[Paper Review] A note on complex interpolation and Calder\'on product of quasi-Banach spaces

Wen Yuan|arXiv (Cornell University)|May 22, 2014

Advanced Banach Space Theory6 references3 citations

TL;DR

This paper establishes that the inner complex interpolation space of two analytically convex quasi-Banach lattices coincides with the closure of their intersection in the Calderón product space, extending Shestakov's 1974 result for Banach lattices to the non-separable, quasi-Banach setting. The proof relies on Gagliardo-Peetre interpolation and a novel analyticity-based construction of interpolating functions.

ABSTRACT

In this paper, we prove that the inner complex interpolation of two quasi-Banach lattices coincides with the closure of their intersection in their Calder\'on product. This generalizes a classical result by Shestakov in 1974 for Banach lattices.

Motivation & Objective

• To resolve the open question of whether the inner and outer complex interpolation methods coincide for non-separable quasi-Banach lattices.
• To extend Shestakov’s 1974 result—valid for Banach lattices—beyond the separable case to general quasi-Banach lattices.
• To establish a precise characterization of the inner complex interpolation space in terms of the Calderón product.
• To provide a new proof technique using Gagliardo-Peetre interpolation and analytic function theory, distinct from Shestakov’s approach.
• To demonstrate the validity of the coincidence result in non-separable settings, such as Morrey spaces, where previous results do not apply.

Proposed method

• Define the inner complex interpolation space $[X_0, X_1]_\theta^i$ via functions in $F_0(X_0, X_1)$, which vanish on the boundary strip and take values in $X_0 \cap X_1$.
• Use the Gagliardo-Peetre interpolation space $\langle X_0, X_1 \rangle_\theta$ as an intermediate tool, defined via finite sums of functions in $X_0 \cap X_1$ with controlled growth in $X_0$ and $X_1$.
• Construct an analytic function $F(z) = \sum_{|k| \leq M} 2^{k(z - \theta)} f_k$ with $f_k \in X_0 \cap X_1$, which interpolates $f$ at $z = \theta$.
• Prove that $F \in F_0(X_0, X_1)$ by verifying analyticity, boundedness in $X_0 + X_1$, and strong continuity of boundary traces.
• Establish norm control: $\|F\|_F \lesssim \|f\|_{\langle X_0, X_1 \rangle_\theta}$, implying $\|f\|_{[X_0, X_1]_\theta^i} \lesssim \|f\|_{\langle X_0, X_1 \rangle_\theta}$.
• Conclude $\langle X_0, X_1 \rangle_\theta \hookrightarrow [X_0, X_1]_\theta^i$, and since $X_0 \cap X_1$ is dense in both, equality holds in the norm of the Calderón product.

Experimental results

Research questions

• RQ1 Does the inner complex interpolation of two analytically convex quasi-Banach lattices coincide with the closure of their intersection in the Calderón product?
• RQ2 Can Shestakov’s 1974 result for Banach lattices be extended to non-separable quasi-Banach lattices?
• RQ3 Is the Gagliardo-Peetre interpolation space equivalent to the inner complex interpolation space in the quasi-Banach lattice setting?
• RQ4 Does the coincidence result hold for non-separable spaces such as Morrey spaces?
• RQ5 Can the proof technique be adapted to avoid reliance on separability assumptions in complex interpolation theory?

Key findings

• The inner complex interpolation space $[X_0, X_1]_\theta^i$ is isometrically isomorphic to the closure of $X_0 \cap X_1$ in the Calderón product $X_0^{1-\theta} X_1^\theta$, i.e., $[X_0, X_1]_\theta^i = (X_0 \cap X_1)_{\|\cdot\|_{X_0^{1-\theta} X_1^\theta}}$.
• The result holds without requiring separability, thus generalizing Kalton and Mitrea’s 1998 result which required separability.
• The Gagliardo-Peetre interpolation space $\langle X_0, X_1 \rangle_\theta$ coincides with the inner complex interpolation space: $\langle X_0, X_1 \rangle_\theta = [X_0, X_1]_\theta^i$.
• The outer complex interpolation space $[X_0, X_1]_\theta$ equals the inner one when both $X_0$ and $X_1$ are either Banach spaces or separable quasi-Banach lattices.
• The result applies to non-separable spaces such as Morrey spaces: $[M_{p_0}^{u_0}, M_{p_1}^{u_1}]^i_\theta = M_p^u$ under the condition $u_0 p_1 = u_1 p_0$.
• The proof introduces a novel analytic function construction $F(z) = \sum 2^{k(z - \theta)} f_k$ that lies in $F_0$, enabling norm estimates without separability.