Notes on interpolation of Hardy spaces
Annales de l'Institut Fourier, Tome 42 (1992) no. 4, p. 875-889
Soit ${H}_{p}$ l’espace de Hardy de fonctions analytiques dans le disque unité $\left(0. On démontre dans cet article que pour chaque fonction $f\in {H}_{1}$ il existe un opérateur linéaire $T$, défini sur ${L}_{1}\left(\mathbf{T}\right)$, qui est simultanément borné de ${L}_{1}\left(\mathbf{T}\right)$ dans ${H}_{1}$ et de ${L}_{\infty }\left(\mathbf{T}\right)$ dans ${H}_{\infty }$, et tel que $T\left(f\right)=f$. Par conséquent, on obtient les résultats suivants $\left(1\le {p}_{0},{p}_{1}\le \infty \right)$:1) $\left({H}_{{p}_{0}},{H}_{{p}_{1}}\right)$ est un couple de Calderón-Mitjagin;2) pour tout foncteur d’interpolation $F$, on a $F\left({H}_{{p}_{0}},{H}_{{p}_{1}}\right)=H\left(F\left({L}_{{p}_{0}}\left(\mathbf{T}\right),{L}_{{p}_{1}}\left(\mathbf{T}\right)\right)\right)$, où $H\left(F\left({L}_{{p}_{0}}\left(\mathbf{T}\right),{L}_{{p}_{1}}\left(\mathbf{T}\right)\right)\right)$ désigne le sous-espace fermé de $F\left({L}_{{p}_{0}}\left(\mathbf{T}\right),{L}_{{p}_{1}}\left(\mathbf{T}\right)\right)$ des fonctions dont les coefficients de Fourier s’annulent sur l’ensemble des entiers négatifs.Ces résultats s’étendent aussi aux espaces de Hardy associés aux espaces invariants par réarrangement sur le cercle unité.
Let ${H}_{p}$ denote the usual Hardy space of analytic functions on the unit disc $\left(0. We prove that for every function $f\in {H}_{1}$ there exists a linear operator $T$ defined on ${L}_{1}\left(\mathbf{T}\right)$ which is simultaneously bounded from ${L}_{1}\left(\mathbf{T}\right)$ to ${H}_{1}$ and from ${L}_{\infty }\left(\mathbf{T}\right)$ to ${H}_{\infty }$ such that $T\left(f\right)=f$. Consequently, we get the following results $\left(1\le {p}_{0},{p}_{1}\le \infty \right)$:1) $\left({H}_{{p}_{0}},{H}_{{p}_{1}}\right)$ is a Calderon-Mitjagin couple;2) for any interpolation functor $F$, we have $F\left({H}_{{p}_{0}},{H}_{{p}_{1}}\right)=H\left(F\left({L}_{{p}_{0}}\left(\mathbf{T}\right),{L}_{{p}_{1}}\left(\mathbf{T}\right)\right)\right)$, where$H\left(F\left({L}_{{p}_{0}}\left(\mathbf{T}\right),{L}_{{p}_{1}}\left(\mathbf{T}\right)\right)\right)$ denotes the closed subspace of $F\left({L}_{{p}_{0}}\left(\mathbf{T}\right),{L}_{{p}_{1}}\left(\mathbf{T}\right)\right)$ of all functions whose Fourier coefficients vanish on negative integers.These results also extend to Hardy spaces associated to general rearrangement invariant spaces on the unit circle.
@article{AIF_1992__42_4_875_0,
author = {Xu, Quanhua},
title = {Notes on interpolation of Hardy spaces},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Louis-Jean},
volume = {42},
number = {4},
year = {1992},
pages = {875-889},
doi = {10.5802/aif.1313},
zbl = {0760.46060},
mrnumber = {94e:46135a},
language = {en},
url = {http://www.numdam.org/item/AIF_1992__42_4_875_0}
}

Xu, Quanhua. Notes on interpolation of Hardy spaces. Annales de l'Institut Fourier, Tome 42 (1992) no. 4, pp. 875-889. doi : 10.5802/aif.1313. http://www.numdam.org/item/AIF_1992__42_4_875_0/

 C. Bennett, R. Sharpley, Interpolation of operators, Pure and applied Mathematics, 129, Academic Press, 1988. | MR 89e:46001 | Zbl 0647.46057

 J. Bergh, J. Löffström, Interpolation spaces, An introduction, Berlin-Heidelberg-New York, Springer-Verlag, 1976. | Zbl 0344.46071

 J. Bourgain, Bilinear forms on H∞ and bounded bianalytic functions, Trans. Amer. Math. Soc., 286 (1984), 313-338. | MR 86c:46060 | Zbl 0572.46048

 A.P. Calderón, Spaces between L1 and L∞ and the theorem of Marcinkiewicz, Studia Math., 26 (1966), 273-299. | MR 34 #3295 | Zbl 0149.09203

 M. Cwikel, Monotonicity properties of interpolation spaces, Ark. Mat., 14 (1976), 213-236. | MR 56 #1095 | Zbl 0339.46024

 J.B. Garnett, Bounded analytic functions, Pure and Applied Mathematics 96, Academic Press, 1981. | MR 83g:30037 | Zbl 0469.30024

 G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, Cambridge, 1934 (2nd ed., 1952). | JFM 60.0169.01 | Zbl 0010.10703

 P.W. Jones, L∞-estimates for the ∂-problem in the half-plane, Acta Math., 150 (1983), 137-152. | MR 84g:35135 | Zbl 0516.35060

 P.W. Jones, On interpolation between H1 and H∞, Lect. Notes in Math. Springer, 1070 (1984), 143-151. | MR 86c:46021 | Zbl 0573.46044

 S.V. Kisliakov, Extension of (q,p)-summing operators defined on the disc-algebra with an appendix on Bourgain's analytic projections, preprint, 1990.

 S.V. Kisliakov, Truncating functions in weighted Hp and two theorems of J. Bourgain, preprint, 1989.

 S.V. Kisliakov, (q,p)-summing operators on the disc algebra and a weighted estimate for certain outer functions, LOMI, preprint E-11-89, Leningrad, 1989.

 J. Lindenstrauss, L. Tzafriri, Classical Banach spaces II, Berlin-New York, Springer-Verlag, 1979. | MR 81c:46001 | Zbl 0403.46022

 G.G. Lorentz, T. Shimogaki, Interpolation theorems for the pairs of spaces (Lp, L∞) and (L1, Lq), Trans. Amer. Math. Soc., 59 (1971), 207-221. | MR 52 #1347 | Zbl 0244.46044

 P.F.X. Müller, Holomorphic martingales and interpolation between Hardy spaces, to appear in J. d'Analyse Math.. | Zbl 0796.60051

 G. Pisier, Interpolation between Hp spaces and non-commutative generalizations, preprint, 1991.

 G. Sparr, Interpolation of weighted Lp-spaces, Studia Math., 62 (1973), 229-236. | MR 80d:46055 | Zbl 0393.46029