no. 4
Notes on interpolation of Hardy spaces
Annales de l'Institut Fourier, Tome 42 (1992) no. 4, pp. 875-889.
corrigé par Erratum : Notes on interpolation of Hardy spaces

Soit H p l’espace de Hardy de fonctions analytiques dans le disque unité (0<p≤∞). On démontre dans cet article que pour chaque fonction f∈H 1 il existe un opérateur linéaire T, défini sur L 1 (T), qui est simultanément borné de L 1 (T) dans H 1 et de L ∞ (T) dans H ∞ , et tel que T(f)=f. Par conséquent, on obtient les résultats suivants (1≤p 0 ,p 1 ≤∞):

1) (H p 0 ,H p 1 ) est un couple de Calderón-Mitjagin;

2) pour tout foncteur d’interpolation F, on a F(H p 0 ,H p 1 )=H(F(L p 0 (T),L p 1 (T))), où H(F(L p 0 (T),L p 1 (T))) désigne le sous-espace fermé de F(L p 0 (T),L p 1 (T)) 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 (0<p≤∞). We prove that for every function f∈H 1 there exists a linear operator T defined on L 1 (T) which is simultaneously bounded from L 1 (T) to H 1 and from L ∞ (T) to H ∞ such that T(f)=f. Consequently, we get the following results (1≤p 0 ,p 1 ≤∞):

1) (H p 0 ,H p 1 ) is a Calderon-Mitjagin couple;

2) for any interpolation functor F, we have F(H p 0 ,H p 1 )=H(F(L p 0 (T),L p 1 (T))), where

H(F(L p 0 (T),L p 1 (T))) denotes the closed subspace of F(L p 0 (T),L p 1 (T)) 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},
     pages = {875--889},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {42},
     number = {4},
     year = {1992},
     doi = {10.5802/aif.1313},
     zbl = {0760.46060},
     mrnumber = {94e:46135a},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1313/}
}
TY  - JOUR
AU  - Xu, Quanhua
TI  - Notes on interpolation of Hardy spaces
JO  - Annales de l'Institut Fourier
PY  - 1992
SP  - 875
EP  - 889
VL  - 42
IS  - 4
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1313/
DO  - 10.5802/aif.1313
LA  - en
ID  - AIF_1992__42_4_875_0
ER  - 
%0 Journal Article
%A Xu, Quanhua
%T Notes on interpolation of Hardy spaces
%J Annales de l'Institut Fourier
%D 1992
%P 875-889
%V 42
%N 4
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1313/
%R 10.5802/aif.1313
%G en
%F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.1313/

[1] C. Bennett, R. Sharpley, Interpolation of operators, Pure and applied Mathematics, 129, Academic Press, 1988. | MR | Zbl

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

[3] J. Bourgain, Bilinear forms on H∞ and bounded bianalytic functions, Trans. Amer. Math. Soc., 286 (1984), 313-338. | MR | Zbl

[4] A.P. Calderón, Spaces between L1 and L∞ and the theorem of Marcinkiewicz, Studia Math., 26 (1966), 273-299. | MR | Zbl

[5] M. Cwikel, Monotonicity properties of interpolation spaces, Ark. Mat., 14 (1976), 213-236. | MR | Zbl

[6] J.B. Garnett, Bounded analytic functions, Pure and Applied Mathematics 96, Academic Press, 1981. | MR | Zbl

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

[8] P.W. Jones, L∞-estimates for the ∂-problem in the half-plane, Acta Math., 150 (1983), 137-152. | MR | Zbl

[9] P.W. Jones, On interpolation between H1 and H∞, Lect. Notes in Math. Springer, 1070 (1984), 143-151. | MR | Zbl

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

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

[12] 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.

[13] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces II, Berlin-New York, Springer-Verlag, 1979. | MR | Zbl

[14] 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 | Zbl

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

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

[17] G. Sparr, Interpolation of weighted Lp-spaces, Studia Math., 62 (1973), 229-236. | MR | Zbl

Cité par Sources :