In this post Terence Tao exposes the tensor power trick, and leaves as an exercise to use this tecnique to prove Riesz-Thorin. This is what I managed to do (I will use the same notation as here):
First, note that $$ ||T||_{p_\vartheta\to q_\vartheta}\le ||T||_{p_0\to q_0}^{1-\vartheta}||T||_{p_1\to q_1}^\vartheta$$ holds for $T$ iff it holds for $cT$. Similarly, it holds for $\mu_2$ iff it holds for $k\mu_2$. Rescaling these two, we can suppose $||T||_{p_0\to q_0}=1=||T||_{p_1\to q_1}$. So we have to prove that $||T||_{p_\vartheta\to q_\vartheta}\le 1$. This is where I get stuck: the idea is to prove that, for $f\in L^{p_0}\cap L^{p_1}$, we have $$\int_{\Omega_2} |Tf|^{q_\vartheta}\le C\|f\|_{p_\vartheta}$$ From this, the tensor power trick implies easily that the ineuqlity still holds for $C=1$ and the result for general $f$ is obtained as in the usual proof of Riesz-Thorin (see e.g. Folland). I do not know how to prove that inequality. My first idea was to do something like $$\int_{\Omega_2}|Tf|^{q_\vartheta}\le \int_{\Omega_2}|Tf|^{q_0}+|Tf|^{q_1}\le ||f||_{p_0}^{q_0}+||f||_{p_1}^{q_1}$$ but this does not seem to lead anywhere.
I would appreciate either hints on how to continue this proof or on how to approach the problem in some different way (using the tensor power trick, I already know the proof with the $3$ lines lemma).