Let $T$ be a an operator of the form $T(\{f_j\}_{j\in\mathbb{N}})=\{M f_j\}_{j\in\mathbb{N}}$, for some sublinear operator $M$. Assume that $T$ is bounded on $L^p(\ell^{q_0})$ and on $L^p(\ell^{q_1})$, for some $p,q_0,q_1\ge 1$.
$\textbf{Question:}$ Does it follow that $T$ is bounded on $L^p(\ell^{q})$ for every $q$ between $q_0$ and $q_1$?
The key point here is that $T$ is only assumed to be sublinear, not necessarily linear. In Grafakos' $\textit{Classical Fourier Analysis}$ (see Exercise 4.5.2 therein) there is a vector-valued version of the Riesz-Thorin interpolation theorem, but it requires $T$ to be linear. On the other hand, one has the classical Marcinkiewicz interpolation theorem, which works for sublinear operators, but only in the scalar-valued case.
I asked a version of this question on MO, but I haven't received any response yet. I would appreciate any hints or perhaps a reference to suitable literature.