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Let D be the unit disc in the complex plane and S be a sequence of points in D. Denote H∞ the set of bounded holomorphic functions on D. We shall say that S is a H∞ interpolating sequence if ∞ ∞ ∀λ ∈??? (S), ∃f ∈ H :: ∀a ∈ S, f(a)= λa. These interpolating sequences where characterized by L. Carleson [Carleson, 1958] and by H. Shapiro and A. Shieds [Shapiro and Shields, 1961] for the Hardy spaces Hp(D) by the same condition: a − b ∀b ∈ S, ≥ δ > 0. 1 − ab¯ a∈S, a6=b Y B Cn Dn Cn In several variables for the unit ball Ω = ⊂ or for the unit polydisc Ω = ⊂ , this characterisation is still an open question, even for the H2(Ω). The aim is to study interpolating sequences in the Hardy spaces in the ball and in the polydisc in Cn. We shall follow here the vectorial point of view, as in [Amar et al., 2019], [Amar, 1977], [Amar, 2018] or in [Agler and MacCarthy, 2002], instead of the functional one, as done classically in [Shapiro and Shields, 1961]. ′ 1 1 ′ For p ≥ 1 we note p its conjugate p′ + p =1. For a B, call B its .

Definition 1.1. Let p> 1 and S := {ea}a∈S a sequence of vectors of one in the Banach space B′. Then S is p-Carleson if it is p′-hilbertian i.e.: ′ ′ p ′ (Hp ) ∃C > 0, ∀λ ∈??? (S), a∈S λaea B′ ≤ Ckλk???p (S). S is p-interpolating if it is p′-besselian i.e.: ′ ′ p P ′ (Bp ) ∃C > 0, ∀λ ∈??? (S), a∈S λaea B′ ≥ ckλk???p (S). It is a p′-Riesz sequence if S is p′-hilbertian and p′-besselian.

P Recall a result of Babenko [Babenko, 1947]: there exists {ea}a∈S a basis of unit vectors in the Hilbert space H which is 2-hilbertian but {ea}a∈S is not a Riesz basis. The same way there exists {ea}a∈S a basis of unit vectors in the Hilbert space H which is 2- besselian but {ea}a∈S is not a Riesz basis. To have a elementary proof of Babenko results, together with an extension of Babenko examples to ???p spaces, see the work by I. Chalendar, B. Chevreau and E. A. [Amar et al., 2019]. We shall get as a by-product of the results here that we have Hardy spaces of the ball or of the polydisc in C2 which are p-hilbertian but not p-besselian, giving new Babenko examples.

We shall apply this definition to the case of B = Hp(Ω), the Hardy space of Ω, with Ω the ball n p′ p ∗ or the polydisc in C , and with ea = ka,p′ ∈ H =(H ) , the normalised reproducing kernel of the point a ∈ Ω. The precise definitions are in the next section. p 0 0 Define the restriction operator Rp : H (Ω) →??? (S), with ??? (S) the set of complex valued sequences, by: p 0 ∀f ∈ H (Ω), Rpf := {hf,ka,p′ i}a∈S ∈??? (S). p p As we shall see in Lemma 4.1, we have that S is H (Ω) interpolating iff we have that Rp(H (Ω)) ⊃ p p p ??? (S), hence, for any sequence λ ∈??? (S), there is a function f ∈ H such that ∀a ∈ S, hf,ka,p′ i = λa. So the vectorial definition of p-interpolating sequences is equivalent to the functional usual one. Now we can set

2 Definition 1.2. Let p ≥ 1 and S ⊂ Ω be a sequence of points in Ω. We say that the sequence S has the linear extension property, LEP, if there is a bounded linear operator E : ℓp(S) → Hp such p p that for any λ ∈ ℓ (S), E(λ) interpolates the sequence λ in H (Ω). I.e. ∀a ∈ S, hE(λ),ka,p′ i = λa. p p p Its range ES is the subspace E(??? (S)) of H . P. Beurling [Beurling and Carleson, 1962] proved that the H∞(D) interpolating sequences in the unit disc have always the LEP. Using a very nice method due to S. Drury [Drury, 1970], A. Bernard [Bernard, 1971] proved the same for interpolating sequences in uniform algebras, hence for H∞(Dn) and H∞(B). For Hp(D) interpolating sequences, 1 ≤ p< ∞, I proved in [Amar, 1983] that they have the LEP, by use of ∂¯ methods. In [Schuster and Seip, 1998] this result is reproved by a different method. This question was open for a while in the several variables case.

Using the vectorial point of view, we shall prove: Theorem 1.3. Let S be a sequence of points in Ω. Suppose that S is Hp(Ω) interpolating with p ≥ 2. Then S has the LEP. We also study strictly Hp interpolating sequences and the case of interpolating sequences for weighted Bergman spaces.

2 Definitions, notation.

Recall the definition of Hardy spaces in the polydisc. Definition 2.1. The Hardy space Hp(Dn) is the set of holomorphic functions f in Dn such that, iθ iθ1 iθn n with e := e ×···×e and dθ := dθ1 ··· dθn the Lebesgue on T : p iθ p kfkHp := sup f(re ) dθ < ∞. Tn r<1 Z ∞ n n The Hardy space H ( D ) is the space of holomorphic and bounded functions in D equipped with the sup norm. p n n The space H (D ) possesses a reproducing kernel for any a ∈ D , ka(z): 1 1 ka(z)= ×···× . (1 − a¯1z1) (1 − a¯nzn) We know that a function f in Hp has almost everywhere boundary values f ∗ in Lp(Tn). And we have the reproducing property: p n ∗ iθ iθ ∀f ∈ H (D ), f(a) := hf,kai = f (e )k¯a(e )dθ1 ··· dθn, Tn where h·, ·i is the scalar product of theZ Hilbert space H2(Dn). We shall use the notation: ((1 − az¯ )) := (1 − a¯1z1)×···×(1 − a¯nzn) hence: 2 2 2 ((1 −|a| )) := (1 −|a1| )×···×(1 −|an| ). Then the normalized reproducing kernel in Hp(Dn) is, for p ≥ 1: ′ ((1 −|a|2))1/p k (z)= . a,p ((1 − az¯ )) The same way for the unit ball B of Cn, we have:

3 Definition 2.2. The Hardy space Hp(B) is the set of holomorphic functions f in B such that, with z = rζ ∈ B, r ∈ (0, 1), ζ ∈ ∂B, and dσ the Lebesgue measure on ∂B: p p kfkHp := sup |f(rζ)| dσ(ζ) < ∞. B r<1 Z∂ The Hardy space H∞(B) is the space of holomorphic and bounded functions in B equipped with the sup norm. p n The space H (B), B ⊂ C possesses a reproducing kernel for any a ∈ B, ka(z) and a normalised one ka,p(z): 1 ka(z) 2 −n/p′ ka(z)= n , ka,p(z)= , with kkakp ≃ (1 −|a| ) , (1 − a¯ · z) kkakp n where we use the notation a¯ · z := j=1 a¯jzj. Again we know that a function f in Hp has almost everywhere boundary values f ∗ in Lp(∂B). And we have P p ∀f ∈ H (B), f(a) := hf,kai = f(ζ)k¯a(ζ)dζ, ∂B where h·, ·i is the scalar product of theZ Hilbert space H2(B).

B Dn −2 Let Ω be either the ball or the polydisc . We shall use the notation: ∀a ∈ Ω, χa := kkakH2 . n 2 2 n Hence in D , χa = ((1 −|a| )) and in B, χa = (1 −|a| ) . Definition 2.3. Let S be a sequence of points in Ω. We say that S is a dual bounded sequence p p in H (Ω) if there is a sequence {ρa}a∈S ⊂ H (Ω) such that, with kb,p′ the normalised reproducing p′ kernel in H (Ω) for the point b ∈ Ω and δa,b := 0 for a =6 b and δa,b := 1 for a = b: ′ ∃ C > 0, ∀a, b ∈ S, hρa,kb,p i = δa,b and kρakHp(Ω) ≤ C. In Section 6 we shall consider the Grammian associated to a sequence of points S in Ω. This is the infinite matrix G given by Ga,b := hka,p′ , kb,pi, p where the ka,p are the normalised reproducing kernels for H (Ω). ′ We consider G as an operator on ???p (S) as ′ p ′ ∀µ ∈??? (S), Gµ := {(Gµ)b}b∈S, (Gµ)b := a∈S µaGa,b = a∈S µaka,p ,kb,p . In order to state some results, we shall need: P P Definition 2.4. Let S be a sequence of points in Ω. We shall say that S is strictly Hp interpolating ′ if the Grammian G is bounded below on ???p (S).

In Section 7 we study the case of the weighted Bergman spaces of the ball with the following definition.

B Cn N p B Definition 2.5. Let f be a in n ⊂ and k ∈ ; we say that f ∈ Ak( ) if p p 2 k kfkk,p := |f(z)| (1 −|z| ) dm(z) < ∞. B Z Where dm is the Lebesgue measure in Cn.

4 3 The results.

Recall that Ω denote either the ball B or the polydisc Dn.

Theorem 3.1. Let S be a sequence of points in Ω. Suppose that S is Hp(Ω) interpolating with p ≥ 2. Then S has the LEP.

Then we get:

Theorem 3.2. Let S be a sequence of points in Ω. Suppose that S is Carleson and S is strictly Hp interpolating. p p Then S is H interpolating and has the LEP with range ES := Span{ka,p, a ∈ S}. And its converse:

Theorem 3.3. Let S be a sequence of points in Ω. Suppose that S is Hp interpolating and has the p LEP with range ES := Span{ka,p, a ∈ S} for a p> 2. Then S is strictly Hp interpolating.

In Section 7 we study the case of the weighted Bergman spaces of the ball. By applying a "Subordination Lemma" in [Amar, 1978] Section 2, p.716 or, for a general form of it [Amar, 2015], we get the same results as for the Hardy spaces.

B Cn p B Theorem 3.4. Let S be a sequence of points in ⊂ . Suppose that S is Ak-interpolating in p for a p ≥ 2. then S has the LEP with range ES := Span{ka,p, a ∈ S}. B p Theorem 3.5. Let S be a sequence of points in . Suppose that S is Carleson for Ak, and S is p strictly Ak interpolating. p p Then S is Ak interpolating and has the LEP with range ES := Span{ka,p, a ∈ S}. And its converse: B p Theorem 3.6. Let S be a sequence of points in . Suppose that S is Ak interpolating and has the p LEP with range ES := Span{ka,p, a ∈ S} for a p> 2. p Then S is strictly Ak interpolating.

2 Finally we prove Babenko examples made of holomorphic functions. With Ω= B2 or Ω= D we get:

Theorem 3.7. There are sequences of reproducing kernels in Hp(Ω) which are p-hilbertian but not p-Riesz sequences.

4 Equivalence with the functional definitions.

Lemma 4.1. Let S be a sequence of points in Ω. Then S is Hp(Ω) interpolating iff: p p RpH ⊃??? (S).

5 Proof. p p • First it is well known that RpH ⊃??? (S) implies:

p ∃c> 0, ∀f ∈ H (Ω), kRpfk???p(S) ≥ ckfkHp(Ω). (4.1)

To see this consider the subspace: p p LK := {λ ∈??? (S), ∃f ∈ H , kfkp ≤ Kkλkp, Rpf = λ}. p p Clearly LK is a closed subspace of ??? (S) and we have K∈N LK =??? (S). Hence by Baire’s Theorem one of the LK contains a non void ball, which gives (4.1). S p p • Second suppose that (4.1) is true and RpH ⊃??? (S). Then:

p ′ ′ ∀f ∈ H , kfkp ≤ 1, µaka,p ≥ f, µaka,p . ′ * + Xa∈S p Xa∈S But

′ 1/p f, µaka,p = µ¯af(a)χa . * a∈S + a∈S X p X ′ p ′ ′ We take f ∈ H such that f(a)χa,p = λa ∈??? (S) with a∈S µ¯af(a)χa,p = kµk???p . This is p p possible because RpH ⊃??? (S) with kfk ≤ Ckλk p . So we get, with kλk p =1, p ??? P ??? 1 µ k ′ ≥ kµk p′ . a a,p C ??? a∈S ′ X p ′ p Which means that {ka,p }a∈S is p-besselian, hence S is H (Ω) interpolating.

p ′ ′ • Third suppose that S is H (Ω) interpolating, i.e. a∈S µaka,p p′ & kµk???p . ′ p We truncate S to its first N terms, call it SN . Then we have that {ka,p }a∈SN has a H dual sequence p P {ρa}a∈SN ⊂ ES := Span{ka,p, a ∈ SN }, i.e. p ∀a ∈ SN , ∃ρa ∈ H :: hρa,kb,p′ i = δab.

p′ p′ ′ p Now we have that the dual space of ESN := Span{ka,p , a ∈ SN } is the quotient space H /Ann(ESN ). ′ p p p By definition the annihilator of ESN is ZSN := {g ∈ H :: ∀a ∈ SN , g(a)=0}. So we have

p ∀λ ∈??? (SN ), λaρa = sup λaρa, f . p′ a∈S p p f∈E ,kfk ′ ≤1 *a∈S + N H /Z SN p N X SN X p′ But f ∈ E means that f = µbkb,p′ hence S b∈ SN

P ′ λaρa = sup λaρa, µbkb,p . p′ ′ a∈SN p p f∈ES ,kfkp ≤1 *a∈SN b∈SN + H /ZS N X N X X So, using Hölder inequalities,

′ λaρa = sup λaµ¯a ≤kλk???p kµk???p p′ ′ a∈SN p p f∈ES ,kfkp ≤1 a∈SN H /ZS N X N X ′ ′ by assumption, kµ k p ≤ C µaka,p ′ = Ck fk ′ , with C independent of N, hence ??? a∈SN p p P

6 λaρa ≤ Ckλk???p . p a∈SN Hp/Z X SN p ′ Now the functions g := a∈SN λaρa + ZS interpolate the right values: ∀a ∈ S, g(a) = λaχa,p . So p p we can find one f in H with norm less than (C + ǫ)kλk??? (SN ). We proved that for any NP: p p p ′ ∀λ ∈??? (SN ), ∃f ∈ H , kfkp ≤ (C + 1)kλk??? (SN ) with ∀a ∈ S, hf,ka,p i = λa. p p Because ??? (SN ) is dense in ??? (S) and none of the constants depend on N, we get the result by letting N →∞. I.e. the restriction operator verifies (4.1). 

Lemma 4.2. Let S be a sequence of points in Ω. Then S is Carleson iff there exists a p ∈ (1, ∞) such that: p (∗) C > 0, ∀f ∈ H (Ω), kRpfk???p(S) ≤ CkfkHp(Ω), ′ i.e., with the measure dµ := a∈S χa,p δa, p p p p ∀f ∈ H , Ω |f| dµ ≤ C kfkp. Which means that µ is a CarlesonP measure in Ω. R Proof. • Suppose first that S verifies (∗) and take any p ∈ (1, ∞). ′ We have, for any µ ∈???p (S),

µaka,p′ = sup f, µaka,p′ . p ′ f∈H ,kfkp≤1 * + Xa∈S p Xa∈S ′ ′ But f, a∈S µaka,p = a∈S µ¯a hf,ka,p i and by Hölder inequalities we get:

P P 1/p p ′ ′ ′ µ¯ahf,ka,p i ≤ |hf,ka,p i| kµk???p (S). (4.2) ! a∈S a∈S X X

S Carleson means that: p p p ′ ∀f ∈ H , |hf,ka,p i| . kfkp, a∈S so we get by (4.2)X that

p ′ ′ µ¯ahf,ka,p i . kfkpkµk???p (S)

Xa∈S which gives µ k ′ ′ . kµk p′ because kfk ≤ 1. a∈S a a,p p ??? p

P • Suppose now that

′ ′ µaka,p . kµk???p . a∈S ′ X p Take f ∈ Hp then

f, µaka,p′ = µ¯ahf,ka,p′ i . * + a∈S a∈S X ′ X p 1/p p ′ ′ ′ Now take µ ∈??? (S), kµk ???p (S) = 1 such that a∈S µ¯ahf,ka,p i = a∈S |hf,ka,p i| . Then we get P P 

7 1/p p ′ ′ ′ ′ |hf,ka,p i| = f, µaka,p ≤kfkp µaka,p . kfkpkµk???p . kfkp ! * + ′ Xa∈S Xa∈S Xa∈S p p′  because kµk??? (S) =1. This means that S is verifies (∗) .

Remark 4.3. Recall that if S is Carleson for a p ∈ [1, ∞[, then it is Carleson for any q ∈ [1, ∞[ as the geometric characterisation of , done for B by Hastings [Hastings, 1975] and done for Dn by Chang [Chang, 1979], proved.

5 Proof of the main result.

S has the LEP for Hp means that there is a E :???p(S) → Hp such that ∀a ∈ S, hE(λ), ka,p′ i = λa. We shall say that S has the LEP with range F := Range(E). Theorem 5.1. Let S be a sequence of points in Ω. Suppose that S is Hp-interpolating in Ω for a p p ≥ 2. then S has the LEP with range ES := Span{ka,p, a ∈ S}. Proof. • The case p =2 is easy. 2 2 2 2 Set ES := Span{ka,2, a ∈ S} and P the orthogonal projection from H onto ES. Let λ ∈??? (S) 2 2 then, because S is H interpolating, there is a function f ∈ H such that ∀a ∈ S, hf,ka,2i = λa and kfk2 . kλk???2(S). Set g := P f then ∀a ∈ S, hg,ka,2i = hPf,ka,2i = hf,Pka,2i = hf,ka,2i = λa 2 and kgk2 = kP fk2 ≤ kfk2 . kλk???2(S). If there is another function h ∈ ES such that ∀a ∈ 2 S, hh, ka,2i = λa then hg − h, ka,2i = 0 hence g − h ⊥ ES ⇒ g − h = 0. So the operator E : λ ∈ 2 2 ??? (S) → g ∈ ES is the extension we are searching for.

• The case p> 2. Using the Theorem 8.2 we have that, if Ω= B, S is Carleson. By Theorem 1.11 p. 3 in [Amar, 2019] we have that, if Ω= Dn, S is Carleson. So in any cases we have that S is Carleson. Because S is Hp interpolating, we have:

′ p ′ ′ ∀µ ∈??? (S), µaka,p & kµk???p . ′ Xa∈S p Because S is Carleson, we have:

′ p ′ ′ ∀µ ∈??? (S), µaka,p . kµk???p . ′ a∈S p X p′ ′ ′ Hence we get that the sequence {k a,p }a∈S is a p-Riesz basis for the space ES := Span{ka,p , a ∈ S}. This easily implies, see for instance Theorem 4.6, p. 15 in [Amar et al., 2019], that there is a p′ ′ ′ p ′ ′ bounded operator Qp : ES →??? (S) with bounded inverse such that ∀a ∈ S, ǫa = Qp ka,p , where p′ ǫa is the canonical basis of ??? (S). Using Theorem 1.5 and Theorem 1.6 p. 179 in [Amar, 2007], we get that if S is Hp interpolating ′ then S is Hq interpolating for q < p. Because p > 2 we get that p′ < 2 < p hence S is also Hp interpolating.

8 ′ So we can apply the above inequalities with p instead of p to get that the sequence {ka,p}a∈S p is a p-Riesz basis for the space ES := Span{ka,p a ∈ S}. And again there is an bounded operator p p Qp : ES →??? (S) with bounded inverse such that ∀a ∈ S, fa = Qpka,p where fa is the canonical basis of ???p(S). The idea is to extend the Proposition 2 p. 13 in [Amar, 1977] done for Hilbert spaces to our case. We get ∗ δab = hǫa, fbi = hQp′ ka,p′ , Qpka,pi = hka,p′ , (Qp′ ) Qpka,pi. ∗ Hence, setting ∀a ∈ S, ρa := (Qp′ ) Qpka,p, we get that {ρa}a∈S is a bounded dual sequence to p ′ {ka,p }a∈S contained in ES. I.e. ′ ∀a ∈ S, hρa,kb,p i = δab and ∃C > 0, ∀a ∈ S, kρakp ≤ C. Moreover we have: p ∗ ∀λ ∈??? (S), λaρa = (Qp′ ) Qp λaka,p .

a∈S p a∈S p But, using that S is Carleson, X we get X

λaka,p . kλk???p(S).

a∈S p So X

p ∗ ∀λ ∈??? (S), λaρa = (Qp′ ) Qp λaka,p ≤

a∈S p a∈S p X X

′ ′ ≤k Qp kkQ pk λaka,p . kQp kk Qpkkλkp. a∈S X p Because Q and Q ′ are bounded. We finally get: p p

p ∀λ ∈??? (S), λaρa . kλkp.

Xa∈S p It remains to define the extension operator: p ∀λ ∈??? (S), E (λ)(z) := λaρa(z) a∈S to end the proof of the theorem.X

Remark 5.2. The Proposition 4.5 in [Amar et al., 2019] says, in a fairly general situation, that p S is H interpolating with the LEP iff we have that the sequence {ρa}a∈S is p-hilbertian, i.e. iff

a∈S λaρa(z) ≤ Ckλk???p(S). This is pretty clear in our case here. P

6 Strictly Hp interpolating sequences.

Let S be a sequence of points in Ω. We consider the Grammian associated to S. This is the infinite matrix G given by Ga,b := hka,p′ , kb,pi. ′ We consider G as an operator on ???p (S) defined as

p′ ∀µ ∈??? (S), Gµ := {(Gµ)b}b∈S, (Gµ)b := µaGa,b = µaka,p′ ,kb,p . a∈S *a∈S + X X

9 ∗ ∗ ¯ The adjoint G of G is the matrix Ga,b = Gb,a hence: ∗ ′ Ga,b = hka,p,kb,p i. Remark 6.1. We have the well known results: ∗ ′ ′ kGk???p→???p = kG k???p →???p ′ ′ ∗ and G is bounded below, i.e. kGµk???p & kµk???p , iff G is bounded below. First we generalise to our setting half of the Proposition 9.5,p. 127of[Agler and MacCarthy, 2002]: Proposition 6.2. Let S be a sequence of points in Ω. Let: (CS) The sequence S is Carleson. ′ (BG) The associated Grammian G is bounded on ???p (S). Then (CS) implies (BG). Proof. We have

(Gµ)b := µahka,p′ , kb,pi = µaka,p′ , kb,p . a∈S *a∈S + Hence X X

¯ ′ hGµ,λi = λb(Gµ)b = µaka,p , λbkb,p . b∈S *a∈S b∈S + X ′X X To get that G is bounded on ???p (S) amounts to prove

¯ ′ λa(Gµ)a . kλk???p(S)kµk???p (S).

a∈S But, using X Hölder inequalities,

µaka,p′ , λaka,p ≤ µaka,p′ λaka,p . (6.3) * + ′ a∈S a∈S a∈S p a∈S p X X X X

Suppose now that the sequence S is Carleson. By definition this means:

′ p ′ ′ ∀µ ∈??? (S), µaka,p . kµk???p a∈S ′ X p and

p ∀λ ∈??? (S), λaka,p . kλk???p . ′ a∈S p X Hence (6.3) gives

′ ′ |hGµ,λi| = µbkb,p , λaka,p . kλk???p(S)kµk???p (S). * + b∈S a∈S Hence G is bounded. X The proof isX complete. 

Definition 6.3. Let S be a sequence of points in Ω. We shall say that S is strictly Hp interpolating ′ if the operator G is bounded below on ???p (S).

′ Remark 6.4. Using Remark 6.1, we have that S is strictly Hp interpolating iff S is strictly Hp interpolating.

10 Theorem 6.5. Let S be a sequence of points in Ω. Suppose that S is Carleson and S is strictly Hp interpolating. p p Then S is H interpolating and has the LEP with range ES := Span{ka,p, a ∈ S}. ′ p p′ For the proof we shall need a lemma, where ZS := {u ∈ H :: ∀a ∈ S, u(a)=0}. Lemma 6.6. If S is strictly Hp interpolating and Carleson then we have:

′ p′ p ′ ′ ∀µ ∈??? (S), ∀u ∈ ZS , u + µaka,p & kµk???p (S). ′ a∈S p X

Proof. We have, for any f ∈ Hp,

′ hf,ui + f, a∈S µaka,p u + µaka,p′ ≥ . ′ kfkp a∈S p P X We choose f := a∈S λ aka,p then hf,ui = a∈S λahka,p,ui =0 because ∀a ∈ S, u(a)=0. Hence we have P P ′ a∈S λaka,p, a∈S µaka,p |hGµ,λi| u + µaka,p′ ≥ = . λ k λ k a∈S ′ P a∈S Pa a,p p a∈S a a,p p X p p ′ We choose λ ∈??? (S) , kλk???p(S) =P 1 such that hGµ,λi = PkGµk???p (S). Now we use that G is bounded below to get ′ ′ hGµ,λi = kGµk???p (S) & kµk???p (S). Because S is Carleson:

p ∀λ ∈??? (S), λaka,p . kλk???p ,

Xa∈S p . p hence a∈S λaka,p p kλk??? =1. So we get P hGµ,λi ′ ′ u + µaka,p ≥ & kµk???p (S). ′ a∈S λaka,p p Xa∈S p The proof of the lemma is complete.  P

Remark 6.7. In fact this lemma proves that if S is strictly Hp interpolating and Carleson then we have:

′ ′ µaka,p & kµk???p (S). ′ a∈S p′ p H /ZS X

Proof of the Theorem. Suppose that S is strictly Hp interpolating. ′ p We truncate S to its first N terms, call it SN . Then we have that {ka,p }a∈SN has a H dual bounded p sequence {ρa}a∈SN ⊂ ES := Span{ka,p, a ∈ SN }. Take λ ∈???p(S ) we want to estimate λ ρ and, because the dual space of Ep is N a∈SN a a p SN ′ ′ Hp /Zp we have SN P

λ ρ , f a∈SN a a λaρa = sup . ′ p′ p kfkp′ a∈SN f∈H /Z P p SN X

11 ′ ′ p′ p p But f ∈ H /ZSN can written, with u ∈ ZSN ,

f = u + µaka,p′ .

a∈SN So we have to computeX

λaρa, f = λaρa, u + µaka,p′ .

*a∈SN + *a∈SN a∈SN + X p X p′ X But a∈SN λaρa ∈ ES and u ∈ ZSN , imply a∈SN λaρa, u =0. So it remains

P ′ P λaρa, µaka,p = λaµ¯a *a∈S a∈S + a∈S XN XN XN because hρa,kb,p′ i = δa,b. So we get

′ ′ λaρa, µaka,p = λaµ¯a ≤kλk???p(S)kµk???p (S). * + a∈SN a∈SN a∈SN Hence X X X

p p′ kλk??? (SN )kµk??? (SN ) λaρa ≤ sup . (6.4) ′ p′ u + µ k ′ a∈S f∈Hp /Z a∈SN a a,p p′ N p SN X P Now we use Lemma 6.6 to get

′ ′ u + µaka,p & kµk???p (S ). N a∈SN p′ X Putting it in (6.4) we deduce that

λaρa . kλk???p .

aX∈SN p N The constant under the . is independent of N, so we get, in particular, that ρa ≤ C. Now we use the diagonal process to let N →∞ and to get that there is a dual bounded sequence {ρ } ⊂ Ep a a∈S S such that p ∀λ ∈??? (S), λaρa . kλk???p .

a∈S p It remains to define the X extension operator:

p ∀λ ∈??? (S), E (λ)(z) := λaρa(z), a∈S to end the proof of the theorem.X

Using Remark 6.4 and Theorem 6.5, we get that if S is strictly Hp interpolating, then S is strictly ′ Hp interpolating, hence S is Hr interpolating with the LEP for r = max(p,p′) ≥ 2. This leads to the converse of Theorem 6.5.

Theorem 6.8. Let S be a sequence of points in Ω. Suppose that S is Hp interpolating and has the p LEP with range ES := Span{ka,p, a ∈ S} for a p> 2. Then S is strictly Hp interpolating.

12 Proof. Because p > 2 then S is Carleson by Theorem 8.2 for the ball B and by Theorem 1.11 p. 3 in [Amar, 2019] for the polydisc Dn. Using Theorem 1.5 and Theorem 1.6 p. 179 in [Amar, 2007], we get that if S is Hp interpolating ′ then S is Hq interpolating for q < p. Because p > 2 we get that p′ < 2 < p hence S is also Hp interpolating.

′ Let hν,Gµi = a,b∈S νaµ¯bhka,p, kb,p i. p ′ S being H interpolating means µ k ′ & kµk p′ . P a∈S a a,p p ??? p′ On the other hand, because S is alsoP H interpolating and Carleson, then:

νaka,p ≃kνk???p . (6.5)

a∈S p X p ′ p To get the norm of a∈S µaka,p by duality, we have to test on f ∈ H /ZS. Such an f can be written: P p f = u + νaka,p with u ∈ ZS. a∈S X S being Carleson means:

p ′ ∀f ∈ H , k{hf,ka,p i}k???p(S) . kfkp. (6.6)

p p So, because S is H interpolating with the LEP with range ES, we have that it exists a dual basis p {ρa}a∈S ⊂ ES such that: p ′ p ′ P f := hf,ka,p iρa ∈ ES ⊂ H with kP fkp . k{hf,ka,p i}k???p(S) Xa∈S hence kP fkp . kfkp by (6.6). Because P f = f on S, we get also P 2 = P. Now

f, µaka,p′ = u + νaka,p, µaka,p′ = νaka,p, µaka,p′ , * a∈S + * a∈S a∈S + *a∈S a∈S + X ′ X X X X because u, a∈S µaka,p =0. Hence P ′ a∈S νaka,p, a∈S µaka,p µaka,p′ = sup . p p ′ f∈H /Z u + a∈S νaka,p a∈S p S P P p X But P (u + a∈S νaka,p)= a∈S λaρa withP a∈S λaρ a . u + a∈S νaka,p ≤ 1. p p Hence, because ν k = λ ρ , we choose f = u + ν k realizing the norm: P a∈S a Pa,p a∈S a a P Pa∈S a a,p ′ a∈S νaka,p, a∈S µaka,p ′ P ′ P P kµk???p ≤ µaka,p . . ′ a∈S λaρa a∈S p P P p X But by (6.5) we get a∈S νa ka,p p ≃kνk??? Pp hence from a∈S νaka,p = a∈S λaρa we get

P P P

13 λaρa = νaka,p ≃kνk???p .

a∈S p a∈S p Hence X X

νaka,p, µaka,p′ ′ ′ a∈S a∈S kµk???p ≤ µaka,p . kνk p a∈S ′ P P??? X p which means exactly that G is bounded below. Hence S is strictly Hp interpolating. 

Remark 6.9. In the case p =2 we retrieve the fact that S is H2 interpolating iff the Grammian G is bounded below.

7 Bergman spaces of the ball.

Let us recall the definition of the Bergman spaces of the ball we are interested in. We shall use the n 2 notation a¯ · z := j=1 a¯jzj hence |z| := z · z.¯ B Cn N p B Definition 7.1. PLet f be a holomorphic function in n ⊂ and k ∈ ; we say that f ∈ Ak( ) if p p 2 k kfkk,p := |f(z)| (1 −|z| ) dm(z) < ∞. B Z Where dm is the Lebesgue measure in Cn. p B B The space Ak( ) possesses a reproducing kernel for any a ∈ , ka(z) and a normalised one ka,p(z): 1 ka(z) ka(z)= n+k+1 , ka,p(z)= . (1 − a¯ · z) kkakp ′ −2 −2 2 n+k+1 −1/p Set χa := kkakk,2 = ka(a) = (1 −|a| ) . Then kkakp ≃ χa . p And we have ∀f ∈ H (B), f(a) := hf,kai, where h·, ·i is the scalar product of the Hilbert space 2 B Ak( ). In the unit disc the Ap(D) interpolating sequences where characterized in the nice papers of K. Seip [Seip, 1993a], [Seip, 1993b]. He used densities to do it, opposite to the product of Gleason distances used for Hp(D) interpolating sequences.

7.1 Links between Bergman and Hardy spaces. We shall use the "Subordination Lemma", see [Amar, 2015] and earlier [Amar, 1978]. n+k+1 We shall write (z,ζ) := (z1, ..., zn,ζ1, ..., ζk+1) to define a point in C . Now on we denote Bk the unit ball in Ck. p B p B The links between Ak( n) and H ( n+k+1) are given by the Subordination Lemma: p B ˜ p B ˜ f ∈ Ak( n) iff f(z,ζ) := f(z) is in H ( n+k+1), and we have kfkk,p ≃ f . Hp p B p B We also have f(z,ζ) ∈ H ( n+k+1) ⇒ f(z, 0) ∈ Ak( n) with kf(·, 0)kk,p ≤ Ck fk Hp . p The reproducing kernels for H (Bn+k+1) are: ˜ B ˜ 1 ˜ ka(z) ˜ 2 −(n+k+1)/p′ ∀a, z ∈ n+k+1, ka(z)= n+k+1 , ka,p(z)= , with ka ≃ (1−|a| ) . (1 − a¯ · z) k˜ p a p Hence, not surprisingly, if a ∈ Bn, we get B ˜ ˜ 2 −(n+k+1)/p′ ∀a, z ∈ n, ka(z)= ka(z), ka,p(z)= ka,p(z), with kk akp ≃ (1 −|a| ) .

14 7.2 Carleson measures for Bergman spaces. p B In [Hastings, 1975] the Carleson measures for the space Ak( ) are defined. See also [Cima and Mercer, 1995], [Zhu, 2005], [Abate and Saracco, 2011] and [Amar, 2015]. B p B Definition 7.2. Let µ be a Borel measure in n then it is called a Carleson measure for Ak( n) if: p B p p ∀p ≥ 1, ∃C > 0, ∀f ∈ Ak( n), |f(z)| dµ(z) ≤ Ckfkk,p. Bn The same for the usual Carleson measuresZ for Hardy spaces: p B p p ∀p ≥ 1, ∃C > 0, ∀f ∈ H ( n), |f(z)| dµ(z) ≤ CkfkHp(B). B Z n By their geometric definition [Amar, 2015], [Zhu, 2005], we already know that if µ is Carleson for a p ∈ [1, ∞[ then it is Carleson for all p ∈ [1, ∞[. Let µ be a measure in Bn and extend it by 0 in Bn+k+1. Call the extended measure µ.˜ We have p B Proposition 7.3. The measure µ is a Carleson measure for Ak( n) iff the measure µ˜ is a Carleson p measure for H (Bn+k+1). Proof. p • Suppose first that µ˜ is Carleson for H (Bn+k+1). p B ˜ ˜ p B Then take f ∈ Ak( n). Extend it as f(z,ζ) := f(z), then f ∈ H ( n+k+1). We have, because µ˜ is supported by Bn: p f˜(z,ζ) dµ˜(z,ζ)= |f(z)|p dµ(z). Bn+k+1 Bn Z p Z p But µ˜ is Carleson for H (Bn+k+1) and f˜ ∈ H (Bn+k+1), so we get p p p ˜ ˜ p |f(z)| dµ(z)= f(z,ζ) dµ˜(z,ζ) ≤ C f ≤ Ckfkk,p. Hp(B ) Bn Bn+k+1 n+k+1 Z pZB Hence µ is Carleson for Ak( n). p B p B • Suppose now that µ is Carleson for Ak( n). Let g(z,ζ) ∈ H ( n+k+1). We still have |g(z,ζ)|p dµ˜(z,ζ)= |g(z, 0)|p dµ(z). Bn+k+1 Bn Z Zp B p B Hence, because µ is Carleson for Ak( n), and g(z, 0) ∈ Ak( n) by the Subordination Lemma: p p p p |g(z,ζ)| dµ˜(z,ζ)= |g(z, 0)| dµ(z) ≤ Ckg(·, 0)k ≤ Ckgk p . k,p H (Bn+k+1) Bn+k+1 Bn Z p Z Hence µ˜ is Carleson for H (Bn+k+1).  Remark 7.4. Proposition 7.3 defines extrinsically the Carleson measures for the Bergman spaces p B Ak( n). As a Corollary of P. Thomas’ Theorem [Thomas, 1987] we get:

B 1 B Theorem 7.5. Let S be a sequence of points in . If the sequence S is Ak( ) interpolating then S p B is a Carleson sequence forAk( n). Proof. The sequence S is contained in the set ζ =0 in Bn+k+1, with z ∈ Bn and (z,ζ) ∈ Bn+k+1. Because 1 S is Ak-interpolating we get

15 1 1 ∀λ ∈??? (S), ∃f ∈ Ak :: ∀a ∈ S, hf, ka,∞i = λa and kfkk,1 . kλk???1(S). 1 1 ˜ ˜ Fix λ ∈??? (S) and f ∈ Ak doing the interpolation. Consider f(z,ζ) := f(z). Then, as we seen, f ∈ 1 1 H (B ), f˜ interpolates the sequence λ ∈??? (S) and kfk 1 . kfk . kλk 1 . Hence n+k+1 H (Bn+k+1) k,1 ??? (S) 1 we get that S is H (Bn+k+1)-interpolating in Bn+k+1. We apply P. Thomas’ Theorem [Thomas, 1987] p to get that S is a Carleson sequence for H (Bn+k+1). Because S ⊂ Bn we deduce that S is a Carleson p B sequence forAk( n). 

7.3 The main result for the Bergman spaces. B p B Theorem 7.6. Let S be a sequence of points in n. Suppose that S is Ak-interpolating in n for a p p ≥ 2. then S has the LEP with range ES := Span{ka,p, a ∈ S}. Proof. The sequence S is contained in the set ζ =0 in Bn+k+1, with z ∈ Bn and (z,ζ) ∈ Bn+k+1. Because p S is Ak-interpolating we get p p ′ ∀λ ∈??? (S), ∃f ∈ Ak :: ∀a ∈ S, hf, ka,p i = λa and kfkk,p . kλk???p(S). p p ˜ Fix λ ∈??? (S) and f ∈ Ak doing the interpolation. Consider f(z,ζ) := f(z). Then, as we seen, p p f˜ ∈ H (B ), f˜ interpolates the sequence λ ∈??? (S) and kfk p . kfk . kλk p . n+k+1 H (Bn+k+1) k,p ??? (S) p Hence we get that S is H (Bn+k+1)-interpolating in Bn+k+1. So we can apply Theorem 5.1 which ˜ gives that S has the LEP in Span{ka,p, a ∈ S}. This gives the result because when a ∈ Bn, we have p that the reproducing kernels k˜a,p for H (Bn+k+1) are the same as the reproducing kernels ka,p for p ˜ Ak i.e. ka,p(z,ζ)= ka,p(z). The proof is complete. 

p 7.4 Strictly Ak interpolating sequences.

As we did for Hardy spaces, we define the Grammian for a sequence of points in Bn: Ga,b := hka,p′ , kb,pi. ′ Again this define an operator on ???p (S):

p′ ∀µ ∈??? (S), Gµ := {(Gµ)b}b∈S, (Gµ)b := µaGa,b = µaka,p′ ,kb,p . a∈S *a∈S + We still have X X

Proposition 7.7. Let S be a sequence of points in Bn. Let: (CS) The sequence S is Carleson for Ap. k ′ (BG) The associated Grammian G is bounded on ???p (S). Then (CS) implies (BG).

Proof. We lift everything on Bn+k+1 and, because the reproducing kernels agree, we get that G˜a,b = Ga,b. p p On the other hand S Carleson for Ak means that the associated measure µS is Carleson for Ak, hence p its extension µ˜S is Carleson for H (Bn+k+1) by Proposition 7.3. So we get, applying Proposition 6.2, ′ that G˜ is bounded on ???p (S). p′ Because G˜a,b = Ga,b we get that G is bounded on ??? (S). 

16 p Definition 7.8. Let S be a sequence of points in Bn. We shall say that S is strictly A interpolating ′ k if the operator G is bounded below on ???p (S). Now we have, as for the Hp case: B p Theorem 7.9. Let S be a sequence of points in n. Suppose that S is Carleson for Ak, and S is p strictly Ak interpolating. p p Then S is Ak interpolating and has the LEP with range ES := Span{ka,p, a ∈ S}. Proof. Lifting the situation from Bn to Bn+k+1, because the reproducing kernels agree, we get that G˜a,b = p p B Ga,b. Hence S strictly Ak interpolating implies that S is strictly H ( n+k+1) interpolating. It p remains to apply Theorem 6.5 to have that S is H (Bn+k+1) interpolating and has the LEP with ˜p ˜ B range ES := Span{ka,p, a ∈ S}. But S ⊂ n then the reproducing kernels agree, so we get that S p p is Ak interpolating and has the LEP with range ES := Span{ka,p, a ∈ S}. The proof is complete. 

As for Hardy spaces we have the converse of the Theorem 7.9. B p Theorem 7.10. Let S be a sequence of points in n. Suppose that S is Ak interpolating and has p the LEP with range ES := Span{ka,p, a ∈ S} for a p> 2. p Then S is strictly Ak interpolating. Proof. For the proof we repeat exactly the arguments we use for proving Theorem 7.9. 

We also have, still using the Subordination Lemma: p s Theorem 7.11. Let S be a dual bounded sequence in Ak. Then S is Ak interpolating with the LEP, for any s ∈ [1,p[. Proof. p We lift the situation to Bn+k+1 which leads to deal with the Hardy space H (Bn+k+1) and we apply s Theorem 1.4 p. 482 in [Amar, 2009]. Hence we get the result that S is H (Bn+k+1) interpolating with the LEP, for any s ∈ [1,p[. It remains to go back to Bn to end the proof of this theorem. 

8 Babenko examples in the class of holomorphic functions.

Recall the result of Babenko [Babenko, 1947]: there exists {ea}a∈S a basis of unit vectors in the Hilbert space H which is 2-hilbertian but {ea}a∈S is not a Riesz basis. The same way there exists {ea}a∈S a basis of unit vectors in the Hilbert space H which is 2- besselian but {ea}a∈S is not a Riesz basis. Hence a natural question is: is it possible to have Babenko examples in the case of systems of reproducing kernels? In the case of the unit disc the answer is no because of the following remarkable property of the disc algebra. Let S be a finite sequence of distinct points in the unit disc. Let {ka,2}a∈S be the normalized Cauchy kernels and set ES := Span{ka,2, a ∈ S}. Set also {ρa}a∈S the dual basis of {ka,2}a∈S in ES and {ρa,2}a∈S this normalized dual basis. Then we have [Amar, 1977]:

17 Theorem 8.1. Let S be a finite sequence of points in D. The following anti-linear is true

λaka,2 = λ¯aρa,2 .

a∈S H2(D) a∈S H2(D) X X

This Theorem easily implies that if {ka,2}a∈S is hilbertian or Besselian for any sequence of points in D, then it is a Riesz sequence. Hence the Babenko phenomenon cannot exist in the case of the Cauchy kernels in the disc. We have the following Theorem: Theorem 8.2. ( [Thomas, 1987]) Let S be a sequence of points in B. If the sequence S is H1(B) interpolating then S is a Carleson sequence. The Theorem by P. Thomas is valid also for a class of harmonic functions in the ball. See [Amar, 2008] for an easier proof using Wirtinger inequalities, but working only for holomorphic functions. Using the link between interpolating sequences and besselian systems and between Carleson sequences and hilbertian systems, we have that, in Hp(B), if a sequence of reproducing kernels is p-besselian then it is automatically a p-Riesz sequence. We also have: Theorem 8.3. ( [Amar, 2019]) Let S be a sequence of points in Dn. Suppose that S is Hp(Dn) interpolating with a p> 2. Then S is Carleson. Then, the same way as above, we have that, in Hp(Dn), p> 2, if a sequence of reproducing kernels is p-besselian then it is automatically a p-Riesz sequence. But we have: Theorem 8.4. The are sequences of reproducing kernels in Hp(Ω) which are p-hilbertian but not p-Riesz sequences. Proof. By the characterisation of K. Seip [Seip, 1993a], [Seip, 1993b], we know that, for any p ≥ 1, q > p, there are p-interpolating sequences for the Ap(D) which are not q-interpolating for Aq(D). Take such a sequence S ⊂ D with q>p> 2. Then, using the subordination lemma, we have that p S˜ := {(a, 0), a ∈ S}. can be seen as a p-interpolating sequence in H (B2), hence it is a p-Carleson p q sequence in H (B2). So it is also a q-Carleson sequence in H (B2). By Proposition 7.3, we get that S is also q-Carleson sequence in Aq(D), but it is not a q-interpolating sequence. Hence the normalised associated reproducing kernels in Aq(D) make a q-hilbertian sequence but not a q-Riesz sequence. q q We have the same result in H (B2) for the normalised associated reproducing kernels in H (B2) associated to the sequence S˜ := {(a, 0), a ∈ S}. To deal with Hp(D2) we set the sequence S in the diagonal of the bi-disc: S˜ := {(a, a), a ∈ S}. Then again we know that S˜ is Hp(D2) but not Hq(D2) interpolating, see [Amar and Menini, 2002] and the references therein. Because p> 2, we get that S˜ is p-Carleson, hence q-Carleson. Because S˜ is not a q-interpolating sequence, the normalised associated reproducing kernels in Hq(D2) make a q-hilbertian sequence but not a q-Riesz sequence. 

18 This leads to the natural conjectures, still with Ω being the ball or the polydisc:

• If S is dual bounded in H2(Ω) (resp. in Hp(Ω)), then it is H2(Ω) (resp. Hp(Ω)) interpolating.

Of course these conjectures are true in one variable, because dual boundedness is easily seen to be equivalent to the Carleson condition. Hence it implies interpolation.

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