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

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.

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.

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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. https://aif.centre-mersenne.org/articles/10.5802/aif.721/

[1] Eric Amar, Interpolation Lp, to appear.

[2] D. Burkholder, R. Gundy and M. Silverstein, A maximal function characterization of the class Hp, Trans. A.M.S., 157 (1971), 137-157. | MR | Zbl

[3] L. Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math., 80 (1958), 921-930. | MR | Zbl

[4] L. Carleson and J. Garnett, Interpolating sequences and separation properties, Jour. d'Analyse Math., 28 (1975), 273-299. | Zbl

[5] R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103 (1976), 611-635. | MR | Zbl

[6] R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. A.M.S., 83 (1977), 569-645. | MR | Zbl

[7] P. R. Duren, Theory of Hp Spaces, Academic Press, New York, 1970. | MR | Zbl

[8] C. Fefferman and E. Stein, Hp spaces of several variables, Acta Math., 129 (1972), 137-193. | MR | Zbl

[9] J. Garnett, Interpolating sequences for bounded harmonic functions, Indiana U. Math. J., 21 (1971), 187-192. | MR | Zbl

[10] L. Hörmander, Lp estimates for (pluri-) subharmonic functions, Math. Scand., 20 (1967), 65-78. | Zbl

[11] E. M. Stein, Boundary Behavior of Holomorphic Functions of Several Complex Variables, Princeton University Press, Princeton, 1972. | MR | Zbl

[12] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970. | MR | Zbl

[13] N. Varopoulos, Sur un problème d'interpolation, C.R. Acad. Sci. Paris, Ser. A, 274 (1972), 1539-1542. | MR | Zbl

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