I am having trouble calculating limit of a certain integral expression.
In some computation, I need to show that $\|f_n * g\|_{L^p} \to \|g\|_{L^1}$ as $n\to\infty$, where $f_n=(2n)^{-1/p}\chi_{(-n,n)}$ and $g$ is a positive integrable function.
Please help, thank you.
By a well known inequality (Young), $$ \|f_n*g\|_p\le\|f_n\|_p\|g\|_1, $$ and $$ \|f_n\|_p^p=\int_{\Bbb R}((2n)^{-1/p}\chi_{(-n,n)})^p=1. $$ So, is enough to prove that $$ \|f_n*g\|_p\ge\|g\|_1-\epsilon $$ for $n$ large enough.
Now, $$ \|f_n*g\|_p^p= {1\over 2n}\int_{\Bbb R}\left(\int_{x-n}^{x+n} g\right)^p\, dx $$ and the idea is: the interior integral $$\int_{x-n}^{x+n} g\approx\|g\|_1$$ when $n$ is large and $x$ is not very far form zero. Namely, if $\varepsilon>0$, then $\exists n_\varepsilon\in{\Bbb N}$ s.t. for $x\in[-n_\varepsilon,n_\varepsilon]$: $$ \|g\|_1-\epsilon/2\le\int_{-n_\epsilon}^{n_\epsilon}g $$
Now, let be $n\ge n_\epsilon$. Then, $$ n-n_\epsilon\le x\le n-n_\epsilon \implies [-n_\epsilon,n_\epsilon]\subset[x-n,x+n] $$ and $$||f_n * g||_p^p= \frac{1}{2n}\int_{\mathbb{R}}\left(\int_{x-n}^{x+n}g\right)^p dx\ge \frac{1}{2n}\int_{n_\epsilon-n}^{n-n_\epsilon}\left(\int_{x-n}^{x+n}g\right)^p dx\ge $$ $$ \frac{1}{2n}\int_{n_\epsilon-n}^{n-n_\epsilon}\left(\int_{-n_\epsilon}^{n_\epsilon}g\right)^p dx\ge \frac{1}{2n}\int_{n_\epsilon-n}^{n-n_\epsilon}(||g||_1-\epsilon)^p dx = {n-n_\epsilon\over n}(||g||_1-\epsilon/2)^p. $$ Now, make $n$ large enough.
EDIT: Interesting consequence: this proves the sharpness of Young's inequality in this case: Equality in Young's inequality for convolution.
It seems like Martin's answer would give a gap of 2. Choosing $n_\epsilon$ such that $$\|g\|_1-\varepsilon\le\int_{-n_\varepsilon/2}^{n_\varepsilon/2}g\le \int_{x-n_\varepsilon}^{x+n_\varepsilon}g$$ for $x\in[-n_{\epsilon}/2,n_{\epsilon}/2]$, we have that $$||f_{n_\epsilon} * g||_p^p=\frac{1}{2 n_\epsilon} \int_{\mathbb{R}}\left(\int_{x-n_\epsilon}^{x+n_\epsilon} g(y) dy\right)^p dx \geq \frac{1}{2 n_\epsilon} \int_{-n_\epsilon/2}^{n_\epsilon/2}\left(\int_{x-n_\epsilon}^{x+n_\epsilon} g(y) dy\right)^p dx $$ $$\geq \frac{1}{2 n_\epsilon} \int_{-n_\epsilon/2}^{n_\epsilon/2}\left(||g||_1-\epsilon\right)^p dx = \frac{1}{2} (||g||_1 - \epsilon)^p$$
where the first inequality is by non-negativity, second by the choice of $n_\epsilon$ so that the inequality in Martin's answer holds.
