SAMPLING SEQUENCES FOR BERGMAN SPACES FOR p < 1. Alexander P. Schuster and Dror Varolin

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1 SAMPLING SEQUENCES FOR BERGMAN SPACES FOR p < Alexander P. Schuster and ror Varolin Abstract. We provide a proof of the sufficiency direction of Seip s characterization of sampling sequences for Bergman spaces for p < based on the methods of Berndtsson and Ortega-Cerdà.. Introduction For 0 < p < and φ a function subharmonic in = {z C : z < }, define F p φ the set of functions analytic in satisfying { f φ,p = } /p f(z p e φ(z z da(z <, to be where da denotes Lebesgue area measure. We say that a sequence Γ = {γ n } of distinct points in the disk is a sampling sequence for F p φ if there exist positive constants K and K such that K f p φ,p n f(γ n p e φ(γ n ( γ n K f p φ,p for all f F p φ. Letting φ(z = log z, we obtain the standard Bergman space A p and the corresponding sampling sequences, which were characterized by Seip [5] for p = using methods that were extended to the case p < by the first named author [4]. Berndtsson and Ortega-Cerdà [] showed, using an altogether different proof, that a variation of Seip s density condition from [5] is actually sufficient to give sampling sequences in Fφ. While Seip s proof relies on a duality argument and most likely cannot be modified to work for A p when 0 < p <, the techniques of [], as was conjectured in [] (p. 7, can be adapted to F p φ (and hence Ap for 0 < p <. The purpose of this note is to show how this can be done. We introduce the definitions necessary for the statement of the theorem we will prove. The sequence Γ = {γ n } is said to be uniformly discrete if δ(γ = inf n m φ γ n (γ m > 0, 99 Mathematics Subject Classification. 30H05, 46E5. Key words and phrases. Bergman space, sampling. Typeset by AMS-TEX

2 where φ ζ (z = ζ z ζz is the standard involutive Möbius transformation. The disk of centre ζ and radius r in this metric will be denoted by (ζ, r. In the disk it is useful to consider the invariant Laplacian = ( z / z z, and for a measure µ and a function g, the invariant convolution µ g, defined by (µ g(z = dµ(ζ g(φ z (ζ π ( ζ. Consider now the measure ν = π n ( γ n δ γn, where δ z is the irac-delta measure at the point z, and for / < r < the function ξ r (ζ = { c r log ζ if / < ζ < r, 0 otherwise, da(ζ where c r is such that ξ r(ζ π( ζ =. We are now in a position to state the main theorem of this note. Main Theorem. Suppose a sequence Γ is uniformly discrete, and φ is a C subharmonic function with uniformly bounded invariant Laplacian φ. If there exists r < and δ > 0 such that (ν ξ r (z > p φ(z + δ for all z, then Γ is a sampling sequence for F p φ. Seip [5] introduces the following definitions. For Γ uniformly discrete, z and / < r <, let /< γ n <r (Γ, r = log γ n log r and (Γ = lim inf r We then have the following theorem, as stated in [4]. inf (φ z(γ, r. z Theorem A. Let p <. A uniformly discrete sequence Γ is a sampling sequence for A p if and only if (Γ > /p. A calculation shows that (φ z (Γ, r (ν ξ r (z = c r log r as r. Moreover, since φ(z = if φ(z = log z, the sufficiency direction of Theorem A will follow from the Main Theorem, for 0 < p <.

3 As mentioned above, the proof is based on the techniques used in []. Our main interest lies in proving the Main Theorem when 0 < p <, thus completing Theorem A, but the proof works, without modification, when p <. With the reader of the paper [] in mind, we will employ the same notation as in that article. We remark that the proof of the necessity part of Theorem A for 0 < p < differs only slightly from the case p < and will not be repeated here. The paper is organized as follows. In the next section we recall some of the notation from [] that is required for the Main Theorem, which we then prove given a collection of technical lemmas. Finally, we complete these technicalities, some of which were claimed without proof in [], so we have included details here for the convenience of the reader. For 0 < t, ɛ <, consider the functions. Proof of the Main Theorem χ ɛ = t ɛ χ (0,ɛ and ν ɛ = ν χ ɛ. Note that ν ɛ da ξ r ν ξ r = (ν ξ r da χ ɛ ν ξ r, which approaches 0 as ɛ 0 and t. Here we have used the fact that fda g = gda f and µ (hda g = (µ hda g ( whenever h is radial. We can therefore choose r and t close to and ɛ close to 0 so that ν ɛ da ξ r (z > p φ(z + δ for all z. Consider now the function v = p (ν ɛ ν ɛ da ξ r da E, where E(z = log z. Since E is the fundamental solution of the invariant Laplacian (with respect to the invariant convolution, we see that v = p (ν ɛ ν ɛ da ξ r, so that the function ψ = φ + v satisfies ψ p ν ɛ δ. We require the following four lemmas to complete the proof of the theorem. 3

4 Lemma. There are positive constants C r and C ɛ such that C ɛ v(z 0 for all z. ( Moreover, v(z t log ɛ p C r (3 for all z with ρ(z, γ n < ɛ for some n. Lemma. δ h(z p e ψ(z ( z da(z t ɛ n (γ n,ɛ e ψ(z h(z p ( z da(z (4 for all h F p φ. Lemma 3. There is a constant C > 0 such that for each h F p φ h a F p φ such that h a (a = h(a and and a, there exists C e φ(a h a (z p h(z p e φ(z Ce φ(a h a (z p for all z (a, /. Lemma 4. There is a constant C > 0 such that ɛ g(z p da(z C g(a p ( a + Cɛ p g(z p da(z (a,ɛ (a,/ for all g F p φ. 4

5 We take these lemmas as given and proceed with the proof. Suppose that h F p φ. Then δ h(z p e ψ(z z da(z t ɛ n = t ɛ n (γ n,ɛ Ctɛ pt n Ctɛ pt n (γ n,ɛ h(z p e φ(z e v(z z da(z h(z p e φ(z (γ n,ɛ γ n Ctɛ pt e φ(γ n γ n n = Ctɛ pt e φ(γ n γ n n Ctɛ pt n Ctɛ pt n Ctɛ pt n e φ(γ n γ n (γ n,ɛ (γ n,ɛ (γ n,ɛ z da(z h(z p e ψ(z z da(z h(z p e φ(z da(z h n (z p da(z h n (z p da(z {C h n (γ n p ( γ n + Ctɛ p e φ(γn ( γ n h(γ n p + Ctɛ p pt n e φ(γn ( γ n h(γ n p + Ctɛ p pt (γ n,/ (γ n,/ h n (z p da(z } h(z p e φ(z z da(z h(z p e φ(z z da(z. The first line is Lemma, the third follows from Lemma, while the fourth follows from the fact that γ n z 4 ɛ for all z (γ n, ɛ. The fifth line is a consequence of Lemma 3, where h n = h γn, while the seventh follows from Lemma 4 and the eighth from Lemma 3. If we take ɛ small enough, we arrive at the lower sampling inequality. The upper inequality follows from the fact that Γ is uniformly discrete. 3. Technicalities We consider now the proofs of the four lemmas. Proof of Lemma. We enumerate Γ once and for all, and let Γ N be the first N terms of Γ. We then set ν N = π γ Γ N ( γ δ γ and w N = ν N E (ν N EdA ξ r. 5

6 Several applications of ( yield w N (z = γ Γ N ( log ϕ z (γ ξ r (ϕ ζ (γ log ϕ z (ζ da(ζ π( ζ Our first claim is that for each compact set K there exists an integer N = N(K such that for all z K and M > N, w M (z = w N (z. Indeed, fix γ Γ and z such that ϕ γ (z > r. The function ζ log ϕ γ (ζ is harmonic in the domain V r (γ = {ζ : < ϕ γ(ζ < r}, and thus ξ r (ϕ ζ (γ log ϕ z (ζ da(ζ π( ζ = ξ r (ϕ ζ (γ log ϕ z (ζ da(ζ V r (γ π( ζ = ξ r (u log ϕ z ϕ γ (u da(u π( u = log ϕ z(γ. /< u <r The second equality follows from the invariance of the measure da(ζ/(π( ζ, while the third equality comes from the mean value property for harmonic functions and the fact that ξ r (ζda(ζ/(π( ζ is a radial probability measure. We therefore have that w N (z = γ Γ N (z,r ( log ϕ z (γ. ξ r (ϕ ζ (γ log ϕ z (ζ da(ζ π( ζ The claim thus follows from the fact that for a given compact set K, there is only a finite number N of points γ Γ such that for some z K, ϕ γ (z r. We set w = lim N w N, where the limit is taken in the locally uniform topology. In other words, w is the ordered sum w(z = ( log ϕ z (γ ξ r (ϕ ζ (γ log ϕ z (ζ da(ζ γ Γ π( ζ. Since (ν N E = ν N is a positive measure, ν N E is subharmonic. Therefore, again since ξ r (ζda(ζ/(π( ζ is a radial probability measure, ν N E ν N E ξ r, i.e., w N 0. It follows that w 0. Since v = p (wda χ ɛ, this implies the right hand side of (. Turning our attention now to (3, we wish to show first that there exists a constant E r > 0 such that for every γ Γ, w(z log ϕz (γ Er. 6

7 whenever z (γ, σ, where σ = δ(γ/. By the above remarks, and since Γ is uniformly discrete, there exists an integer N, depending only on r, such that w(z = N j=0 ( log ϕ z (γ j ξ r (ϕ ζ (γ j log ϕ z (ζ da(ζ π( ζ where γ 0 = γ and γ,..., γ N are the members of Γ in (γ, r+σ +rσ. It follows that w(z log ϕ z (γ = ξ r (ϕ ζ (γ log ϕ z (ζ da(ζ π( ζ N + log ϕ z (γ j j= N j=, ξ r (ϕ ζ (γ j log ϕ z (ζ da(ζ π( ζ. Now, for any t Γ the integral I t = ξ r (ϕ ζ (t log ϕ z (ζ da(ζ π( ζ may be estimated as follows: I t = log ϕ ζ (t log ϕ z (ζ da(ζ c r < ϕ ζ(t <r π( ζ log 4 log ϕ z (ζ da(ζ c r π( ζ + log 4 da(ζ ξ r (ϕ ζ (t π( ζ = c r log 4 (z,/ / We thus obtain that 0 s log s ( s ds + log 4 =: r. w(z log ϕz (γ Iγ + N log ϕ z (γ j + j= N I γj j= r (N + + N log σ =: E r. Next, we need to estimate the convolution product F ɛ (z = ( log ϕ γ ( t da χ ɛ (z = ɛ log ϕ γ ϕ z (ζ da(ζ π( ζ. It is easy to verify that, with u = ϕ z (γ, (0,ɛ ϕ γ ϕ z (ζ = λϕ u (ζ, 7

8 where λ =. Thus, changing variables, we have F ɛ (z = t ɛ (u,ɛ log ζ da(ζ π( ζ. Then F ɛ (z t log ɛ = t ɛ log ζ da(ζ (u,ɛ ɛ π( ζ + t da(ζ log 4ɛ ɛ (u,ɛ π( ζ t log ɛ = 4t log ζ da(ζ ɛ (u,ɛ π( 4ɛ ζ + t ɛ log ɛ 4ɛ t log ɛ ɛ = 4t log ζ da(ζ ɛ (u,ɛ π( 4ɛ ζ + t log 4 ɛ + log ɛ tɛ ɛ. The second line follows from a change of variables and the fact that the hyperbolic area ɛ of a disk of radius ɛ is ɛ. Since u < ɛ, ɛ (u, ɛ, and so the absolute value of the integral in the last line is bounded by 4ɛ log ζ da(ζ, which is seen to converge. We thus have a constant C such that for sufficiently small ɛ, log φ γ ( da χ ɛ (z t log ɛ C, whenever φ γ (z < ɛ. Therefore, we have that for ϕ γ (z < ɛ, wda χɛ (z t log ɛ wda χɛ (z log ϕ γ ( da χ ɛ (z r + M =: C r. + log ϕγ ( da χ ɛ (z t log ɛ Since v = p (wda χ ɛ, the proof of (3 is complete. Finally, we wish to prove the left inequality of (. First, if ϕ z (γ < δ(γ/ for some γ Γ, then (3 gives a universal lower bound for v(z. On the other hand, if z is isolated away from Γ, then a look at the way w was estimated above shows that v(z is bounded from below by a negative number of even smaller modulus. This completes the proof. Proof of Lemma. Let us note that the right hand side of (4 is h(z p e ψ(z ν ɛ (z z da(z. We set U = h p e ψ. Then log U + ψ = p log h is a subharmonic function. Thus 0 log U + ψ = U U U U z + ψ U + ψ. U 8

9 It follows from the nonnegativity of U and the estimate ( on ψ above, that U U ψ p Uν ɛ + U δ. ividing by z and integrating yields δ U(z z da(z p U(z z ν ɛ(zda(z + ( z U(zdA(z. We would like to show that the second integral on the right is nonpositive. If U were compactly supported, we could use integration by parts to shift the Laplacian to z. Since ( z = and U 0, we would be done. So instead, one cuts things off as follows. Let χ t 0, 0 << t < be a function which is identically one on [0, t], supported compactly in [0,, with additional properties to be described shortly. Then ( z χ t ( z U(zdA(z = Recalling that on radial functions, ( ( z χ t ( z U(zdA(z. = 4 ( r + r r, one computes that ( ( z χ t ( z = χ t ( z z χ t( z + ( z χ t ( z. One then has ( z χ t ( z U(zdA(z = χ t ( z U(zdA + J t, where ( J t = π ( z z χ t( z U(z + ( z χ da(z t ( z π( z Now, by Lemma and the hypotheses on h, U is integrable with respect to the Poincaré area, and thus with respect to Euclidean area. Hence as t, χ t ( z U(zdA(z U(zdA(z 0. We claim that with a good choice of χ t, the integral J t 0 as t. To see this, simply choose χ t so that it has bounded invariant Laplacian, uniformly in t. (Examples of this 9

10 are easy enough to construct. For instance, let f be a smooth function on the nonnegative real line, which is supported on [0, /] and is identically on [0, /4]. Then just take ( z t χ t ( z := f +t. The boundedness of the invariant Laplacian is easy to check. Because χ t is radial, this will also give a bound on the gradient. One can then apply the dominated convergence theorem. This completes the proof. Proof of Lemma 3. Since φ is subharmonic with bounded invariant Laplacian, we may apply the Riesz decomposition theorem. Thus, if G is a fixed Green operator, one has φ = G( φ + f a for some harmonic function f a in a neighborhood of the closed disc (a, /. Let g a be a holomorphic function whose real part is f a, and set q a = g a g a (a. Then Now, let h a (z = h(ze p q a (z. Then φ φ(a R(q a K. h(z p e φ(z = h a (z p e φ(z+r(q a(z Ce φ(a h a (z p, where C = e K. The other inequality in the lemma follows similarly. Proof of Lemma 4. Let z (a, ɛ and write g(z g(a = z a g (udu, so that z g(z g(a + g (udu a g(a + z a sup u (a,ɛ g (u, which implies that } g(z p { g(a p p + z a p sup g (u p u (a,ɛ { g(a p p + C z a p sup ( u p g(ζ p da(ζ u (a,ɛ (u,ɛ { } p g(a p + C z a p ( a p g(ζ p da(ζ. (a,/ The second line follows from the standard estimate (see [3], for example g (u p C( u p g(ζ p da(ζ. 0 (u,ɛ }

11 Therefore, ɛ g(z p da(z (a,ɛ C ɛ g(a p (a,ɛ da(z + C ɛ ( a p (a,/ g(ζ p da(ζ a z p A(z. (a,ɛ Since a z p A(z ɛ p (a,ɛ (a,ɛ az p da(z ɛ p+ ( a p+ Cɛ p+ ( a p+, (0,ɛ az 4 p da(z the result follows. References. B. Berndtsson and J. Ortega-Cerdà, On interpolation and sampling in Hilbert spaces of analytic functions, J. Reine Angew. Math. 464 (995, H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman spaces, Springer-Verlag, Berlin, Luecking, Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives, Amer. J. Math. 07 (985, A. Schuster, On Seip s description of sampling sequences for Bergman spaces, Complex Variables Theory Appl. 4, K. Seip, Beurling type density theorems in the unit disk, Invent. Math. 3 (994, -39. epartment of Mathematics, San Francisco State University, San Francisco, CA 943 address: schuster@sfsu.edu epartment of Mathematics, University of Michigan, Ann Arbor, MI address: varolin@math.lsa.umich.edu