Random interpolating sequences in Dirichlet spaces
Nikolaos Chalmoukis, Andreas Hartmann, Karim Kellay, Brett Wick
RRANDOM INTERPOLATING SEQUENCES INDIRICHLET SPACES
N. CHALMOUKIS, A. HARTMANN, K. KELLAY, AND B. D. WICK
Abstract.
We discuss random interpolating sequences in weightedDirichlet spaces D α , 0 ≤ α ≤
1. Our results in particular implythat almost sure interpolating sequences for D α are exactly thealmost sure separated sequences when 0 ≤ α < / H = D ), and they are exactly the almost surezero sequences for D α when 1 / < α <
1. We show that this lastresult remains valid in the classical Dirichlet space D = D whenone considers a weaker notion of interpolation, so-called simple in-terpolation. As a by-product we improve a sufficient condition byRudowicz for random Carleson measures in Hardy spaces. Introduction
Understanding interpolating sequences is an important problem incomplex analysis in one and several variables. The characterization ofwhen a sequence of points is an interpolating sequence finds many appli-cations to different problems in signal theory, control theory, operatortheory, etc. In classical spaces like Hardy, Fock and Bergman spaces,interpolating sequences are now well understood objects, at least inone variable [16, 25, 27]. In Dirichlet spaces, it turns however outthat getting an exploitable description of such interpolating sequencesis a notoriously difficult problem related to capacities. Crucial workhas been undertaken in the 90s by Bishop and Marshall-Sundberg (seemore precise indications below). However, while easier checkable suffi-cient conditions were given by Seip in the meantime, no real progress
Mathematics Subject Classification.
Key words and phrases.
Interpolating sequences, separation, Carleson measure,random sequences.The research of the first author is supported by the fellowship INDAM-DP-COFUND-2015 ”INdAM Doctoral Programme in Mathematics and/or Applicationscofund by Marie Sklodowska-Curie Actions” a r X i v : . [ m a t h . C V ] J u l N. CHALMOUKIS, A. HARTMANN, K. KELLAY, AND B. D. WICK in the understanding of these sequences has been made since thoseworks. In such a situation, a probabilistic approach can lead to a newvision of these interpolating sequences. Note that besides the Hardyand Bergman spaces, the Dirichlet space, and its weighted companions,are beyond the most prominent spaces of analytic functions on the unitdisk. They appear naturally in problems on classical function theory,potential theory, as well as in operator theory when one investigatesfor instance weighted shifts.Here we consider random sequences of the following kind. Let Λ( ω ) = { λ n } with λ n = ρ n e iθ n ( ω ) where θ n ( ω ) is a sequence of independentrandom variables, all uniformly distributed on [0 , π ] (Steinhaus se-quence), and ρ n ∈ [0 ,
1) is a sequence of a priori fixed radii. Dependingon distribution conditions on ( ρ n ) as will be discussed below, we askabout the probability that Λ( ω ) is interpolating for Dirichlet spaces D α ,0 ≤ α ≤
1. Recall that the weighted Dirichlet space D α , 0 ≤ α ≤
1, isthe space of all analytic function f on the unit disc D such that (cid:107) f (cid:107) α := | f (0) | + (cid:90) D | f (cid:48) ( z ) | (1 − | z | ) − α d A ( z ) < ∞ , where d A ( z ) = d x d y/π stands for the normalized area measure on D (we refer to [15] for Dirichlet spaces). If α = 0, D is the Hardy space H , and the classical Dirichlet space D corresponds to α = 1.Recall that in a Hilbert space H of functions analytic in the unitdisk D equipped with a reproducing kernel k λ , i.e. f ( λ ) = (cid:104) f, k λ (cid:105) H for every λ ∈ D and f ∈ H (a so-called reproducing kernel Hilbertspace), a sequence Λ of distinct points in D is called (universal) inter-polating if { ( f ( λ ) / (cid:107) k λ (cid:107) H ) λ ∈ Λ : f ∈ H } = (cid:96) (for the difference betweeninterpolating and universal interpolating sequences see below). Con-cerning the deterministic case of interpolation in the classical Dirichletspace D , in unpublished work Bishop [7] and, independently, Marshall-Sundberg [17] characterized the interpolating sequences. The first pub-lished proof was given by Bøe [10] who provides a unifying scheme thatapplies to spaces that satisfy a certain property related to the so-calledPick property (see [2, 25]), and Dirichlet spaces fall in this category.For these spaces Λ is a (universal) interpolating sequence if and onlyif Λ is H -separated (i.e sup λ (cid:54) = λ ∗ ∈ Λ |(cid:104) k λ / (cid:107) k λ (cid:107) H , k λ ∗ / (cid:107) k λ ∗ (cid:107) H (cid:105)| < δ Λ < µ = (cid:80) λ ∈ Λ δ λ / (cid:107) k λ (cid:107) H is a Carleson measure for H (i.e, (cid:82) D | f | d µ ≤ C Λ (cid:107) f (cid:107) H ). Recently, Aleman, Hartz, McCarthy and Richter [1] haveshown that this characterization remains valid in arbitrary reproduc-ing kernel Hilbert spaces satisfying the complete Pick property. Ste-genga [26] characterized Carleson measures for Dirichlet spaces, but ADOM INTERPOLATIN IN DIRICHLET SPACES 3 this characterization is based on capacities which are notoriously dif-ficult to estimate for arbitrary unions of intervals. There are othercharacterizations of Carleson measures in Dirichlet spaces, see [3, 4],as well as [6, 15] and references therein, but which are not easily inter-preted geometrically for interpolating sequences. Finally, we mentionrelated work by Cohn [14] based on multipliers.In [25], Seip gave simple sufficient geometric conditions on a sequenceto be (universal) interpolating for the Dirichlet spaces see Theorems 3.3and 5.1, which, surprisingly, will allow us to obtain sharp results forrandom interpolating sequences in D α for α ∈ (0 , α (cid:54) = 1 /
2. For α = 1, the result by Bishop will give us the sharp result at least forsimple interpolating sequences in D .We also would like to observe that more generally, when the de-terministic frame does not give a full answer to a problem, or if thedeterministic conditions are not so easy to check, it is interesting tolook at the random situation. In particular, it is interesting to ask forconditions ensuring that a sequence picked at random is interpolatingalmost surely or not (i.e., which are in a sense “generic situations”?).In this context, it is also worth mentioning the huge existing literaturearound gaussian analytic functions which investigates the zero distri-bution in classes of such functions [22].The problems we would like to study in this paper are inspired byresults by Cochran [12] and Rudowicz [23] who considered randominterpolation in the Hardy space. Since interpolation in this space ischaracterized by separation (in the pseudohyperbolic metric) and bythe Carleson measure condition (note that the Hardy space was thepioneering space with a kernel satisfying the complete Pick property),those authors where interested in a 0-1 law for separation, see [12], anda condition for being almost surely a Carleson measure [23], which ledto a 0-1 law for interpolation. It is thus natural to discuss separation,Carleson measure type conditions and interpolation in Dirichlet spaces.Concerning separation in Dirichlet spaces D α , 0 < α <
1, this turnsout to be the same as in the Hardy spaces (see [25, p.22]), so that inthat case Cochran’s result perfectly characterizes the situation. Theseparation in the classical Dirichlet space, however, is much more del-icate than in the Hardy space. We establish here a 0-1-type law forseparation in D . While our proof of this fact is inspired by Cochran’sideas, our proof requires a careful adaptation to the metric in thatspace.Concerning Carleson measure type results in Dirichlet spaces, 0 <α <
1, we will first discuss the situation in the Hardy space and improve
N. CHALMOUKIS, A. HARTMANN, K. KELLAY, AND B. D. WICK
Rudowicz’ result simplifying his proof. Our new proof carries overto the Dirichlet situation and allows, together with a 1-box conditionby Seip (which requires itself separation), to discuss the results oninterpolation in D = D . For the spaces D α , < α < α : for 0 ≤ α < /
2, almost sure separation correspondsto almost sure interpolation, while for 1 / < α <
1, almost sure zerosequences correspond to almost sure interpolating sequences. Partialresults are given also for α = 1 / α = 1 of the scale of Dirichlet spaces underconsideration here follows more or less the same scheme as above, butrequires to work with a weaker notion of interpolation if one asks foran if and only if statement. More precisely, in the classical Dirichletspace D we are able to show that, almost surely, a sequence is simple interpolating (see definition below) if and only if it is almost surely azero sequence for D . For universal interpolation (see definition below)we still obtain an almost optimal result. We insist that the conditionfor almost sure separation in D is much weaker than the Carleson-measure type condition so that we cannot hope for optimality from theseparation (as was the case in the Hardy space). It can already be seenfrom our improvement of the sufficient condition for Carleson measuresin Hardy spaces, that the Carleson measure condition is not sufficientto obtain a 0-1 law in H .Since zero sequences are of some importance as we have just seen,another central ingredient of our discussion is a rather immediate adap-tion of Bogdan’s result on almost sure zero sequence in the Dirichletspace to the case of weighted Dirichlet spaces which we add for com-pleteness in an annex.As usual, the definition of interpolating sequences is based on thereproducing kernel of D α :(1.1) k z ( w ) = zw log 11 − zw if α = 1 , − ¯ zw ) − α , if 0 ≤ α < . Contrarily to the Hardy space situation, it turns out that in certainspaces (e.g. the Dirichlet space) there exist two notions of interpolationdepending on whether the restriction operator R Λ : H −→ (cid:96) , R Λ f =( f ( λ ) / (cid:107) k λ (cid:107) H ) λ ∈ Λ takes values in (cid:96) or not. ADOM INTERPOLATIN IN DIRICHLET SPACES 5
Definition.
Let ≤ α ≤ . A sequence Λ of distinct points in D saidto be • a (simple) interpolating sequence for D α if R Λ : f → ( f ( λ ) / (cid:107) k λ (cid:107) α ) λ ∈ Λ is onto (cid:96) , i.e. the interpolation problem f ( λ ) = a λ has a so-lution f ∈ D α for every sequence ( a λ ) with ( a λ / (cid:107) k λ (cid:107) α ) λ ∈ Λ ∈ (cid:96) (Λ) . • a universal interpolating sequence for D α if it is interpolatingand moreover R Λ is well defined from D α into (cid:96) . Sequences which are interpolating for the Dirichlet space but notuniversally interpolating were discovered by Bishop in [7], and theywere further analyzed in [5] and [13].Throughout this paper, even when not stated explicitly, when speak-ing about interpolation we mean universal interpolation. In the onlycase we work with simple interpolating sequences this will be stated ex-plicitly. In all other cases, in our theorems on interpolation, sufficientconditions always imply universal interpolation, while for necessaryconditions it suffices to only impose standard interpolation.We will now discuss in details the results we have obtained in thispaper.1.1.
Back to the Hardy space.
As pointed out in the introduction,before considering the situation in the Dirichlet space, it seems ap-propriate to re-examine the situation in the Hardy space. Recall thatCochran established a 0-1 law for (pseudohyperbolic) separation (seeTheorem 1.3 below) and Rudowicz showed that Cochran’s conditionfor separation implies almost surely the Carleson measure condition.This implies that interpolation is characterized by the condition ensur-ing almost sure separation. As it turns out the situation in Dirichletspaces is quite different. So, in order to get a better understanding westart stating an improvement of Rudowicz’ results on random Carlesonmeasures in the Hardy space which will help to better understand thecase of Dirichlet spaces.Recall that the measure d m Λ = (cid:80) λ ∈ Λ (1 − | λ | ) δ λ is called a Carlesonmeasure if there is a constant C such that for every interval I ⊂ T , m Λ ( S I ) ≤ C | I | , where S I = (cid:8) z = r e it ∈ D : e it ∈ I, − | I | ≤ r < (cid:9) is the usual Carleson window (see [16]). We will prove that a weakercondition than Rudowicz’ leads to Carleson measures almost surely in N. CHALMOUKIS, A. HARTMANN, K. KELLAY, AND B. D. WICK the Hardy space. We first need to introduce a notation: N n = { λ ∈ Λ( ω ) : 1 − − n ≤ | λ | ≤ − − n − } , n = 0 , , . . . . Theorem 1.1.
Let β > and suppose (cid:88) n ≥ − n N βn < + ∞ . Then the measure d m Λ is a Carleson measure almost surely in theHardy space. As a result, the Carleson measure condition alone is not sufficient togive a 0-1 law for interpolation in the Hardy space.Note that Rudowicz [23] showed that the above condition with β = 2is sufficient. There is still a gap remaining between the above condi-tion and the Blaschke condition which corresponds to β = 1. So apriori there might be sequences which are almost surely zero sequenceswithout giving rise to almost sure Carleson measures.1.2. Interpolation in Dirichlet spaces D α , < α < . For ourpaper, a key result for interpolation in Dirichlet spaces is [25, Theorem4, p.38] which shows that pseudohyperbolic separation (see definitionsbelow) and a certain 1-box condition are sufficient for (universal) inter-polation in D α , 0 < α <
1. As already mentioned in the introductionabove, separation in this case is as in the Hardy space, so that we es-sentially need to discuss the Carleson measure part of Seip’s theorem.1.2.1.
Random zero sequences in Dirichlet spaces.
A central rolein our interpolation results will be played by random zero sequences.Indeed, for an interpolating sequence in the Dirichlet space it is neces-sary to be a zero sequence (interpolation implies that there are func-tions vanishing on the whole sequence except for one point λ , andmultiplying this function by ( z − λ ) yields a function in the Dirichletspace vanishing on the whole sequence). We recall some results onrandom zero set in Dirichlet spaces. Carleson proved in [11] that when(1.2) (cid:88) λ ∈ Λ (cid:107) k λ (cid:107) − α < ∞ then the Blaschke product B associated to Λ belongs to D α , 0 < α < α = 0 this corresponds to the Blaschke condition for the Hardyspace). When α = 1, Shapiro–Shields proved in [24] that the condition(1.2) is sufficient for { λ } λ ∈ Λ to be a zero set for the classical Dirichletspace D , see also [25, Theorem 1]. Note that if 0 ≤ α < (cid:88) λ ∈ Λ (cid:107) k λ (cid:107) − α (cid:16) (cid:88) λ ∈ Λ (1 − | λ | ) − α (cid:16) (cid:88) n − (1 − α ) k N n ADOM INTERPOLATIN IN DIRICHLET SPACES 7 and if α = 1 then (cid:88) λ ∈ Λ (cid:107) k λ (cid:107) − (cid:16) (cid:88) λ ∈ Λ | log(1 − | λ | ) | − (cid:16) (cid:88) n n − N n . On the other hand, it was proved by Nagel–Shapiro–Shields in [20]that if { r n } ⊂ (0 ,
1) does not satisfy (1.2), then there is { θ n } suchthat { r n e iθ n } is not a zero set for D α . Bogdan [10, Theorem 2] gives acondition on the radii | λ n | for the sequence Λ( ω ) to be almost surelyzeros sequence for D :(1.3) P (Λ( ω ) is a zero set for D ) = (cid:26)
10 if and only if (cid:88) n n − N n (cid:26) < ∞ = ∞ . Bogdan’s arguments carry over to D α , α ∈ (0 , Theorem 1.2.
Let ≤ α < . Then (1.4) P (Λ( ω ) is a zero set for D α ) = (cid:26) if and only if (cid:88) n − (1 − α ) n N n (cid:26) < ∞ = ∞ . Interpolation in Dirichlet spaces D α . As pointed out ear-lier, interpolation is intimately related with separation conditions andCarleson measure type conditions. Recall that a sequence Λ is called(pseudohyperbolically) separated ifinf λ,λ ∗∈ Λ λ (cid:54) = λ ∗ ρ ( λ, λ ∗ ) = inf λ,λ ∗∈ Λ λ (cid:54) = λ ∗ | λ − λ ∗ || − λλ ∗ | ≥ δ Λ > . Since in Dirichlet spaces D α , 0 ≤ α <
1, the natural separation ( D α -separated sequence) is indeed pseudohyperbolic separation [25, p.22],we recall Cochran’s separation result on pseudohyperbolic separation. Theorem 1.3 (Cochran) . A sequence Λ( ω ) is almost surely (pseudo-hyperbolically) separated if and only if (1.5) (cid:88) n − n N n < + ∞ . We should pause here to make a crucial observation. We have alreadymentioned that interpolating sequences are necessarily zero-sequences.Also separation is another necessary condition for interpolation. Nowthe condition for zero sequences (1.4) depends on α while the separa-tion condition does not, and it follows that depending on α , it is onecondition or the other which is dominating. From (1.4) and (1.5) it N. CHALMOUKIS, A. HARTMANN, K. KELLAY, AND B. D. WICK is not difficult to see that this breakpoint is exactly at α = 1 / α = 1 /
2, (1.4) still implies (1.5)). This motivates already the necessaryconditions of our next result. For the sufficiency we will need to appealto Seip’s one-box condition [25, Theorem 5, p.38].
Theorem 1.4. (i)
Let < α < / , then P (cid:0) Λ( ω ) is interpolating for D α (cid:1) = (cid:26) if and only if (cid:88) n − n N n (cid:26) < ∞ = ∞ . (ii) Let α = 1 / . If there exists β > such that (cid:88) k − n N βn < ∞ then P (Λ( ω ) is interpolating for D / ) = 1 .Conversely, If P (Λ( ω ) is interpolating for D / ) = 1 then (cid:88) n − n/ N n < ∞ . (iii) Let / < α < . Then P (cid:0) Λ( ω ) is interpolating for D α (cid:1) = (cid:26) if and only if (cid:88) n − (1 − α ) n N n (cid:26) < ∞ = ∞ . Our techniques, based on Seip’s one-box condition, do not provide acomplete answer in the case α = 1 / Corollary 1.5.
The following statements hold: (1)
Let ≤ α < / . The sequence Λ( ω ) is almost surely interpo-lating for D α if and only if it is almost surely separated . (2) Let / < α < . The sequence Λ( ω ) is almost surely interpo-lating for D α if and only if it is almost surely a zero sequence. Interpolation in the classical Dirichlet space.
For the clas-sical Dirichlet space we will first establish a result on separation, andthen use again a 1-box condition by Seip as stated in [25, Theorem 5,p.39].
ADOM INTERPOLATIN IN DIRICHLET SPACES 9
Separation in the Dirichlet space.
In the case α = 1, theseparation is given in a different way. Let ρ D ( z, w ) = (cid:115) − | k w ( z ) | k z ( z ) k w ( w ) , z, w ∈ D . A sequence Λ is called D –separated ifinf λ,λ ∗∈ Λ λ (cid:54) = λ ∗ ρ D ( λ, λ ∗ ) > δ Λ > δ Λ <
1. This is equivalent to (see [25, p.23])(1.6) (1 − | λ | )(1 − | λ ∗ | ) | − λλ ∗ | ≤ (1 − | λ | ) δ , λ, λ ∗ ∈ Λ . For separation in the Dirichlet space D we obtain the following 0-1law. Theorem 1.6. P (Λ( ω ) is D –separated for ) = , if ∃ γ ∈ (1 / , such that (cid:88) n − γn N n < ∞ , , if ∀ γ ∈ (1 / , such that (cid:88) n − γn N n = ∞ . We observe that in both conditions we can replace the sum by asupremum (this amounts to replacing γ by a slightly bigger or smallervalue). The lower bound 1 / γ is not very important, since it is thebehavior close to the value 1 which counts.1.3.2. Interpolation in the Dirichlet space D . Recall that Bogdanshowed that Λ( ω ) is almost surely a zero sequence for D if and only if (cid:80) n n − N n < + ∞ . This motivates already the necessary part of the fol-lowing complete characterization of almost surely simple interpolatingsequences for D . Theorem 1.7. P (cid:0) Λ( ω ) is simple interpolating for D (cid:1) = (cid:26) if and only if (cid:88) n n − N n (cid:26) < ∞ = ∞ . We can reformulate the above result in the same spirit as Corollary1.5
Corollary 1.8.
A sequence is almost surely simple interpolating for D if and only if it is almost surely a zero sequence for D . Translating Theorem 1.1 to the Dirichlet space, we get the followingresult for universal interpolation which is optimal in a sense.
Theorem 1.9.
If there exist γ ∈ (0 , such that (1.7) (cid:88) n n − γ N n < ∞ , then P (cid:0) Λ( ω ) is an interpolating sequence for D (cid:1) = 1 . Organization of the paper.
This paper is organized as follows.In the next section we present the improved version of the Rudowiczresult concerning random Carleson measures in Hardy spaces which isthe guideline for the corresponding result in the Dirichlet space. In-deed, this largely clarifies and simplifies not only the situation in theHardy space, but also indicates the direction of investigation for theDirichlet space. In Section 3 we prove the sufficient condition for in-terpolation in D α , 0 < α <
1. Here, we will also prove Corollary 1.4.In the following section we show the 0-1 law on separation in the clas-sical Dirichlet space. This requires a subtle adaption of the Cochrandiscussion in the Hardy space to the much more intricate geometryin the Dirichlet space. Section 4 is devoted to the characterization ofseparated random interpolating sequences in the Dirichlet space. Theproofs of the results on interpolating sequences in the classical Dirichletspace are contained in Section 5. Actually, as in the Hardy space, thecore of the proof being probabilistic, we are able to get rid of analyticfunctions. In the final Section 6, we give some indications to the 0-1law on zero-sequences in weighted Dirichlet spaces based on Bogdan’sproof in the classical Dirichlet space.A word on notation. Suppose A and B are strictly positive expres-sions. We will write A (cid:46) B meaning that A ≤ cB for some positiveconstant c not depending on the parameters behind A and B . By A (cid:39) B we mean A (cid:46) B and B (cid:46) A . We further use the notation A ∼ B provided the quotient A/B → Carleson condition in the Hardy space
Before considering Carleson measure conditions in the Dirichlet space,we will discuss the situation in the Hardy space, in particular we willprove here Theorem 1.1.2.1.
Proof of Theorem 1.1.
We start introducing some notation.Let I n,k = { e πit : t ∈ [ k − n , ( k + 1)2 − n ) } n ∈ N , k = 0 , , . . . , n ADOM INTERPOLATIN IN DIRICHLET SPACES 11 be dyadic intervals and S n,k = S I n,k the associated Carleson window.In order to check the Carleson measure condition for a positive Borelmeasure µ on D it is clearly sufficient to check the Carleson measurecondition for windows S n,k : µ ( S n,k ) ≤ C | I n,k | (cid:39) C − n , for some fixed C > n ∈ N , k = 0 , . . . n . Given n , k and m ≥ k let X n,m,k be the number of points of Λ contained in S n,k ∩ A m (we stratify the Carleson window S n,k into a disjoint union of layers S n,k ∩ A m ). Since A m contains N m points and the (normalized) lengthof S n,k is 2 − n , we have X n,m,k ∼ B (2 − n , N m ) (binomial law). In orderto show that dm Λ is almost surely a Carleson measure we thus have toprove the existence of C such that m Λ ( S n,k ) = (cid:88) m ≥ n − m X n,m,k ≤ C − n almost surely, in other words we have to provesup n,k n (cid:88) m ≥ n − m X n,m,k ≤ C almost surely (in ω ). The estimate above had already been investigatedby Rudowicz [23]. Here we will proceed in a different way with respectto Rudowicz’ argument to obtain an improved version of his result andwhich allows to better understand the Dirichlet space situation. Proof of Theorem 1.1.
In view of our preliminary remarks, we need tolook at the random variable Y n,k = 2 n + ∞ (cid:88) m = n − m X n,m,k , where, as said above, X n,m,k ∼ B (2 − n , N m ). Hence, saying that Y n,k ≥ A means that there are Carleson windows for which the Carleson mea-sure constant is at least A . Also denote by G Y n,k the probability gen-erating function of the random variable Y n,k , i.e. G Y n,k ( s ) = E ( s Y n,k ).It is well known that for a random variable X which follows binomialdistribution with parameters p, N we have that G X ( s ) = (1 − p + ps ) N .By the hypothesis, for n sufficiently large, N n ≤ (1 − (cid:15) ) n , (cid:15) = 1 − /β . Introduce now two parameters A, s >
Markov’s inequality we have thatlog P ( Y n,k ≥ A ) = log P ( s Y n,k ≥ s A ) ≤ log (cid:0) s A G Y n,k ( s ) (cid:1) = (cid:88) m ≥ n N m log(1 − − n + 2 − n s n − m ) − A log( s ) ≤ − n (cid:88) m ≥ n N m ( s n − m − − A log( s )= (cid:88) m ≥ N n + m − ( n + m ) m ( s − m − − A log( s ) . At this point notice that x ( a /x − ≤ a, for all x ≥ , a >
0, whichtogether with the hypothesis on N n giveslog P ( Y n,k ≥ A ) ≤ (cid:88) m ≥ − (cid:15) ( n + m ) s − A log( s ) = 2 (cid:15) (cid:15) − s − (cid:15)n − A log( s ) . Now set s = 2 (cid:15)n , A = (cid:15) in the last inequality to getlog P ( Y n,k ≥ A ) ≤ (cid:15) (cid:15) − − (cid:15)n − n log(2) . Hence, P ( Y n,k ≥ (cid:15) ) ≤ C ( (cid:15) )2 − n .In view of an application of the Borel-Cantelli Lemma we compute (cid:88) n ≥ n (cid:88) k =1 P ( Y n,k ≥ A ) ≤ C ( (cid:15) ) (cid:88) n ≥ n × − n < ∞ . Hence, by the Borel-Cantelli Lemma, the event Y n,k ≥ A can happenfor at most a finite number of indices ( n, k ). In particular the Carlesonmeasure constant of dm Λ is almost surely at most A except for a finitenumber of Carleson windows. (cid:3) Proof of Theorem 1.4
We start with the following elementary lemma which is well knownin probability theory and which will be very useful in the proof belowof Theorem 3.2. (it is essentially approximation of the binomial law byPoisson law). We refer for instance to [8] for the material on probabilitytheory — essentially elementary — used in this paper.
ADOM INTERPOLATIN IN DIRICHLET SPACES 13
Lemma 3.1. If X is a binomial random variable with parameters p, N ,then for every s = 0 , , . . . , lim N →∞ pN → P ( X = s )( pN ) s = lim N →∞ pN → P ( X ≥ s )( pN ) s = 1 s ! . The proof of Theorem 1.4 is based on the following seemingly moregeneral result.
Theorem 3.2.
Let < α < . Suppose Λ( ω ) is almost surely separatedand almost surely a zero sequence. If there exists β > − α − α such that (3.1) (cid:88) n − n N βn < ∞ then P (cid:0) Λ( ω ) is interpolating D α (cid:1) = 1 . As we have already mentioned in Subsection 1.2.2, imposing simul-taneously Λ being a zero sequence and separated is rather artificial.One could be tempted to repeat the proof used above in the Hardyspace. However, it turns out that this does not give the best result, inparticular when α > /
2. Our proof of Theorem 3.2 will be based onthe following one-box condition introduced by Seip ([25, Theorem 4,p.38]), and on Tchebychevs inequality instead of Markovs. Note thatRudowicz already used Tchebychevs inequality, but his reasoning inthe Hardy space worked only for β ≥ Theorem 3.3 (Seip) . A separated sequence Λ in D is a universal in-terpolating sequence for D α , < α < , if there exist < κ < − α and C > such that for each arc I ⊂ T (cid:88) λ ∈ Λ ∩ S I (1 − | λ | ) κ ≤ C | I | κ . (3.2)Observe in particular that the above condition implies (cid:88) n − nκ N n < + ∞ , so that in particular (cid:80) n − n (1 − α ) N n < + ∞ , and thus that (3.2) impliesthat Λ is a zero sequence almost surely. We have to show that the conditions of Theorem 3.2 imply (3.2)almost surely for some fixed κ ∈ (0 , n,k κn (cid:88) m ≥ n − κm X n,m,k ≤ C almost surely. Again X n,m,k ∼ B (2 − n , N m ), where we can now assume N m ≤ m/β for some β > (3 − α ) / (2 − α ) (at least for sufficientlylarge m ).The key idea here is now to split this sum into two pieces. The firstpiece can be estimated with the aid of binomial law, and for the secondone we will use Tchebychevs inequality. For γ > Y n,k = 2 κn n (1+ γ ) (cid:88) m = n − κm X n,m,k (cid:124) (cid:123)(cid:122) (cid:125) Y n,k + 2 κn + ∞ (cid:88) m = n (1+ γ )+1 − κm X n,m,k (cid:124) (cid:123)(cid:122) (cid:125) R n,k . We also use the notation Z n,k := n (1+ γ ) (cid:88) m = n X n,m,k , so that now we get P ( Y n,k ≥ A/ ≤ P ( Z n,k ≥ A/ ∼ ( pN ) A/ ( A/ − n (cid:80) n (1+ γ ) m = n N m ) A/ ( A/ . It can be observed that we now have less points N m as in the Hardyspace situation (where N m ≤ m/β with β > /
2, while now we have N m ≤ m/β with β > (3 − α ) / (2 − α ) > / A we again conclude P ( Y n,k ≥ A/ ≤ − n . To bemore precise, with N m ≤ m/β in mind, we see that 2 − n (cid:80) n (1+ γ ) m = n N m (cid:46) − n (1 − (1+ γ ) /β ) so that we get the same condition on γ as before: (1 + γ ) /β < γ < β − . In that case, setting η = 1 − (1 + γ ) /β > P ( Y n,k ≥ A/ ≤ − η × A/ ≤ − n for sufficiently large A . ADOM INTERPOLATIN IN DIRICHLET SPACES 15
We estimate R n,k using Tchebychev’s inequality: P ( R n,k ≥ A/ (cid:46) Var( R n,k ) = 4 κn (cid:88) m ≥ n (1+ γ ) − κm V ar ( X n,m,k ) (cid:46) κn (cid:88) m ≥ n (1+ γ ) − κm × − n N m ≤ (2 κ − n (cid:88) m ≥ n (1+ γ ) − κm + m/β ≤ n (2 κ − − κ (1+ γ )+(1+ γ ) /β ) = 2 − n (1+2 κγ − (1+ γ ) /β ) . We will need η := 1+2 κγ − (1+ γ ) /β >
1, equivalently 2 κγ > (1+ γ ) /β ,or 2 κβγ > γ , i.e.(3.4) γ > κβ − . Now the two conditions on γ , (3.3) and (3.4), require12 κβ − < β − ⇐⇒ < κβ − β − κβ + 1 ⇐⇒ β (1 + 2 κ ) < κβ ⇐⇒ β > κ κ = 1 + 12 κ . (3.5)Observe that Seip’s result requires 0 < κ < − α and that the aboveexpression is decreasing in κ so that:1 + 2 κ κ > − α )2(1 − α ) = 3 − α − α . By assumption, we have β > (3 − α ) / (2 − α ) so that there exists κ ∈ (0 , − α ) with (3.5). And for that κ we can find γ satisfying (3.3)and (3.4). From here, the proof follows the same lines as in the Hardyspace discussed in the preceding section. This proves Theorem 3.2. (cid:3) Let us give the proof of Theorem 1.4.
Proof of Theorem 1.4. (i) Let 0 < α < / (cid:80) n − n N n < + ∞ .If (cid:80) n − n N n < + ∞ , then Λ is almost surely separated. Moreover,we can pick β = 2 in (3.1). Hence3 − α − α = 1 + 12(1 − α ) < β. Also, in this case (cid:80) n − (1 − α ) N n < + ∞ , which gives (1.4) and hence,by Theorem 1.2 that Λ is a zero sequence almost surely. We concludefrom Theorem 3.2 that Λ is almost surely interpolating.(iii) Consider the case 1 / < α < If Λ is interpolating almost surely, then it is a zero sequence almostsurely, which implies (cid:80) n − (1 − α ) n N n < + ∞ by Theorem 1.2.Suppose (cid:80) n − (1 − α ) n N n < + ∞ . Then Λ is a zero sequence almostsurely by Theorem 1.2. Also it is clear that the condition implies that (cid:80) n − n N n < + ∞ , which further implies that the sequence is almostsurely separated. Pick β = 1 + 12(1 − α ) + (cid:15) > − α ) = 3 − α − α , for some (cid:15) > N n ≤ (1 − α ) n (at least for sufficiently large n ). Hence N β − n ≤ (1 − α ) × ( β − n = 2 (1 − α ) × ( − α ) + (cid:15) ) n = 2 (1 / − α ) (cid:15) ) n . Then 2 − αn × N β − n = 2 ( − α +1 / − α ) (cid:15) ) n . Now since α > /
2, there exists (cid:15) > − α +1 / − α ) (cid:15) < (cid:88) n − n N βk = (cid:88) n − (1 − α ) n N n × − αn N β − n < ∞ . We conclude from Theorem 3.2 that Λ is interpolating almost surely.If (cid:80) n − (1 − α ) n N n = + ∞ . Then Λ is not a zero sequence almostsurely by Theorem 1.2, and hence it is almost surely not interpolating.Conversely, we need to check that if Λ is almost surely not inter-polating, then we have (cid:80) n − (1 − α ) n N n = + ∞ . By contraposition, ifthe sum is bounded, we have to show that the probability that Λ isinterpolating is stricly positive. But we already know that in that casethis probability is 1 > α = 1 /
2: In this case we only have a sufficient condition whichfollows by direct application of Theorems 1.2 and 3.2. (cid:3) Separated random sequences for the Dirichlet space
We will now prove the separation result in D . Proof of Theorem 1.6. Separation with probability 0 . Assume that forall γ ∈ (1 / ,
1) we have sup k − γk N k = ∞ . As it turns out, underthe condition of the Theorem, separation already fails in dyadic annuli(without taking into account radial Dirichlet separation).Assume now that γ l → l → ∞ and sup k − γ l k N k = ∞ for every l . For each k = 1 , . . . , let I k = [1 − − k +1 , − − k ). DefineΩ ( l ) k = { ω : ∃ ( i, j ) , i (cid:54) = j with ρ i , ρ j ∈ I k and | θ i ( ω ) − θ j ( ω ) | ≤ π − γ l k } . ADOM INTERPOLATIN IN DIRICHLET SPACES 17
In view of (1.6), if ω ∈ Ω ( l ) k , this means that in the dyadic annulus A k there are at least two points close in the Dirichlet metric. To bemore precise, if ω ∈ Ω ( l ) k , then there is a pair of distinct point λ i ( ω )and λ j ( ω ) such that | λ i | , | λ j | ∈ I k and | arg λ i ( ω ) − arg λ j ( ω ) | ≤ π − γ l k .Hence (1 − | λ i | )(1 − | λ j | ) | − λ i λ j | ≥ c − k − k + π − γ l k , where the constant c is an absolute constant. Hence(1 − | λ i | )(1 − | λ j | ) | − λ i λ j | ≥ c
11 + π k (1 − γ l ) ≥ c (cid:48) − k (1 − γ l ) ≥ c (cid:48)(cid:48) (1 − | λ i | ) − γ l . Absorbing c (cid:48)(cid:48) into a suitable change of the power δ l := 1 − γ l into δ (cid:48) l (which can be taken by choosing for instance 2 δ l > δ (cid:48) l > δ l provided k is large enough), then by (1.6) ρ D ( λ i ( ω ) , λ j ( ω )) ≤ δ (cid:48) l . Our aim is thus to show that for every l ∈ N , we can find almost surely λ i ( ω ) (cid:54) = λ j ( ω ) such that ρ D ( λ i ( ω ) , λ j ( ω )) ≤ δ (cid:48) l , i.e. P (Ω ( l ) k ) = 1. (Notethat δ (cid:48) l → l → + ∞ .)Let us define a set E := { j : 2 − γ l j − N j ≤ } . Observe that when k / ∈ E , then at least two points are closer than π − γ l k (this is completelydeterministic), so that in that case P (Ω ( l ) k ) = 1. Hence if E (cid:36) N , thenwe are done.Consider now the case E = N , and let k ∈ E = N . We will use theLemma on the probability of an uncrowded road [12, p. 740], whichstates P (Ω ( l ) k ) = 1 − (1 − N k − γ l k − ) N k − (since E = N this is well defined).We can assume that N k ≥ (cid:80) k : N k < − γ l k N k < ∞ ).In particular N k / ≤ N k ( N k − ≤ N k . Since log(1 − x ) ≤ − x , we get (cid:88) k : N k ≥ ( N k −
1) log(1 − N k − γ l k − ) ≤ − (cid:88) k : M k ≥ ( N k − N k − γ l k − ≤ − (cid:88) k : N k ≥ N k − γ l k − = −∞ by assumption. Hence, taking exponentials in the previous estimate, (cid:89) k ∈ E,N k ≥ (1 − N k − γ l k − ) N k − = 0 , which implies, by results on convergence on infinite products, that (cid:88) k P (Ω ( l ) k ) = ∞ . Since the events Ω ( l ) k are independent, by the Borel–Cantelli Lemma, P (lim sup Ω ( l ) k ) = 1 , wherelim sup Ω ( l ) k = (cid:92) n ≥ (cid:91) k ≥ n Ω ( l ) k = { ω : ω ∈ Ω ( l ) k for infinitely many k } . In particular, since the probability of being in infinitely many Ω ( l ) k isone, there is at least one Ω ( l ) k which happens with probability one. Sothat again P (Ω ( l ) k ) = 1.As a result, the probability that the sequence is δ (cid:48) l -separated in theDirichlet metric is zero for every l . Since δ (cid:48) l → l → + ∞ , wededuce that P ( ω : { λ ( ω ) } is separated for D ) = 0 . Separation with probability 1.
Now we assume that (cid:80) k − γk N k < + ∞ for some γ ∈ (1 / , η > α ∈ (0 , λ ∈ Λ, sothat for some k , λ ∈ A k . Consider T η,αλ = { z = r e it : (1 − | λ | ) η ≤ − r ≤ (1 − | λ | ) η , | θ − t | ≤ (1 − r ) α } . Figure 1 represents the situation.Our aim is to prove that under the condition (cid:80) k − γk N k < + ∞ ,there exists η > α ∈ (0 ,
1) such that T η,αλ does not containany other point of Λ except λ , and this is true for every λ ∈ Λ withprobability one. For this we need to estimate P ( T η,αλ ∩ Λ = { λ } ) . Let us cover T η,αλ = ηk (cid:91) j = k/η ( T η,αλ ∩ A j ) , and we need that for every j ∈ [ k/η, ηk ] \ { k } , ( T η,αλ ∩ A j ) ∩ Λ = ∅ and( T η,αλ ∩ A k ) ∩ Λ = { λ } . Note that X j = T η,αλ ∩ A j ∩ Λ) ∼ B ( N j , − αj ), j (cid:54) = k , and X k ∼ B ( N k − , − αk ) (since we do not count λ in the latter ADOM INTERPOLATIN IN DIRICHLET SPACES 19
Figure 1.
Dirichlet neighborhood.case). Hence, since the arguments of the points are independent, wehave P ( T η,αλ ∩ Λ = { λ } ) = P (cid:16)(cid:0) ηk (cid:92) j = k/η,j (cid:54) = k ( X j = 0) (cid:1) ∩ ( X k = 1) (cid:17) = j = ηk (cid:89) j = k/η,j (cid:54) = k (cid:16) P ( X j = 0) (cid:17) × P ( X k = 1) . From the binomial law we have P ( X j = 0) = (1 − − αj ) N j , for j ∈{ k/η, ηk }\{ k } . Also, assuming 0 < γ < α <
1, we have N j − αj = o ( j ),so that P ( X j = 0) = (1 − − αj ) N j ∼ − N j − αj . Moreover P ( X k = 1) = N k − αk (1 − − αk ) N k − ∼ N k − αk . Hence we get P ( T η,αλ ∩ Λ = { λ } ) ∼ exp (cid:16) j = ηk (cid:88) j = k/η,j (cid:54) = k ln( P ( X j = 0) (cid:17) × N k − αk ∼ (1 − j = ηk (cid:88) j = k/η,j (cid:54) = k N j − αj ) × N k − αk . Again we use γ < α < (cid:80) ηkj = k/η,j (cid:54) = k N j − αj is convergent and goes to zero when k → ∞ . This shows in particular that the fact of considering the event of having points in neighboringannuli of A k containing λ can be neglected. Hence P ( T η,αλ ∩ Λ = { λ } ) ∼ N k − αk . We now sum over all λ ∈ Λ by summing over all dyadic annuli A k andthe N k points contained in each annuli: (cid:88) λ ∈ Λ P ( T η,αλ ∩ Λ = { λ } ) ∼ (cid:88) k ∈ N N k × N k − αk = (cid:88) k ∈ N N k − αk . For α > γ , this sum converges by assumption. Using the Borel-Cantellilemma we deduce that T η,αλ ∩ Λ = { λ } for all but finitely many λ withprobability one. Obviously these finitely many neighborhoods T η,αλ contain finitely many points between which a lower Dirichlet distanceexists. This achieves the proof of the separation. (cid:3) It should be observed that the above proof only involves α but not η ,so that it is the separation in the annuli which dominates the situation.5. Proof of Theorem 1.9 and Theorem 1.7
In order to prove Theorems 1.9 and 1.7 we will use Seip’s one boxcondition [25, Theorem 5, p.39] as well as the corollary below whichfollows from a theorem of Bishop ( [7, Theorem 1.3]).
Theorem 5.1 (Seip) . A D -separated sequence Λ in D is a universalinterpolating sequence for D if there exist < γ < and C > suchthat for each arc I ⊂ T (cid:88) λ ∈ Λ ∩ S I (cid:18) log e1 − | λ | (cid:19) − γ ≤ C (cid:18) log e | I | (cid:19) − γ . (5.1) Theorem 5.2 (Bishop) . A D -separated sequence Λ in D is a (simple)interpolating sequence for D if (1) (cid:80) λ ∈ Λ (cid:16) log e1 −| λ | (cid:17) − < + ∞ (zero sequence), and (2) ∃ η ∈ (0 , such that for all ν ∈ Λ , we have (cid:88) λ ∈ Λ ∩ S ( I ην ) (cid:18) log e1 − | λ | (cid:19) − ≤ (cid:18) log e1 − | ν | (cid:19) − where I ην is the interval centered at ν/ | ν | of length (1 − | ν | ) η . Corollary 5.3. A D -separated sequence Λ in D is a (simple) inter-polating sequence for D if there exist C > such that for each arc ADOM INTERPOLATIN IN DIRICHLET SPACES 21 I ⊂ T (cid:88) λ ∈ Λ ∩ S I (cid:18) log e1 − | λ | (cid:19) − ≤ C (cid:18) log e | I | (cid:19) − . (5.2)In order to deduce the corollary from Bishop’s theorem observe thatthe first hypothesis of the Theorem follows immediately by choosing I = T , and the second one from the observation that log(e / | I ην | ) (cid:39) log(e / (1 − | ν | )).Observe that both conditions (5.1) and (5.2) imply in particular (cid:80) n n − N n < ∞ which by Bogdan’s result implies that Λ is a zero se-quence almost surely. Proof of Theorem 1.9 and Theorem 1.7.
We proceed in a similar man-ner as in the proof of Theorem 1.1. Using the usual dyadic discretiza-tion the conditions in Theorem 5.1 and Theorem 5.3 translate to (cid:88) m ≥ n m γ X n,m,k ≤ C n γ almost surely , where 0 < γ < γ = 1 to Bishop’s.As usual we have to estimate the tail of the random variables Y n,k = (cid:88) m ≥ n (cid:16) nm (cid:17) γ X n,m,k . To do that, introduce again two positive parameters s, A . Using theformula for the generating function of a binomially distributed randomvariable and Markov’s inequality we can estimate as followslog P ( Y n,k ≥ A ) = log P ( s Y n,k > s A ) ≤ (cid:88) m ≥ n N m log(1 − − n + 2 − n s ( nm ) γ ) − A log( s ) ≤ − n (cid:88) m ≥ n N m m γ (cid:0) mn (cid:1) γ ( s ( nm ) γ − n γ − A log( s ) ≤ n γ − n s (cid:88) m ≥ n N m m γ − A log( s ) . Setting s = 2 n/ and A = 4 the above calculation gives P ( Y n,k > (cid:46) C − n . Again, an application of the Borel-Cantelli Lemma concludes the proof. (cid:3) Annex : Proof of Theorem 1.2
Carleson proved in [11, Theorem 2.2] that, for 0 < α <
1, if (cid:88) λ ∈ Λ (1 − | λ | ) − α < ∞ then the Blaschke product B associated to Λ belongs to D α . So thesufficiency part of Theorem 1.2 follows immediately from this result(and is moreover deterministically true).For the proof of the converse we will need the following two lemmas.The first one is a version of the Borel-Cantelli Lemma [8, Theorem 6.3]. Lemma 6.1. If { A n } is a sequence of measurable subsets in a proba-bility space ( X, P ) such that (cid:80) P ( A n ) = ∞ and (6.1) lim inf n →∞ (cid:80) j,k ≤ n P ( A j ∩ A k )[ (cid:80) k ≤ n P ( A k )] ≤ , then P (lim sup n →∞ A n ) = 1 . The second Lemma is due to Nagel, Rudin and Shapiro [20, 21] whodiscussed tangential approach regions of functions in D α . Lemma 6.2.
Let f ∈ D α , < α < . Then, for a.e. ζ ∈ T , we have f ( z ) → f ∗ ( ζ ) as z → ζ in each region | z − ζ | < κ (1 − | z | ) − α , ( κ > . Proof of Theorem 1.2.
In view of our preliminary observations, we areessentially interested in the converse implication. So suppose (cid:80) n − (1 − α ) n N n =+ ∞ or equivalently(6.2) (cid:88) n (1 − ρ n ) − α = + ∞ . We have to show that Λ is not a zero sequence almost surely. For this,introduce the intervals I (cid:96) = (e − i (1 − ρ (cid:96) ) − α , e i (1 − ρ (cid:96) ) − α ) and let F (cid:96) = e iθ (cid:96) I (cid:96) .Denoting by m normalized Lebesgue measure on T , observe that m ( F (cid:96) ) = m ( I (cid:96) ) = (1 − ρ (cid:96) ) − α . We have for every ϕ ∈ F (cid:96) , λ (cid:96) ∈ Ω κ,ϕ = { z ∈ D : | z − e iϕ | <κ (1 − | z | ) − α } . By Lemma 6.2 it suffices to prove that | lim sup (cid:96) F (cid:96) | > ADOM INTERPOLATIN IN DIRICHLET SPACES 23 a.s. (the latter condition means that there is a set of strictly posi-tive measure on T to which Λ accumulates in Dirichlet tangential ap-proach regions according to Lemma 6.2, which is of course not pos-sible for a zero sequence). Let E denote the expectation with re-spect to the Steinhaus sequence ( θ n ). By Fubini’s theorem we have E [ m ( F j ∩ F k )] = m ( I j ) m ( I k ), j (cid:54) = k , (the expected size of intersec-tion of two intervals only depends on the product of the length of bothintervals). By Fatou’s Lemma and (6.2) E (cid:104) lim inf n →∞ (cid:80) j,k ≤ n m ( F j ∩ F k )[ (cid:80) k ≤ n m ( F k )] (cid:105) ≤ lim inf n →∞ E (cid:104) (cid:80) j,k ≤ n m ( F j ∩ F k )[ (cid:80) k ≤ n m ( F k )] (cid:105) = lim inf n →∞ (cid:80) j,k ≤ n E [ m ( F j ∩ F k )][ (cid:80) k ≤ n m ( F k )] = lim inf n →∞ (cid:80) j,k ≤ n,j (cid:54) = k m ( I j ) m ( I k ) + (cid:80) k ≤ n m ( I k )[ (cid:80) k ≤ n m ( I k )] = lim inf n →∞ (cid:32) (cid:80) k ≤ n m ( I k )(1 − m ( I k ))[ (cid:80) k ≤ n m ( I k )] (cid:33) . Now, since 1 − m ( I k ) →
1, and by (6.2), keeping in mind that m ( I k ) =(1 − ρ k ) − α , we havelim n →∞ (cid:80) k ≤ n m ( I k )(1 − m ( I k ))[ (cid:80) k ≤ n m ( I k )] = 0 . This implies that (6.1) holds on a set B of positive probability andhence, by the zero-one law, on a set of probability one. From Lemma6.1 we conclude P (lim sup n →∞ F n ) = 1 a.s., which is what we had toshow. (cid:3) References [1]
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ADOM INTERPOLATIN IN DIRICHLET SPACES 25
Dipartimento di Matematica, Universit`a di Bologna, 40126, Bologna,Italy
E-mail address : [email protected] Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400,Talence, France
E-mail address : [email protected]
Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400,Talence, France
E-mail address : [email protected]
Department of Mathematics & Statistics, Washington University –St. Louis, One Brookings Drive, St. Louis, MO USA 63130–4899
E-mail address ::