Let $1 \leq p < \infty$ and $f \in L^p(R)$. Define $f_n(x) = 1_{[n,n+1]}f(x)$ for each $n \in N$. I want to prove that:
(1) $f_n \in L^q$ for every $n$ and $q \in [1,p]$;
(2) $f_n \rightarrow 0$ in $L^q$ for all $q \in [1,p]$.
For (1), I am trying to understand why we need $q \in [1,p]$. In order to have $f_n$ summable, we must have $$\int_R|f_n(x)|^q = \int_R|1_{[n,n+1]}f(x)|^q = \int_R1_{[n,n+1]}^q|f(x)|^q < \infty.$$Now, $1_{[n,n+1]}^q \in L^r$ for all $r \in [1,\infty]$ and $|f(x)|^q \in L^{p/q}$ which can be easily checked. Therefore, we can use Holder's inequality with the exponents $p/q$ and $r$ such that $1/r + q/p = 1$ and we get that $f_n$ is summable for all those combinations of $(q,r)$: $$\int_R|f_n(x)|^q = \int_R |1_{[n,n+1]}|^q|f(x)|^q \leq \|f\|_p^q$$ because $\|1_{[n,n+1]}\|_* = 1$. Now, why is it that we need to have $q \in [1,p]$?
For (2) I am not understanding what tool to use since the bound that I get does not have $n$ in it. Maybe there is something wrong with the estimates.
Thank you in advance.
By the Hölder inequality we get $\|f_n\|_q\le \|f_n\|_p$ for $1\le q\le p.$ Next $$\sum_{n=1}^\infty \|f_n\|_q^p\le \sum_{n=1}^\infty \|f_n\|_p^p\le \|f\|_p^p$$ Hence $\|f_n\|_q\to 0.$