Real interpolation and transposition of certain function spaces

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.