Weak convergence in $L^{p}$ examples

Question

The problem is from Stein and Shakarchi's Functional Analysis:

a) Let $f_{n}(x) = sin(2πnx), f_{n} ∈ L p ([0, 1])$ with $p ∈ [1, ∞)$. Show that $f_{n}$ converges to $0$ weakly in $L p$.

b) $f_{n}(x) = n^{1/p}χ(nx)$ in $Lp(R)$. Then $f_{n} → 0$ weakly if $p > 1$, but not when $p = 1$. Here$ χ$ denotes the characteristic function of $[0, 1]$.

c) $f_{n}(x) = 1 + sin(2πnx)$ in $L1([0, 1])$. Then $f_{n} → 1$ weakly also in $L1([0, 1])$, $||f_{n}||_{L1} = 1$, but $||f_{n} − 1||_{L1}$ does not converge to zero.

I know the solution for part b), I need help in a) and c), we may check them for simple functions.

Answer 1

Use the Riesz Representation theorem: Let $1 ≤ p < \infty.$ Suppose $\Lambda$ is a bounded linear functional on $L^p([0,1])$. Then there is a function $g \in L^q([0,1])\subseteq L^1([0,1])$, where $q$ is the conjugate of $p$, for which $\Lambda f=\int_0^1g(x)f(x)dx$ for each $f\in L^p([0,1])$. Then, $\Lambda f_n=\int_0^1g(x)\sin 2\pi nxdx$. The result now follows from the Riemann–Lebesgue lemma.

To prove that $\|f_{n} -1\|\nrightarrow 0$, note that $\int_0^1|\sin 2\pi nx|dx=2n\int_0^{1/2n}\sin 2\pi nxdx=\frac{2}{\pi}.$

Answer 2

For part a), first prove the claim for bounded smooth functions $\phi\in L^{p'}([0,1])$ using partial integration: $$\begin{align} \int_0^1f_n(x)\phi(x)dx&=-\frac{1}{2\pi n}\cos(2\pi nx)\phi(x)\Bigg \vert_0^1+\frac{1}{2\pi n}\int_0^1\cos(2\pi nx)\phi'(x)dx\\ &=\frac{1}{2\pi n}(\phi(0)-\phi(1))+\frac{1}{2\pi n}\int_0^1\cos(2\pi nx)\phi'(x)dx \end{align}$$ and then since $|\phi|\leq M$ and $\phi'$ is a smooth function on a compact set (thus bounded) so $$\left\vert\int_0^1f_n(x)\phi(x)dx\right\vert\leq \frac{2M}{2\pi n}+\frac{1}{2\pi n}\int_0^1|\phi'(x)|dx\to0$$ as $n\to \infty$.

For a general $g\in L^{p'}([0,1])$ let $(\phi_n)$ be a sequence of smooth and bounded functions such that $\lVert g-\phi_n\rVert_{p'}\to 0$ as $n\to\infty$ and then do the following: $$\begin{align} \left|\int_0^1f_n(x)g(x)dx\right|&=\left|\int_0^1 (f_n(x)\phi_n(x)+f_n(x)(g(x)-\phi_n(x)))dx\right|\\ &\leq \left|\int_0^1f_n(x)\phi_n(x)dx\right|+\int_0^1|g(x)-\phi_n(x)|dx \end{align}$$ And then use Hölder's inequality on the second term.