Grafakos in his $\textit{Classical Fourier Analysis}$ formulates (see Exercise 4.5.2 therein) the following vector-valued version of the Riesz-Thorin interpolation theorem.
$\textbf{Theorem}$
Let $1\le p_0, q_0,p_1,q_1, r_0, s_0, r_1, s_1\le\infty$ and $\theta\in(0,1)$ satisfy \begin{align*} \frac{1-\theta}{p_0}+\frac{\theta}{p_1}&=\frac{1}{p},\qquad \frac{1-\theta}{q_0}+\frac{\theta}{q_1}=\frac{1}{q},\\ \frac{1-\theta}{r_0}+\frac{\theta}{r_1}&=\frac{1}{r},\qquad \frac{1-\theta}{s_0}+\frac{\theta}{s_1}=\frac{1}{s}, \end{align*} and let $T$ be a linear operator mapping $L^{p_0}(\mathbb{R}^n, \ell^{r_0})$ to $L^{q_0}(\mathbb{R}^n, \ell^{s_0})$ and $L^{p_1}(\mathbb{R}^n, \ell^{r_1})$ to $L^{q_1}(\mathbb{R}^n, \ell^{s_1})$. Then $T$ maps $L^{p}(\mathbb{R}^n, \ell^{r})$ to $L^{q}(\mathbb{R}^n, \ell^{s})$.
My $\textbf{question}$ is whether one can weaken the assumption that $T$ is linear? More precisely, does an analogous result hold for an operator of the form $$ T: \{f_j\}_{j\in\mathbb{N}}\rightarrow\{M f_j\}_{j\in\mathbb{N}}, $$ where $M$ is a sublinear operator i.e. $|M(f+g)(x)|\le |Mf(x)|+|Mg(x)|,\quad x\in\mathbb{R}^d$?
In the scalar-valued there is an interpolation theorem for sublinear operators and it goes by the name of Marcinkiewicz-Zygmund.
I would appreciate any hints or perhaps a reference to suitable literature.
