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Carleson Measures and $ H^{P} $ Interpolating Sequences in The Carleson measures and Hp interpolating sequences in the polydisc. Eric Amar Abstract Let S be a sequence of points in Dn. Suppose that S is Hp interpolating. Then we prove that the sequence S is Carleson, provided that p > 2. We also give a sufficient condition, in terms of dual boundedness and Carleson measure, for S to be an Hp interpolating sequence. Contents 1 Introduction and definitions. 1 2 First main result. 4 3 Second main result. 6 3.1 Interpolation of Hp spaces................................. 6 3.2 BMOfunctionsinthepolydisc. ....... 7 4 Appendix. Assumption (AS) 10 arXiv:1911.07038v2 [math.FA] 15 Jun 2020 1 Introduction and definitions. Recall the definition of Hardy spaces in the polydisc. Definition 1.1. 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 measure on T : p iθ p kfkHp := sup f(re ) dθ < ∞. Tn r<1 Z The Hardy space H∞( Dn) is the space of holomorphic and bounded functions in Dn equipped with the sup norm. 1 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) p n And we have ∀f ∈ H (D ), f(a) := hf,kai, where h·, ·i is the scalar product of the Hilbert space H2(Dn). Dn 2 2 2 p 2 −1/p′ For a ∈ , set ((1−|a| )) := (1−|a1| )···(1−|an| ). The H norm of ka is kkakp = ((1−|a| )) hence the normalized reproducing kernel in Hp(Dn) is 2 1/p′ 2 1/p′ (1 −|a1| ) (1 −|an| ) ka,p(z)= ×···× . (1 − a¯1z1) (1 − a¯nzn) Let S be a sequence of points in Dn. 0 p n 0 Let ℓ (S) be the set of sequences on S and define the restriction operator Rp : H (D ) → ℓ (S) by: p n 2 1/p ∀f ∈ H (D ), Rpf := {((1 −|a| )) f(a)}a∈S. p n p n p Definition 1.2. We say that S is H (D ) interpolating if Rp(H (D )) ⊇ ℓ (S). We can state, for 1 ≤ p< ∞: p n Definition 1.3. We say that the sequence S is Carleson if the operator Rp is bounded from H (D ) to ℓp(S), i.e. p Dn ∃C > 0, ∀f ∈ H ( ), kRpfkℓp(S) ≤ CkfkHp(Dn). Definition 1.4. Let µ be a Borel measure in Dn. We shall say that µ is a Carleson measure if ∃p, 1 ≤ p< ∞, such that: p Dn p ∃C > 0, ∀f ∈ H ( ), f ∈ L (µ) and kfkLp(µ) ≤ CkfkHp(Dn). n n For z ∈ D define the rectangle Rz := {ζ ∈ T :: |ζj − zj/ |zj|| ≤ 1 −|zj| , j =1, ..., n}. n n Now, for any open set Ω in T , the (generalised) Carleson region is ΓΩ := {z ∈ D :: Rz ⊂ Ω}. Definition 1.5. The measure µ is a rectangular Carleson measure in Dn if: Dn ∃C > 0 :: ∀z ∈ , |µ| (ΓRz ) ≤ C |Rz| . The "natural" generalisation of the Carleson embedding Theorem from the disc to the poly- disc would be that: µ is a Carleson measure iff it is a rectangular Carleson measure. But Car- leson [Carleson, 1974] gave a counter example to this and A. Chang [Chang, 1979] gave the following characterisation. Theorem 1.6. The measure µ is a Carleson measure in Dn iff, for any open set Ω: ∃C > 0 :: ∀Ω, |µ| (ΓΩ) ≤ C |Ω| . Because this characterisation does not depend on p, if µ is p-Carleson for a 1 ≤ p < ∞ then it is q-Carleson for any q. This justifies the absence of p in the definition of Carleson measure. For S a sequence of points in Dn, define the measure: 2 χS := ((1 −|a| ))δa. a∈S X Then we can see easily that the sequence S is Carleson iff the measure χS is a Carleson measure. 2 In one variable the interpolating sequences were characterized by L. Carleson [Carleson, 1958] for H∞(D) and by H. Shapiro and A. Shieds [Shapiro and Shields, 1961] for Hp(D) by the same condition: ∀b ∈ S, dG(a, b) ≥ δ > 0, a∈S, a6=b Ya−b where dG(a, b) := 1−ab¯ is the Gleason distance. In several variables for the unit ball Ω= B or for the unit polydisc Ω= Dn, this characterisation is still an open question, even for H2(Ω). Nevertheless we have already some necessary conditions. Theorem 1.7. ( [Varopoulos, 1972]) If the sequence S is H∞(Dn) interpolating then the measure χS is rectangular Carleson. We shall need Definition 1.8. We say that the Hp(Ω) interpolating sequence S has the linear extension prop- erty, LEP, if there is a bounded linear operator E : ℓp(S) → Hp(Ωn) such that kEk < ∞ and for any λ ∈ ℓp(S), E(λ)(z) interpolates the sequence λ in Hp(Ω) on S. ∞ n Theorem 1.9. ( [Amar, 1980]) If the sequence S is H (D ) interpolating then the measure χS is Carleson and for any p ≥ 1 the sequence S is Hp(Dn) interpolating with the LEP. For the ball we have a better result by P. Thomas [Thomas, 1987]. See also [Amar, 2008a]. 1 Theorem 1.10. If the sequence S is H (B) interpolating then the measure χS is Carleson. The first main result of this work is an analogous result for the polydisc: Theorem 1.11. Let p> 2 and suppose that the sequence S of points in Dn is Hp(Dn) interpolating. Then the sequence S is Carleson. We also have some sufficient conditions. Let Ω be the ball or the polydisc. Theorem 1.12. ( [Berndtsson, 1985] for the ball; [Berndtsson et al., 1987] for the polydisc) Let S be a sequence of points in Ω. Let (a), (b) and (c) denote the following statements: (a) There is a constant δ > 0 such that ∀b ∈ S, dG(a, b) ≥ δ > 0. a∈S, a6=b (b) S is an interpolatingY sequence for H∞(Ω). (c) The sequence S is separated and is a Carleson sequence. Then (a) implies (b), (b) implies (c). However, the converse for each direction is false for n ≥ 2. Now we shall need the following important definition. Definition 1.13. 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 ′ kernel in Hp (Ω) for the point b ∈ Ω: ′ ∃ C > 0, ∀a, b ∈ S, hρa,kb,p i = δa,b and kρakHp(Ω) ≤ C. 3 Using this definition we have, with Ω the ball B or the polydisc Dn: Theorem 1.14. ( [Amar, 2008b]) Let S be a sequence of points in Ω. Suppose S is Hp(Ω) dual bounded with either p = ∞ or p ≤ 2. Moreover suppose that S is Carleson. Then S is Hq(Ω) interpolating with the LEP for any 1 ≤ q < p. For the ball we have a better result. Theorem 1.15. ( [Amar, 2009]) Let S be a sequence of points in B. Suppose S is Hp(B) dual bounded. Then S is Hq(B) interpolating with the LEP for any 1 ≤ q < p. The second main new result here is the extension of Theorem 1.15 to the bidisc. p 2 Theorem 1.16. Let S be a dual bounded sequence in H (D ), such that the associated measure χS is Carleson. Then S is Hs(D2) interpolating with the LEP, for any s ∈ [1,p[. 2 First main result. We have the easy lemma. Lemma 2.1. Let S be a dual bounded sequence in Hp(Dn), then for any 1 ≤ q ≤ p, S is dual bounded in Hq(Dn). Proof. p n To see this, just take ρa := γaka,r where γa is the dual sequence in H (D ). Then, −1 ′ ′ hρa,kb,q i = hγaka,r,kb,q i = γa(b)ka,r(b)kkbkp′ . But −1 ′ hγa,kb,p i = γa(b)kkbkp′ = δab, using −1 ′ δa,b = hγa,kb,p i = kkbkp′ γa(b). 2 −1/p′ −2 2 −1 Recall that kkakp = ((1 −|a| )) hence defining χa := kkak2 = ((1 −|a| )) we get kkakp′ = 2 1/p 1/p ((1 −|a| )) = χa .So we get 1 1 1 1 1 1/q q − p q − p + r′ ′ hρa,kb,q i = γa(b)ka,r(b)χb = δabχa ka,r(a)= δa,bχa ka(a). 2 −1 But ka(a)= kkak2 = χa so finally: 1 1 1 q − p + r′ −1 hρa,kb,q′ i = δa,bχa = δab 1 1 1 provided that r = q − p which is possible because 1 ≤ q ≤ p. p Dn It remains to prove that ∃C > 0 :: ∀a ∈ S, kρakHq (Dn) ≤ C. But because γa ∈ H ( ) and r Dn 1 1 1 ka,r ∈ H ( ) and q = r + p we get kρakq ≤kγakpkka,rkr ≤ C using that ′ ∀a ∈ S, kγakp ≤ C, kka,rkr ≤ C . This finishes the proof of the lemma. Now we are in position to prove our first main result. 4 Theorem 2.2. Let p> 2 and suppose that the sequence S of points in Dn is Hp(Dn) interpolating. Then the sequence S is Carleson. Proof. Suppose that the sequence S ⊂ Dn is Hp(Dn) interpolating. 2 n Because p ≥ 2 we have that S is dual bounded in H (D ) by Lemma 2.1. Call {ρa}a∈S the dual sequence to the normalised reproducing kernels ka,2. We have ∃C > 0 :: ∀a ∈ S, kρak2 ≤ C.
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