First Problem: Let $\{f_k\}$ be a sequence on $L^p(\mathbb{R})$ for $p \in [1,\infty)$ of the form $f_k (x) = k^{1/p} g(kx)$ for a fixed $g \in L^p(\mathbb{R})$. Show that $f_k \to 0$ weakly but $f_k$ does not converge strongly.
Idea:
We want to show that $\int_{\mathbb{R}} k^{1/p} g(kx) \phi(x) dx \to 0$ as $k \to \infty$ where we can without loss of generality assume that $\phi$ is a continuous compactly supported function (by density of $C_0(\mathbb{R}) \subset L^q (\mathbb{R})$ where $\frac{1}{p} + \frac{1}{q} = 1$). Then we have \begin{align*} \int_{\mathbb{R}} k^{1/p} g(kx) \phi(x) dx \leq k^{1/p} \max_{x \in \mathbb{R}} {[\phi(x)]} \int_{\mathbb{R}} g(kx) dx. \end{align*} Now I am unsure what to do. My guess is that since $g \in L^p(\mathbb{R}) \implies g \in L^1 (\mathbb{R})$, by the second fundamental theorem of calculus (for Lebesgue integrable functions) we can define \begin{align*} G(x) := \int_{[-\infty, x]} g(x) dx \end{align*} which is continuous and almost-everywhere differentiable with $G'(x) = g(x)$. In other words, we can compute the antiderivative of $g(kx)$ by using the reverse chain rule ($u$ substitution), which gives us \begin{align*} k^{1/p} \max_{x \in \mathbb{R}} {[\phi(x)]} \int_{\mathbb{R}} g(kx) dx = \frac{k^{1/p} \max_{x \in \mathbb{R}} {[\phi(x)]}}{k} \int_{\mathbb{R}} g(u) du \to 0 \text{ as } k \to \infty. \end{align*} My main concern is my integral is over $\mathbb{R}$ instead of $[-\infty, x]$, but is this a big deal? It seems that one can just split $\int_{\mathbb{R}} = G_1(x) + G_2 (x)$ where $G_1(x) = G(x)$ and $G_2(x) = \int_{[x, + \infty]} g(x) dx$ with $G_2'(x) = g(x)$ by the fundamental theorem of calculus again.
To show that $f_k$ does not converge strongly,just set $g(x) = 1_{E}(x)$ for a measurable set $E$ so that $f_k(x) = k^{1/p} 1_{E}(x)$ does not go to zero as $k \to \infty$.
Second problem: Let $\{f_k\}$ be a sequence on $L^p(\mathbb{R})$ for $p \in [1,\infty)$ of the form $f_k (x) = g(k + x)$ for a fixed $g \in L^p(\mathbb{R})$. Show that $f_k \to 0$ weakly but $f_k$ does not converge strongly.
Idea: By the same logic as before, \begin{align*} \int_{\mathbb{R}} g(k + x) \phi(x) dx \leq \max_{x \in \mathbb{R}}[\phi(x)] \int_{\mathbb{R}} g(k + x) dx, \end{align*} where $\phi$ is a continuous compactly supported function. The problem is doing $u$-substitution doesn't really do anything. My gut feeling tells me to approximate $g$ using continuous compactly supported functions, so that the shift sideways of g $g(k + x)$ goes to a place where $g = 0$.