Non-existence of surjections from $L^p$ to $L^q$ Let $1 \leq p, q \leq \infty$ and suppose there exists a continuous linear surjection
$$ T : L^p[0,1] \longrightarrow L^q[0,1]. $$
Does it necessarily follow that $q \leq p?$

In the case of sequence spaces $\ell_p$ the result holds if we swap $p,q$ by Pitt's theorem, which asserts that for $q<p$ any bounded linear operator $T:\ell_p \to \ell_q$ is compact. In the $L^p$ context I suspect the notions of type/cotype may relevant, but as a non-specialist it isn't obvious to me how these can be applied. This question is mainly out of curiosity, and any references would be appreciated.

It is true that for $1<p<\infty$, $L^p[0,1]$ contains a complemented subspace isomorphic to $L^2[0,1]$. The subspace is the closed linear span of the Rademacher functions. This result is due to Khintchine inequality.
For $p = 1, q > p$, it is true that there is no linear surjection from $L^p[0,1]$ onto $L^q[0,1]$. This is because $L^1[0,1]$ cannot contain infinite dimensional reflexive subspaces (a consequence of $L^1[0,1]$ having the Dunford Pettis Property)
