How to show that $f \in L^p[0,1]$ for all $1 \leq p <2 $ and $\lVert f \rVert_p$ uniformly bounded implies $f \in L^2[0,1]$?

Question

I have searched for several posts and it seems to me that if $f \in L^p[0,1]$ for all $1 \leq p <2 $ and $\lVert f \rVert_p$ is uniformly bounded with respect to $p$, then $f \in L^2[0,1]$.

However, I cannot find a way to prove it myself.

Could anyone please help me?

Answer 1

Apply Fatou's lemma with the sequence $$ f_n(x)= |f(x)|^{t_n}, $$ for any sequence $\{t_n\} \subset [1, 2)$, with $t_n \to 2$ as $n \to \infty$. Evidently, one has $\lim_n f_n = f^2$. You will conclude that $$ \int_0^1 f^2(x) \leq \liminf_n \|f\|_{t_n}^{t_n} \leq \sup_{p \in [1, 2)} \|f\|_p^p < \infty. $$ Hence $f \in L^2[0, 1]$.