Following the book by Bennett and Sharpley, "Interpolation of Operators" (1988, Academic Press), I have found the following result, which is a version of the Riesz-Thorin Convexity Theorem applied to operators defined in the $L^p$-spaces, rather than on the simple functions.
Corollary 2.3. Suppose $1\le p_0,p_1,q_0,q_1\le\infty$. Suppose $0\le\theta\le 1$ and let $$\dfrac{1}{p}=\dfrac{1-\theta}{p_0}+\dfrac{\theta}{p_1},\qquad\dfrac{1}{q}=\dfrac{1-\theta}{q_0}+\dfrac{\theta}{q_1}.$$ Let $T$ be an admissible operator for the couples $(L^{p_0},L^{p_1})$ and $(L^{q_0},L^{q_1})$ and let $M_k$ denote the operator norm of $T$ considered as a bounded operator from $L^{p_k}$ to $L^{q_k}$, $k=0,1$. Thus, $$\lVert Tf\rVert_{q_k}\le M_k\lVert f\rVert_{p_k},\quad(f\in L^{p_k},k=0,1).$$ Then, $T$ is a bounded operator from $L^p$ to $L^q$: $$\lVert Tf\rVert_{q}\le M\lVert f\rVert_{p},\quad(f\in L^{p}),$$ and its operator norm $M$ satisfies $$M\le M_0^{1-\theta}M_1^\theta.$$
In this case by $L^p$ and $L^q$ the authors mean the spaces $L^p(S,\mu)$ and $L^q(R,\nu)$, where $(S,\mu)$ and $(R,\nu)$ are $\sigma$-finite measure spaces.
Moreover, $T:X_0+X_1\to Y_0+Y_1$ is said to be an admissible operator for the couples $(X_0,X_1)$ and $(Y_0,Y_1)$ if for every $i=0,1$ the restriction of $T$ to $X_i$ maps $X_i$ into $Y_i$ and it is a bounded operator from $X_i$ to $Y_i$.
My question is the following: does this result continue to hold when I replace $L^p$ with the Bochner space $L^p(S,X)$, $X$ being a Banach space? More precisely, if $T$ is an admissible operator for the couples $(L^{p_0}((S,\mu),X),L^{p_1}((S,\mu),X))$ and $(L^{q_0}((R,\nu),X),L^{q_1}((R,\nu),X))$ with operator norms $M_0$ and $M_1$, is it still true that $T$ is a bounded operator from $L^p((S,\mu),X)$ to $L^q((R,\nu),X)$ satisfying the same norm estimate?
I could not find any reference in which such a result was explicitely written. I would also be glad if someone could recommend some references about vector-valued interpolation.