Notes on interpolation of Hardy spaces

Xu, Quanhua

Annales de l'Institut Fourier, Tome 42 (1992) no. 4, pp. 875-889.

Soit $H_p$ l’espace de Hardy de fonctions analytiques dans le disque unité $(0<p\le \infty )$. 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 $(1\le p_0,p_1\le \infty )$:

1) $(H_{p_0},H_{p_1})$ est un couple de Calderón-Mitjagin;

2) pour tout foncteur d’interpolation $F$, on a $F(H_{p_0},H_{p_1})=H\left(F\right(L_{p_0}\left(\mathbf{T}\right),L_{p_1}\left(\mathbf{T}\right)\left)\right)$, où $H\left(F\right(L_{p_0}\left(\mathbf{T}\right),L_{p_1}\left(\mathbf{T}\right)\left)\right)$ désigne le sous-espace fermé de $F\left(L_{p_0}\right(\mathbf{T}),L_{p_1}(\mathbf{T}\left)\right)$ 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 $(0<p\le \infty )$. 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 $(1\le p_0,p_1\le \infty )$:

1) $(H_{p_0},H_{p_1})$ is a Calderon-Mitjagin couple;

2) for any interpolation functor $F$, we have $F(H_{p_0},H_{p_1})=H\left(F\right(L_{p_0}\left(\mathbf{T}\right),L_{p_1}\left(\mathbf{T}\right)\left)\right)$, where

$H\left(F\right(L_{p_0}\left(\mathbf{T}\right),L_{p_1}\left(\mathbf{T}\right)\left)\right)$ denotes the closed subspace of $F\left(L_{p_0}\right(\mathbf{T}),L_{p_1}(\mathbf{T}\left)\right)$ 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.

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