If $g \in L^p(\mathbb{R})$ be a given non-trivial function, show that following sequences converge weakly in $L^p$ but not strongly in $L^p$.
(a) $g_k(x)=k^{1/p}g(kx)$.
(b) $h_k(x)=g(x+k)$.
I need to show that for every $f \in L^q$ where q is the conjugate exponent to $p$, we have $$\int k^{1/p}g(kx)f(x)\rightarrow \int u(x)f(x)$$ for some $u\in L^p$.
Similarly for part (b). I used the technique of changing of variables but I could not simplify the integral.
You need $1<p<\infty$. The arguments for a) and b) are similar. Here are some hints for a): Claim: the weak limit is $0$. Start with $\int k^{1/p} g(kx)f(x)dx=\int k^{1/p-1} g(y)f(\frac y k)dy$. Split the integral into integral over $|y| \leq M$ and $|y| >M$. Use Holder's inequality for the second part. Observe that the norm of $k^{1/p} g(x)$ in $L^{p}$is same as that of $g$. In view of this it is enough to prove that result for $f$ in some dense subset of $L^{q}$. The set of functions in $L^{q}$ that vanish in some neighborhood of $0$ is dense and for these functions the first term tends to $0$ (in fact it is identically $0$ for $k$ sufficiently large).