[Paper Review] Interpolation between H^p spaces and non-commutative generalizations, I

Gilles Pisier|ArXiv.org|Jun 4, 1991

Advanced Harmonic Analysis Research9 references25 citations

TL;DR

This paper presents an elementary proof that Hardy spaces $H^p$ on the unit disc are the real interpolation spaces between $H^1$ and $H^\infty$, using only the boundedness of the Hilbert transform and classical factorization. The method extends to non-commutative settings, establishing that $H^p(C_p)$ and $T_p$ are interpolation spaces between $H^1(C_1)$, $H^\infty(C_\infty)$ and $T_1$, $T_\infty$, respectively, with norms equivalent via the same K-functional as in $L_p$ spaces.

ABSTRACT

We give an elementary proof that the $H^p$ spaces over the unit disc (or the upper half plane) are the interpolation spaces for the real method of interpolation between $H^1$ and $H^\infty$. This was originally proved by Peter Jones. The proof uses only the boundedness of the Hilbert transform and the classical factorisation of a function in $H^p$ as a product of two functions in $H^q$ and $H^r$ with $1/q+1/r=1/p$. This proof extends without any real extra difficulty to the non-commutative setting and to several Banach space valued extensions of $H^p$ spaces. In particular, this proof easily extends to the couple $H^{p_0}(\ell_{q_0}),H^{p_1}(\ell_{q_1})$, with $1\leq p_0, p_1, q_0, q_1 \leq \infty$. In that situation, we prove that the real interpolation spaces and the K-functional are induced ( up to equivalence of norms ) by the same objects for the couple $L_{p_0}(\ell_{q_0}), L_{p_1}(\ell_{q_1})$. In another direction, let us denote by $C_p$ the space of all compact operators $x$ on Hilbert space such that $tr(|x|^p)

Motivation & Objective

• To provide a new, elementary proof of Peter Jones' theorem on $H^p$ as real interpolation space between $H^1$ and $H^\infty$.
• To extend the interpolation result to non-commutative settings, including operator spaces $C_p$ and triangular matrices $T_p$.
• To show that the real interpolation spaces for $H^p(C_p)$ and $T_p$ are characterized by the same K-functional as in $L_p$ spaces.
• To generalize the result to Banach space-valued $H^p$ spaces and establish equivalence of norms via interpolation.

Proposed method

• Use the 'square/dual/square' argument: prove K-closure for $(H^{2p}, H^{2q})$ implies K-closure for $(H^p, H^q)$ via pointwise product boundedness into $(H^p, H^q)_{1/2,\infty}$.
• Apply duality: $(H^p, H^q)$ is K-closed if and only if $(H^{p'}, H^{q'})$ is K-closed, with $1/p + 1/p' = 1$.
• Leverage Marcel Riesz's theorem on uniform $L_p$-boundedness of the Hilbert transform for $1 < p < \infty$ to establish K-closure for $(H^4, H^2)$, then descend via the square/dual/square chain.
• Use classical factorization: any $f \in H^p$ can be written as $gh$ with $g \in H^{2p}$, $h \in H^{2q}$, $1/p = 1/(2p) + 1/(2q)$, to link interpolation to product structures.
• Extend the proof to non-commutative settings using isometric embeddings $K_q: C_q(H) \to C_{q,\infty}(H \otimes \ell_2)$ and duality arguments on quotient spaces $T_1/S_1$, $T_\infty/S_\infty$.
• Apply the same framework to Banach space-valued $H^p(B)$, showing $H^p(B) = (H^1(B), H^\infty(B))_\theta$ via factorization and interpolation of multiplication operators.

Experimental results

Research questions

• RQ1 Can the real interpolation property of $H^p$ between $H^1$ and $H^\infty$ be proven without complex analysis or advanced tools, using only basic harmonic analysis?
• RQ2 Does the real interpolation structure of $H^p$ spaces extend to non-commutative operator spaces such as $C_p$ and $T_p$?
• RQ3 Is the K-functional for $H^p(C_p)$ equivalent to that of $L_p(C_p)$, and does it match the K-functional of the underlying $L_p$ couple?
• RQ4 Can the interpolation result for $H^p(B)$ with Banach space $B$-valued functions be established via factorization and bounded multiplication operators?
• RQ5 Can the distance to the subspace of upper triangular matrices in $C_1$ and $C_\infty$ be simultaneously realized by the same operator, as suggested by Kaftal-Larson-Weiss?

Key findings

• The space $H^p$ for $0 < p < \infty$ is the real interpolation space $(H^1, H^\infty)_\theta$ with $\theta = 1/p$, up to equivalence of norms.
• The proof relies solely on the boundedness of the Hilbert transform on $L_p$ for $1 < p < \infty$ and the classical factorization of $H^p$ functions into products of $H^{2p}$ and $H^{2q}$ functions.
• The real interpolation space $(H^1(C_1), H^\infty(C_\infty))_\theta$ is isomorphic to $H^p(C_p)$ for $1/p = 1 - \theta$, with norms equivalent to those in $L_p(C_p)$.
• The space $T_p$ of upper triangular matrices is the real interpolation space $(T_1, T_\infty)_\theta$, with the same equivalence of norms as in $L_p$ spaces.
• For any separable Hilbert space $H$, $H^p(C_p(H)) = (H^1(C_1(H)), H^\infty(B(H)))_\theta$ with $\theta = 1 - 1/p$, and the same holds for the $\widetilde{H}^p$-spaces.
• The distance to the subspace of upper triangular matrices in $C_1$ and $C_\infty$ can be simultaneously realized by the same operator, extending a result of Kaftal-Larson-Weiss to the full scale $p \in [1, \infty]$.