$\int_{0}^{2\pi } \sin (nx) p(x) dx\to 0$?

Question

Let $f\in L^{p} (0, 2 \pi) (1\leq p < \infty.)$

How to show: $I_n = \int_{0}^{2\pi } \sin (nx) f (x) dx\to 0$ as $n\to \infty.$ (In other words, $f_n(x) =\sin (nx)$ converges to 0 weakly in $L^{p'}$)

My Thoughts: I guess, I should prove for first for the dense class in $L^p.$ If suppose, $I_n = \int_{0}^{2\pi } \sin (nx) p(x) dx\to 0$ as $n\to \infty,$ where $p\in A$, $\bar{A}=L^p$ (dense class). But then ,how the general case follows?

How should I show: $\int_{0}^{2\pi} |\sin (nx)|^p dx >0 $?

Motivation: This says that $f_n$ converges to 0 in $L^p$ weakly but not strongly.

Answer 1

For he first question, prove it first for $\phi$ a $C^1$ function with compact support (integrate by parts.) Then $$\begin{align} \Bigl|\int_{0}^{2\pi}\sin(n\,x)\,f (x)\,dx\,\Bigr|& \le\Bigl|\int_{0}^{2\pi}\sin(n\,x)\phi(x)\,dx\,\Bigr|+\int_0^{2\pi}|f(x)-\phi(x)|\,dx\\ &\le\Bigl|\int_{0}^{2\pi}\sin(n\,x)\phi(x)\,dx\,\Bigr|+(2\,\pi)^{1-1/p}\Bigl(\int_0^{2\pi}|f(x)-\phi(x)|^p\,dx\Bigr)^{1/p}. \end{align} $$ For the second, $$ \int_{0}^{2\pi} |\sin(n\,x)|^p\,dx=\frac1n\int_{0}^{2n\pi} |\sin t|^p\,dt=\int_{0}^{2\pi} |\sin t|^p\,dt. $$