that sup z∈C λ∈Λ Proof. By the separation and Lemma 4, it is enough to see that for m big enough Take δ > 0 such that the balls {B(λ, δ)} λ∈Λ are pairwise disjoint. By Corollary 3 Lemma 5(b) implies that the integral is bounded.
For later use, we state a refinement that follows similarly from Remark 2.
Corollary 7. Let Λ be a ρ-separated sequence. There exists m ∈ N such that We will need to partition the plane in rectangles of constant mass. We do that by adapting a general result of [Yul85] to our setting (see also [Dra01, Theorem 2.1]).
Theorem 8. Let µ be a positive doubling measure non-identically zero. There exists a "partition" of C in rectangles R k with sides parallel to the coordinate axis such that: (a) µ = k µ k , where µ k := µ |R k satisfy µ k (C) = 1. (b) R k are quasi-squares: there exists e > 1 depending only on C µ such that the ratio of sides of each R k lies in the interval [1/e, e]. (c) There exists C < 0 such that Remark 3. Dividing the original measure by s ∈ R + we obtain a partition of C into quasi-squares of mass s.
Proof. It is enough to partition the plane in quasi-squares of constant entire mass, because by an stopping-time argument of [OC99] these can then be split into quasi-squares of mass 1.
We construct our partition recursively. We start with a rectangle centred at 0 of entire mass, and with sidelengths l ≤ L so that l ≥ L/2 and l 1−β ≥ 12 √ 2C 0 , where β and C 0 are given in (5) (rectangle ABCD in the picture). Consider next a square Q 1 centred at 0 of sidelength 3L (A 1 B 1 C 1 D 1 in the picture) and define R as the quasi-square with vertices ABB ′ A ′ , where A ′ and B ′ are points on the same side of Q 1 taken so that 0 / ∈ R. We want to make R a little bigger, to make sure that its mass is entire, and we want to do that keeping control on the ratio of sides. Consider the rectangle ABBÃ, whereÃ,B are taken with |AÃ| = |BB| = 2|AA ′ |. Denote by Since the sides of R ′ have length bigger or equal than l we deduce that R ′ contains a disk of centre λ and radius ρ(λ), hence its mass is at least 1. This shows that there exists a rectangle R 1 (AA ′′ B ′′ B in the picture) of entire mass between the original R and the "doubled" R ′ .
We finish the first step of the process by constructing the analogous quasi-square R 2 of entire mass at the opposite side of R (CC ′′ D ′′ D in the picture).

AB D ′′ C ′′
Consider next the rectangle Q 2 limited by the segments We iterate the process above to each of the rectangles All in all, we obtain a new quasi-square Q 3 := A 3 B 3 C 3 D 3 with ratio of sides lying in [1/2, 2] which is a disjoint union of 5 quasi-squares of entire mass. From here we repeat the process, taking Q 3 in place of the original R, and continue recursively to obtain the "partition" of C. By construction we have (a), (b) and (d).
To prove (c) assume that R is a quasi-square of mass 1, centre a and sidelengths l, L. Here R ⊂ D(a, L √ 2), hence ρ(a) L diam(R). Also, D(a, l) ⊂ R and diam(R) l ρ(a).
Lemmas 1 and 2 give control on how big a disc D r (ζ) can be when ζ ∈ D R (z). We will need another result along the same lines.
Given a doubling measure µ and given z ∈ C and 0 < r < R, consider the associated regions . Let γ be the constant given by Lemma 1, and ε, k the constants in (4).
Proof. Applying Lemma 1 to D r (ζ) and D R (z), and using (4), we have Corollary 10. Let {R k } k be a partition of C, as in Theorem 8. Define There exists a positive function ǫ(R) with lim R→∞ ǫ(R)/R = ∞ and such that for all z ∈ C and R > 0 Proof. As the previous Lemma, using Theorem 8(c).
We finish with a result showing that the measure of a disk cannot be too concentrated near its border.
Lemma 11. Let ǫ(r) be a positive function such that lim r→∞ ǫ(r)/r = 0. Then The proof is based in the following projection of the measure µ.
Lemma 12. For every z ∈ C define the measure ν z on R + by Then ν z is doubling and there exists K independent of z such that C νz ≤ KC µ .
Proof. Given x ∈ R + and r > 0 let I r (x) = (x − r, x + r) ∩ R + . We want to see that for all z ∈ C, x ∈ R + and r > 0.
Being µ doubling there exists Since theS j 's are disjoint and ∪ jSj ⊂ A r z (x), we get Proof of Lemma 11. It is enough to see that and by the corresponding version of Lemma 1 for doubling measures in R + , and by Lemma 12, there exists K > 0 independent of z such that Remark 4. An analogous result is true if in the definition of ν z we use, instead of a radial projection with respect to z, a projection associated to quasi-squares of a fixed ratio α ∈ [e −1 , e] (e is the constant of Theorem 8(b)). Let Q r α (z) denote the rectangle with vertices z + r(1 + iα), uniformly in z.

Flat weights.
In this section we describe the weights ω appearing in the spaces F p φ,ω .
Definition 6. A positive measurable function ω is called a flat weight for φ if there exists C > 0 such that for all z, ζ ∈ C The class of flat weights associated to φ will be denoted by Ω φ .
Besides the obvious ω = 1, important examples of flat weights for φ are the functions ω = ρ α , α ∈ R. This is seen applying Lemma 2 and Lemma 4. Furthermore, the weights ω ∈ Ω φ can be assumed to satisfy If the original weight ω does not satisfy this condition, replace it by the regularisatioñ It is clear, by (7), that there exists C > 0 such that C −1 ≤ |ω/ω| ≤ C, hence the spaces of functions and sequences associated to the weights ω andω are the same. On the other hand Assuming that d φ (z, ζ) ≤ 1, from (7), (6) and Lemma 4(a) we deduce that 2.3. Local behaviour and regularisation of φ. Let us start with a result comparing the values of φ in a disk with the value on its centre.

Lemma 13. For every
where G is the Green function of the disc D K (z) and h z is a harmonic function in D K (z) such that h z (z) = 0. By Lemma 5(a) and the result holds.
We have seen in the previous section that ρ φ (z) is Lipschitz (see (6)). Also, because of Lemma 1, φ is Hölder continuous of some positive order on every bounded subset of C (see [Chr91, Lemma 2.8]). More regularity can be attained by taking a suitable weight ψ equivalent to φ.
In the proof of this result we will need to partition C and discretize the measure.
Lemma 15. Let µ be a positive doubling measure in C. Fix m ∈ N. There exist k ∈ N and C > 0 such that for any partition {R p } p as in Theorem 8 with µ(R p ) = mk there are points λ have the same first m moments.
Proof. By Lemma 5 of [OC99], there exists k ∈ N such that for all measure µ p supported in a rectangle R p with total mass mk, there are points σ have the same first m moments.
In order to get a separated sequence replace each σ k j (p) by m points γ the measures µ p and j,l δ γ (p) j,l have still the same first m moments. We will be done as soon as we see that the τ (p) j can be chosen uniformly bounded and so that Λ = {γ (p) j,l } is ρ-separated. For this we use a Besicovitch's lemma: the family {R p } p can be split in q families {R 1 p } p∈I 1 , . . . {R q p } p∈Iq such that two rectangles of the same family are far apart, in the sense that MR l p ∩ MR l p ′ = ∅, p = p ′ , for some large constant M. For the first family {R 1 p } p∈I 1 , it is easy to choose τ By Theorem 8(c) there exists r > 0 such that CR p ⊂ D r (λ (p) j ) for any p ∈ N and i ≤ k. Furthermore, by construction of {R p } p there exists q ∈ N such that any z ∈ C lies in at most q disks D r (λ We now regularise ν p by settingν , where X is a smooth non-negative cut-off function of one real variable such that Notice thatν p and µ p have the first M moments. Indeed, by the mean value property We claim thatν is a doubling measure. The proof of this fact is a bit technical and will be deferred to the end.
By definition ∆ψ =ν. Also,ν(z) is a sum of at most q terms of order 1/ρ 2 (λ Let us show next that |ψ − φ| ≤ C for some C > 0.
Let a p denote the centre of R p . Assume z ∈ R p 0 and let the analogous estimate for |z − a p 0 | and Lemma 4. This yields We split and estimate each sum separately.
Let p M denote the M-th Taylor polynomial of log(|z − ζ|/ρ(z)). Sinceν p − µ p have vanishing moments of order less or equal to M, we can estimate Taking M big enough and using (10) and Lemmas 4(a) and 5(b), For the remaining term we use again the moment condition together with the fact that for Thus By Lemma 5(a) this is finite.
We prove now thatν is doubling. We first show that it is doubling for big balls, i.e. there exist R 0 > 0 and a constant C depending only on the doubling constant C ∆φ of ∆φ such that for all R > R 0 we haveν(D R (a)) ≤ Cν(D R/2 (a)).
Lemma 11 shows that the quotient converges to 1 as R goes to infinity uniformly in a, so there exists R 0 such thatν(D R (a)) ≤ 2C ∆φν (D R/2 (a)) for all R ≥ R 0 .
2.4. The multiplier. A basic tool in our approach is the use of the so-called multiplier: an entire function g such that |g| ≃ e φ outside a neighbourhood of the zeros of g.
Theorem 16. Let φ be a subharmonic function such that ∆φ is a doubling measure. There exists an entire function g such that The function g can be chosen so that, moreover, it vanishes on a prescribed z 0 ∈ C. We say that g is a multiplier associated to φ.
Proof. Take a partition {R p } of C with µ(R p ) = 2πmN and consider the sequence Λ given by Lemma 15. For the sake of clarity we write R p instead of CR p (C is the constant of Lemma 15). Note that now {R p } p is not a partition, although there exists a uniform constant q such that all points of C lie in at most q quasi-squares R p . As in Lemma 15, denote µ p = (1/2π)µ |Rp and let ν p be the sum of the λ ∈ Λ associated to R p . Recall that µ p and ν p have the same first m moments.
Let g be a holomorphic function satisfying which exists because the Laplacian of the term at the right hand side is a sum of Dirac masses. By definition Z(g) = Λ, and the previous construction ensures that (a) holds.
Let us prove (b). Assume that z ∈ R p 0 and let I p 0 denote the set of indices p such that R p ∩ R p 0 = ∅. As in the previous proof, split Again as in the proof of Theorem 14, using the Taylor expansion of log |z − ζ| together with the moment condition one sees that |S 1 (z)| is bounded.
For the second sum notice that there exits γ > 0 such that ∪ p∈Ip 0 R p ⊂ D γ (z). Hence, denoting |z − Λ| = inf λ∈Λ |z − λ|, we get On the other hand, using the ρ-separation of Λ Since #I p 0 is uniformly bounded, this and the estimate of S 1 give: The result is then immediate from Lemma 4(a).
Next we state a useful application of the multiplier. It is a result about peak functions. These functions attain value 1 at a given point and decay very fast away from the point. They are very useful in the estimates of the Bergman kernel and in the construction of solutions to thē ∂ equation. Another proof of the following Lemma, using estimates for the∂-equation, can be found in an Appendix. This second proof is along the lines of [FS89, Theorem 2.1], where a related result is proved.
Let us see first that there exists c > 0 independent of η with c −1 ≤ c η ≤ c. Since |η − σ i | ≃ ρ(η), then We split the estimate of |P η (z)| into several regions. Let ε > 0 be such that that the balls B(σ i , ε) and B(η, ε) are pairwise disjoint. Consider This and Lemma 4(b) solve the case ω = 1.
Thus the result follows from the previous construction taking M big enough and using again Lemma 4(b).

BASIC PROPERTIES OF FUNCTIONS IN F p φ,ω
Here we study the behaviour of functions in F p φ,ω and related topics. We prove the estimates with norms · F p φ,ω on the solutions to the∂ equation (Theorem C) and provide estimates of the Bergman Kernel of F 2 φ,ω on the diagonal. We also introduce a scaled translation in the plane that gives rise to a translated weight and to an isometry between the spaces of functions for the original and the translated weight. This will be used when studying the properties of weak limits (Section 3.5).

Pointwise estimates. Let us first see what is the natural growth of functions in
Lemma 18. Let 1 ≤ p < ∞ and ω ∈ Ω φ . For any r > 0 there exists C > 0 such that for any f ∈ H(C) and z ∈ C: Proof. Let H z be a holomorphic function with Re H z = h z , where h z is the harmonic function in D r (z) given in Lemma 13.
By Lemma 13 we have then which together with (11) concludes the proof.
(c) As (a), using the subharmonicity of |f e −Hz | p .
Lemma 19. Let 1 ≤ p < ∞ and ω ∈ Ω φ . For any entire function g with g(λ) = 0 we have Proof. Lemma 18(c) with r = 1/2 and R = 1 applied to the function g(z)/(z − λ) yields 3.2. Hörmander type estimates. This section is devoted to the proof of the∂-estimates of Theorem C in the introduction.
Theorem C. Let φ be a subharmonic function such that ∆φ is a doubling measure. For any Proof. By Lemma 18(b), there exists r > 0 such that |P η (z)| e ε(φ(z)−φ(η)) on D r (η), for all η ∈ C. Take a sequence Λ such that {D r (λ)} λ∈Λ covers C and the disks {D r/5 (λ)} λ∈Λ are pairwise disjoint, which exist by a standard covering Lemma, see [Mat95, Theorem 2.1]. Let . By Theorem 17, for any λ there exists an entire function m λ (z) = P λ (z)e −φ(λ) such that The radius r has been chosen so that |m λ (ζ)| e φ(ζ) if ζ ∈ D r (λ). Define Clearly∂u λ = f λ , thus u = λ∈Λ u λ is as a solution to∂u = f . We must prove the size estimates. As we have used a linear operator to construct u from the datum f , we only need to check that ue −φ ω L ∞ f e −φ ωρ L ∞ and ue −φ ω L 1 f e −φ ωρ L 1 . The estimates for 1 < p < ∞ follow then by Marcinkiewicz interpolation theorem.
Assume that z ∈ D r (λ) and take K, On the other hand, if z / ∈ D K (λ) Therefore, applying Lemma 6 In the L 1 norm we get Reversing the order of integration we immediately get ue −φ ω L 1 f e −φ ωρ L 1 .

By definition
Lemma 20. There exists C > 0 such that Proof. We use the identity The estimate K φ,ω (z, z) e φ(z) /ω(z) is immediate from Lemma 18(a). In order to prove the reverse estimate we construct f ∈ F 2 φ,ω with f F 2 φ,ω ≤ 1 and |f (z)| ≥ Ce φ(z) /ω(z), for some constant C independent of z.
By Theorem 17, for every m ∈ N there exists P z entire such that , with C independent of z. Define f z (ζ) = c 0 e φ(z) /ω(z) P z (ζ), where c 0 is a positive constant to be chosen later. Now f z (z) = c 0 e φ(z) /ω(z) and hence by Lemma 5(b) there exist c 0 and C independent of z so that f z F 2 φ,ω ≤ 1.

Scaled translations and invariance.
In this section we introduce the scaled translation and its main properties.
Given φ consider the class W φ of subharmonic functions ψ such that An important property of W φ is that there exists η such that ∆ψ(z) |z| 2η for all regular ψ ∈ W φ . This is a consequence of (5) and the fact that ∆ψ ≃ 1/ρ 2 ψ . Fix q > 2η + 1 and consider the kernel where P q is the Taylor polynomial of degree q of log(1 + x) around x = 0, and its associated integral operator

This operator solves the Poisson equation, that is ∆K[f ] = f .
For every x ∈ C, consider the scaled translation the associated subharmonic function

and the associated weight
Define also h x := φ • τ x − φ x . It is clear that h x is harmonic. Take then H x holomorphic having h x as real part and consider the scaled translation operator

Lemma 21.
For every x ∈ C, (a) The subharmonic function φ x belongs to W φ , and the weight ω x satisfies ω x (0) = 1 and Proof. Note first that from the identity This implies that the mapping τ x is actually an isometry between C endowed with the distance d φx and C with d φ , that is (a) By definition φ x (0) = 0, and by (12) ρ φx (0) = 1. This gives properties (ii) and (iii) of W φ .
The case p = ∞ is straightforward from (12).
Given a sequence Λ and x ∈ C let Given a sequence Λ and z ∈ C, denote n Λ (z, r) = #(Λ ∩ D(z, r)), for any r > 0.

Lemma 22.
Let Λ be a sequence in C. Proof. (a) is an immediate consequence of (12).

(b) is a consequence of Lemma 21 and the identity
By a change of variables, it is clear that Taking the supremum over z ∈ C and passing to the limsup we get the result for the upper density. The lower density is dealt with similarly.

Weak limits.
In this section we study weak limits of sequences Λ and their properties.

Definition 7. A sequence of closed sets
denotes the Fréchet distance between Q and R. We say that Q j converges compactwise to Q, denoted Q j ⇀ Q, if for every compact set K we have (Q j ∩K)∪∂K → (Q ∩ K) ∪ ∂K.
Given a ρ-separated sequence Λ, and a sequence {x n } n∈N it is always possible to extract a subsequence of Λ xn j such that Λ xn j ⇀ Λ * for some Λ * . We need also a normal family argument for the translated weights that define the space.
Proof. Take η and q > 2η + 1 as in the definition of the kernel κ (see previous section). Denote µ n = ∆φ xn .
Since |∇µ n | ρ −3 φx n (Theorem 14) and ρ φx n (0) = 1, for any compact set K there exits C K > 0 such that |∇µ n (z)| ≤ C K . By the Arzelà-Ascoli theorem, we can extract a subsequence {µ n k } k converging uniformly on compact sets of C to a function µ * . It follows immediately that the measure with density µ * is doubling and C µ * ≤ C µn = C ∆φ . Furthermore, this implies that ρ φx n → ρ * uniformly on compacts.
Let I 2 be the second integral in the estimate above. We have For all z ∈ D(0, R) and ζ ∈ D(0, t) \ D(0, 1) we have |P q (z/ζ)| ≤ C(R, t), hence the uniform convergence of µ p implies that for p big enough the second integral here is smaller than ε. It remains to prove the convergence of the first term. Take C(t) such that D(0,t) | log |z − ζ/ζ dm(ζ) ≤ C(t) and choose p big enough so that |µ p (ζ) − µ * (ζ)| ≤ ε/C(t) uniformly on D(0, t). Then the estimate follows.
We know that the sequence of distance functions d φx n has a subsequence converging to d φ * uniformly on compact sets of C × C, because the ρ xn k converge uniformly. By construction ω xn (0) = 1. On the other hand, the definition of flat weight implies that they are equibounded on any compact. Moreover, the regularity given by (8) makes them equicontinous on compact sets. We can thus extract again a convergent subsequence.
Let us prove now the stability of the upper and lower densities with respect to weak limits.

PRELIMINARY PROPERTIES OF SAMPLING AND INTERPOLATING SEQUENCES
This section is devoted to prove auxiliary results on interpolating and sampling sequences. A main result is that there do not exist sequences which are simultaneously sampling and interpolating. We also prove some results on inclusions between spaces of sampling and interpolating sequences for various weights.
An easy consequence of Lemma 18 is that we only need to deal with ρ-separated sequences.
(b) As in the proof of [Beu89, Theorem 2, p. 344], using here Lemma 18(b) instead of Bernstein's theorem, we get (c) It is enough to show that there exists r > 0 and M such that #(D η (z) ∩ Λ) ≤ M for all z ∈ C. To this end, consider the function f z (ζ) = e φ (z)/ω(z)P z (ζ), where P z is given by Theorem 17 (with ε = 1 and ω = 0). We have f z F p φ,ω ≤ C, and for r small enough |f z (ζ)| e φ (ζ)/ω(ζ) in D r (z). So the left sampling inequality (see (2)) yields (d) It is enough to see that for R big enough Λ ∩ D R (z) = ∅ for all z ∈ C.
Take f z as in (c). Let ε > 0 be the ρ-separation of Λ. Since , Lemma 18(a) and Lemma 9 lead to According to Remark 2 this tends to 0 uniformly in z as R goes to ∞. Thus, for R big enough the sampling inequality gives In particular Λ ∩ D R (z) = ∅, as desired.

Weak limits and interpolating and sampling sequences.
In this section τ φ x will denote the scaled translation associated to the weight φ, as described in Section 3.4. The main result is as follows.
For every n consider R n big enough so that if D n := D Rn φ * (0) then f n |D n F p φ * ,ω * ≥ 1 − ε n . Set D n := D R 2 n φ * (0). We claim that there exists a smooth cut-off function X n such that X n (ζ) = 1 in D n , X n (ζ) = 0 in C \ D n and |∂X n | ≤ ε n /ρ φ * . To see this start with a smooth X n depending linearly on |ζ| on R n ≤ |ζ| ≤ R 2 n . Then .
Take now j n big enough so that ρ φ jn /ρ φ * ≤ 2 on D n and Define g n = f n X n . Then∂g n is supported on C n := {R n ≤ |ζ| ≤ R 2 n } and |∂g n (ζ)| ≤ ε n |f n (ζ)|/ρ φ * (ζ), so by Theorem 1 there exists u n solution to∂u n =∂g n with u n F p φ jn ,ω jn ∂ g n ρ φ jn F p φ jn ,ω jn ε n f n | D n F p φ jn ,ω jn ε n .
The function G n = g n − u n is holomorphic and satisfies We will check now that G n |Λ jn is small. Split Λ jn into Λ jn = Λ jn ∩ {D n ∪ (C \ D n ))} and Λ jn = Λ jn \ Λ jn . On the one hand From u n | Λ jn ℓ p φ jn ,ω jn ( Λ jn ) ≤ u n F p φ jn ,ω jn ≤ ε n (by Lemma 18 for the case p < ∞, since u is holomorphic in D n ∪ (C \ D n )) we deduce that G n | Λ jn ℓ p φ jn ,ω jn ( Λ jn ) ε n . On the other hand This together with the above and the fact that the sampling constants of Λ and Λ jn coincide (Lemma 22(b)) leads to contradiction.
(b) Assume that Λ * = {λ * k } k , and let v ∈ ℓ p φ,ω (Λ * ) with v ℓ p φ,ω (Λ * ) ≤ 1. Let also Λ j = {λ j k } k be such that Λ j → Λ * uniformly on compact sets. For ε n decreasing to zero and R n big enough (to be chosen later) there exists j n such that v ℓ p φ jn Since the interpolation constant M(Λ j ) does not depend on j there exist f n ∈ F p φ jn ,ω jn with f n F p φ jn ,ω jn ≤ 2M(Λ) and We will now use the same technique as in (a) to modify f n so that it falls in F p φ * ,ω * . Take the cut-off function X n constructed above, define g n = f n X n and consider a solution u n tō ∂u n = f n∂ (X n ) such that: According to Theorem C and (15) such a solution always exists.
The entire function G n = f n∂ (X n ) − u n is F p φ * ,ω * and G n F p φ * ,ω * ≤ CM. By Montel's theorem we may assume that G n converges to a function G ∈ F p φ * ,ω * . Notice that G n (λ jn k ) = v k − u n (λ jn k ) for λ jn k ∈ D Rn φ * (0), and by the L ∞ estimates, |u n (λ jn k )| tends to zero as n goes to infinity. Therefore G interpolates v.

Non-existence of simultaneously sampling and interpolating sequences. An important result in the proof of Theorems A and B is the following.
Theorem 30. There is no sequence Λ both sampling and interpolating for F p φ,ω , p ∈ [1, ∞].
According to Lemma 26(d) there exists r > 0 with C = ∪ λ∈Λ D r (λ). Also, there exists r 0 > 0 depending on r such that, We may now finish by taking a big disk D(0, M) and λ * M ∈ D(0, M) in such a way that ρ(λ * M ) ≥ ρ(λ) for all λ ∈ Λ ∩ D(0, M). In this case This is a contradiction, since lim M →∞ ρ(λ * M )/M = 0 and the left hand side of the previous inequality tends to ∞ as M goes to ∞.

Corollary 31. Any sequence obtained by deleting a finite number of points of
We want to prove next an analogue for interpolating sequences: adding a finite number of points to an interpolating sequence gives again an interpolating sequence.
Given Λ and a point z define, following [Beu89, Notice first that if Λ is interpolating and z / ∈ Λ this is strictly positive. Indeed, Λ is not a uniqueness sequence, otherwise Λ would be also sampling, contradicting Theorem 30. Thus there exists f ∈ F p φ,ω , f = 0 with f |Λ ≡ 0 and, eventually dividing f by a power of (ζ − z), Proof. As in the proof of [Beu89, Lemma 4, p.233], we have .

Inclusions between various spaces of interpolating sequences.
We want to study next the relationship between the spaces of interpolating sequences for various weights. We will use the techniques already exploited in [MT00].
The remaining inequality is immediate from (c).
Let p < ∞. On the one hand, Lemma 18(a) gives On the other hand, (c) and Lemma 5(b) show that for m big enough The left inequalities are proved similarly to (b), for For p = ∞ and v ∈ ℓ ∞, (1+ε)φ,ω (Λ) Lemma 6 and (c) yield Let now p < ∞. Using the estimate (c) and Jensen's inequality for convex functions (which is legitimate thanks to Lemma 6) we have . Now we apply Lemma 5(b) and obtain (e) This follows from (c) and Remark 2, since .

Inclusions between various spaces of sampling sequences.
In this section we want to prove some inclusions between various spaces of sampling sequences. Unlike in the corresponding result for interpolating sequences, for the spaces of sampling sequences there is a gain, in the sense that any sampling sequence is actually sampling for a slightly bigger space. This will allow us to pass from the non-strict to the strict inequality of Theorem A.
Proof. The proof is divided in three steps.
(a) If Λ ∈ Samp F p φ,ω , then Λ ∈ Samp F ∞ φ,ω . We know from Proposition 27 that for all weak limit (Λ * , φ * , ω * ) the sequence Λ * is in Samp F p φ * ,ω * , and by Lemma 29 it will be enough to see that all weak limit Λ * is a uniqueness set for F ∞ φ * ,ω * . If this is not the case, there exists f ∈ F ∞ φ * ,ω * with f |Λ * ≡ 0, f = 0. We claim that for m large enough It is clear that Lemma 19 gives the p-integrability on ∪ m j=1 D(λ * j ). On the other hand, by Lemma 18 Since ∆φ * is doubling there exists m such that this integral converges (Lemma 5(b)).
As f vanishes on this sequence we deduce that f ≡ 0, which is a contradiction.
There is a sequence of functions {g(z, λ)} λ∈Λ such that for all f ∈ F ∞,0 and λ |g(z, λ)| ≤ K uniformly in z. This is so by a duality argument, because is a bounded linear functional from a closed subspace of ℓ ∞,0 (1+ε)φ,ω (Λ) whose norm is bounded independently of z. This is an argument from [Beu89, (see also [Sei93,p.36]).
Proof. The construction of f is made with quasi-squares R p of µ(R p ) = 2πmN and mN associated points in a dilated CR p that made up Λ. Thus, for z ∈ C and t > 0: By Corollary 10, The result is then an application of Lemma 11.
Proof. By Theorems 34 and 35, it is enough to consider the case ω = ρ.
Let f be a multiplier associated to φ such that Λ = Z(f ).
Let us start by proving that Λ is interpolating. By Theorem 34 it is enough to prove that .
Consider also a multiplier g associated to εφ * . In particular |g(z)| ≃ e εφ * (z) d φ * (z, Z(g)). In order to see that Λ * is a uniqueness sequence assume that h ∈ F ∞ (1−ε)φ * ,1 and h|Λ * = 0. Then hg ∈ F ∞ φ * ,1 , by construction. On the other hand, the function F := hg/f * is entire, because h vanishes on Λ * . It is also bounded when z is far from Λ * , since |hg| e φ * and |f * | e φ * . By the maximum principle F is bounded globally, and by Liouville's theorem there exists c ∈ C such that hg = cf * . Since g vanishes in some points outside Λ * we have c = 0, hence h ≡ 0.

SUFFICIENT CONDITIONS FOR SAMPLING
We prove here the sufficiency part of Theorem A. Assume that D − ∆φ (Λ) > 1/2π. By Lemma 26 we can assume that Λ is ρ-separated, and according to Theorem 35 it will be enough to prove that Λ ∈ F ∞ φ,ω . By Corollary 29 this will be done as soon as we show that every weak limit Λ * is a uniqueness sequence for F ∞ φ * ,ω * . Recall the notation n Λ (z, r) = #[Λ ∩ D(z, r)].

NECESSARY CONDITIONS FOR SAMPLING
This section contains the proof of the necessity part of Theorem A. By Lemma 26(b) and Theorem 35 it will be enough to prove the following result.
We use a result comparing the densities between interpolating and sampling sequences, as in [RS95]. We do that by adapting Lemma 4 in [OCS98] to our setting.
Proof. The proof is as in [OCS98,Lemma 4] with minor modifications, so we keep it short.
According to our definition, if S is sampling then {k(z, s) = K φ,α (z, s)e −φ(s) ω(s)} s∈S is a frame in F 2 φ,ω (K φ,ω denotes the Bergman kernel, as in Section 3.3). That is, for f ∈ F 2 φ,ω A consequence is that wherek(z, s) is the dual frame of k(z, s).

SUFFICIENT CONDITIONS FOR INTERPOLATION
Taking into account Theorem 34 and Lemma 37, in order to prove the sufficiency part of Theorem B it is enough to prove the following.
In the proof of this result we need to express the density condition in terms of the quasi-squares appearing in Theorem 8. this will be done in Theorem 42; before we need some preliminaries.
It is clear that F r (w, s) ⊂ R s α (w) ⊂ G r (w, s). Similarly to the proof of Lemma 9, there exists ǫ(s) such that R s−ǫ(s) α (w) ⊂ F r (w, s) and G r (w, s) ⊂ R s+ǫ(s) α (w). Proof of Theorem 40. Take an entire function g vanishing exactly on Λ. We will construct a sequence Σ and an entire function h such that for some ε > 0, (ii) h vanishes exactly on Σ.
Accepting this we reach the result by taking f = gh. This is so because the separateness of Λ ∪ Σ and (iii) imply that f is a multiplier for (1 − ε)φ.
Letμ(R k ) = m k n, with m ≤ m k ≤ M. Notice that m k ∈ N, since µ(R k ) ∈ N. Applying Lemma 15 we obtain a sequence Σ made of points σ k 1 , . . . , σ k m k n ∈ CR k so that the first m moments of the measures ν k =μ k − m k n j=1 δ σ k j vanish. Furthermore, it is clear that we can choose the τ k j in the proof of Lemma 15 so that Λ ∪ Σ is ρ-separated. Let In order to prove (iii) consider v = (1 − ε)φ − log |g| − w, where w(z) = C log |z − ζ| dν(ζ).
Let C > 0 be the constant of Lemma 15. Since the first m moments of ν k 0 vanish, C log |z − ζ|dν k 0 (ζ) = C log |z − ζ| r 0 ρ(z) dν k 0 (ζ) The other integral is estimated using the moment condition for each ν k , as in the estimate of I 1 in Theorem 14.

NECESSARY CONDITIONS FOR INTERPOLATION
Let us start by proving the non-strict density inequality. By Theorem 34, it is enough to consider the case p = 2.

APPENDIX. ALTERNATIVE CONSTRUCTION OF PEAK FUNCTIONS.
As seen at the end of the proof of Theorem 33, it is enough to consider the case ω = ρ. Also, it will be enough to prove that for any φ there exist C, δ > 0 such that for all η ∈ C there is P η holomorphic with P η (η) = 1 and |P η (z)| ≤ Ce φ(z)−φ(η) min 1, ρ(η) |z − η| δ , since then we can apply this to εδ/m φ(z), take the m-th power and use Lemma 4 to conclude.
Proof. First we show that there exists a solution u as in the statement but satisfying an analogous L 2 estimate instead of the L ∞ one. We use Hörmander's theorem [Hör94]: for every ψ subharmonic in C there exists a solution u to∂u =∂F such that C |u| 2 e −2ψ ≤ C C |∂F | 2 e −2ψ ∆ψ .
We will be done as soon as we prove that This is consequence of [Ber97, Lemma 3.1] applied to the function V (ζ) = u(ρ(z)ζ + z).