Assume that $\sup_{p \in (1,2)}\|f\|_p<\infty$, then $f \in L^2$ and $\lim_{p \to 2^-}\|f\|_p=\|f\|_2$

Question

Assume that $\sup_{p \in (1,2)}\|f\|_p<\infty$, then $f \in L^2$ and $\lim_{p \to 2^-}\|f\|_p=\|f\|_2$.

I think I have a proof but I want to make sure it works. Consider $p_n \to 0$ where $p_n\geq 0$ Then the limit is equivalent to showing $$\lim_{n \to \infty} \int |f|^{2-p_n}=\int |f|^2$$Now we simply consider two sets, where $f>1$ and its complement. On both sets (individually) the sequence is monotonic, thus my MCT we get the equality of the limits. Now we know this limit is finite as $\limsup \int |f|^{2-p_n} <\infty$

Is this correct?

Answer 1

There are some errors but they can be fixed. You have to take $(p_n)$ decreasing to $0$ to use monotonicity. A more serious error is in applying Monotone Convergence Theorem for decreasing sequences. That is not valid, but if $f_n\geq 0$ decreases to $f$ and $\int f_n d\mu <\infty$ for some $n$ then $\int f_n d \mu\to \int f d\mu$. To see that this can be used here note that $2-p_n \geq 1.5$ for $n$ sufficiently large. That makes $\int_{|f|\leq 1} |f|^{2-p_n} d \mu <\infty$ for such $n$ and now you can use Monotone Convergence Theorem in the case $|f| \leq 1$.

Answer 2

We can use Fatou's Lemma to get $\int|f|^2<\infty$ by letting $p\to 2^-:$

$$\int|f|^2 = \int \lim |f|^{p} \le \liminf \int|f|^{p}.$$

The $\liminf$ on the right is finite because the $\|f\|_p$ are given to be uniformly bounded.