Convergence in two $L^p$ spaces

Question

How do you argue that for a sequence of functions $\{f_n\}$, convergence in $L^{p_1}$ and $L^{p_2}$ implies convergence in $L^{p}$ for all $p \in [p_1,p_2]$?

I know that we can pass to a $\mu$-almost everywhere convergent subsequence $\{n_k\}$ so that $f_{n_k} \to f$, and I know that by Fatou's lemma we have that

$$\lVert f \rVert_p^p = \int |f^p| \leq \liminf_{k \to \infty} \int |f_{n_k}^p| \leq \liminf_{k \to \infty} \int |f_{n_k}^{p_1}|+ |f_{n_k}^{p_2}|,$$ where the latter term is finite, but I don't know how to conclude the argument. Is it the more gneeral version of the Dominated Convergence Theorem? I would greatly appreciate any help.

Edit: I don't see how the linked post answers my question, and I would appreciate some more detail as to how this argument can be applied and if the approach I suggested is dead wrong or not.

Does this answer your question? Proof of the log-convexity of $L^p$ norms — Anne Bauval, Dec 06 '22 at 16:29
@AnneBauval I don't immediately see how it does, and I would appreciate some more detail as to how this argument can be applied and if the approach I suggested is dead wrong or not. — i like math, Dec 06 '22 at 16:32
Are you assuming convergence to the same limiting $f$ in both spaces, or is that part of your question? — Yuval Peres, Dec 06 '22 at 16:35
@AnneBauval How do I get from this to an argument about sequences? — i like math, Dec 06 '22 at 16:44
It does, he already aknowledged. And ilikemath, note that (contrarily to what @AinvAchor wrote after my last hint) it answers your question not only for $\Bbb R$ but for any measured space, and also if $p_2=\infty.$ I think quasiduplicating an answer to a quasiduplicate question is not a good idea. — Anne Bauval, Dec 06 '22 at 19:02

score 2 · Accepted Answer · answered Dec 06 '22 at 16:50

I assume what you are asking is that if $f_n \to f$ in $L^{p_1}(\mathbb{R})$, and $f_n \to f$ in $L^{p_2}(\mathbb{R})$, then can we assert that $f_n \to f$ in $L^p(\mathbb{R})$ for $p_1 \leq p \leq p_2$. I believe that this is true (for the finite case, I did not check the infinite case). This is because $$\|f_n-f\|_p^p = \int |f_n-f|^p = \int_{\{|f_n-f| > 1\}} |f_n-f|^p + \int_{\{|f_n-f| \leq 1\}} |f_n-f|^p \\ \leq \int_{\{|f_n-f| > 1\}}|f_n-f|^{p_2} + \int_{\{|f_n-f| \leq 1\}} |f_n-f|^{p_1} \\ \leq \int |f_n-f|^{p_2} + \int |f_n-f|^{p_1} \\ = \|f_n-f\|_{p_2}^{p_2} + \|f_n-f\|_{p_1}^{p_1} \to 0,$$ as $n \to \infty$.

Convergence in two $L^p$ spaces

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