Prove that, for $p> 1$, $\lim_{n\to\infty} \|f_n\|_p = +\infty$

Question

I need some help with this one, a hint would be greatly appreciated:

Let $f_n \geq 0$ and $f_n \in L^1\quad \forall n \in \mathbb{N}$ such that $\|f_n\|_1 =1$ for all $n$.

Suppose also that for each $\delta >0$: $$\lim_{n \to \infty} \int_{\{|t|>\delta\}} f_n = 0$$

Prove that, for $p> 1$, $$\lim_{n\to\infty} \|f_n\|_p = +\infty$$

p.s: I had asked a similar question yesterday, but there were some typos in my textbook and the statement was false. This is the correct statement, according to my professor.

Answer 1

You can start from here: let $\delta > 0$ be fixed. Then $$ 1 = \int f_n = \int_{\{|t|\leq \delta\}} f_n + \int_{\{|t| > \delta\}} f_n \leq (2\delta)^{1/p'} \left(\int_{\{|t| \leq \delta\}} |f_n|^p\right)^{1/p} + \int_{\{|t| > \delta\}} f_n $$ $$ \leq (2\delta)^{1/p'} \|f_n\|_p + \int_{\{|t| > \delta\}} f_n. $$ If $L := \liminf_n \|f_n\|_p$ you get that, for every $\delta > 0$, $$ 1 \leq (2\delta)^{1/p'} L, $$ which clearly implies $L = +\infty$.