no. 4
Harmonic interpolating sequences, L p and BMO
Annales de l'Institut Fourier, Tome 28 (1978) no. 4, pp. 215-228

Let (z ν ) be a sequence in the upper half plane. If 1<p≤∞ and if

yν1/pf(zν)=aν,ν=1,2,...(*)

has solution f(z) in the class of Poisson integrals of L p functions for any sequence (a ν )∈ℓ p , then we show that (z ν ) is an interpolating sequence for H ∞ . If f(z ν )=a ν , ν=1,2,... has solution in the class of Poisson integrals of BMO functions whenever (a ν )∈ℓ ∞ , then (z ν ) is again an interpolating sequence for H ∞ . A somewhat more general theorem is also proved and a counterexample for the case p≤1 is described.

Soit (z ν ) une suite de points du demi-plan supérieur ; si, pour p tel que 1<p≤∞, et pour toute suite (a ν ) ν∈N dans ℓ p (N) il existe une fonction f, intégrale de poisson d’une fonction de L p (R) qui vérifie :

yν1/pf(zν)=aν,ν=1,2,...(*)

alors nous montrons que (z ν ,ν∈N) est une suite d’interpolation pour H ∞ . De même, si on fait l’hypothèse qu’il existe une solution f, intégrale de Poisson d’une fonction de BMO qui vérifie (*) avec p=+∞ et (a ν ) dans ℓ ∞ (N), (z ν ) est encore une suite d’interpolation pour H ∞ .

Un théorème un peu plus général est prouvé et on donne un contre-exemple dans le cas où p≤1.

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     author = {Garnett, John B.},
     title = {Harmonic interpolating sequences, $L^p$ and {BMO}},
     journal = {Annales de l'Institut Fourier},
     pages = {215--228},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {28},
     number = {4},
     year = {1978},
     doi = {10.5802/aif.721},
     zbl = {0377.46044},
     mrnumber = {80g:30024},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.721/}
}
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Garnett, John B. Harmonic interpolating sequences, $L^p$ and BMO. Annales de l'Institut Fourier, Tome 28 (1978) no. 4, pp. 215-228. doi: 10.5802/aif.721

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