Real interpolation and transposition of certain function spaces

TL;DR

This paper provides a new proof for a lemma related to the $K$-functional in interpolation theory, characterizes certain function spaces, and extends the Varopoulos Lemma with potential applications in non-commutative analysis.

Contribution

It offers a novel proof of a generalized Varopoulos lemma, describes the $K$-functional for specific interpolation couples, and extends the lemma to conditional expectations, with implications for non-commutative spaces.

Findings

New proof of a generalized Varopoulos lemma

Characterization of the $K$-functional for specific interpolation couples

Extension of the Varopoulos Lemma to conditional expectations

Abstract

Our starting point is a lemma due to Varopoulos. We give a different proof of a generalized form this lemma, that yields an equivalent description of the $K$-functional for the interpolation couple $(X_0,X_1)$ where $X_0=L_{p_0,\infty}(\mu_1; L_q(\mu_2))$ and $X_1=L_{p_1,\infty}(\mu_2; L_q(\mu_1))$ where $0<q<p_0,p_1\le \infty$ and $(\Omega_1,\mu_1), (\Omega_2,\mu_2)$ are arbitrary measure spaces. When $q=1$, this implies that the space $(X_0,X_1)_{\theta,\infty}$ ($0<\theta<1$) can be identified with a certain space of operators. We also give an extension of the Varopoulos Lemma to pairs (or finite families) of conditional expectations that seems of independent interest. The present paper is motivated by non-commutative applications that we choose to publish separately.