Exercise 8, § 6

Let $\mu$ be Lebesgue measure on the interval $X = [0, +\infty[$. a) Let $k > 1$, $h < k$ be two real numbers. Show that for every $n \geq 1$, and every number $p$ such that $1 \leq p \leq +\infty$, the functions $f_n(x) = n^h / (x + n)^k$ belong to $\mathscr{L}^p$; show that $N_p(f_n)$ tends to 0 with $1/n$ for $p > 1/(k - h)$, but that the sequence of the $N_p(f_n)$ is not bounded for $p < 1/(k - h)$. b) Let $k$ be a number $< 1$; show that for every $n > 1$ and every number $p$ such that $1 \leq p \leq +\infty$, the functions $g_n(x) = n^k e^{-nx}$ belong to $\mathscr{L}^p$; show that $N_p(g_n)$ tends to 0 with $1/n$ for $p < 1/k$, but that the sequence of the $N_p(g_n)$ is not bounded for $p > 1/k$. Deduce from a) and b) that if $1 \leq p < q \leq +\infty$, the topologies induced on $\mathscr{L}^p \cap \mathscr{L}^q$ by those of $\mathscr{L}^p$ and $\mathscr{L}^q$ are not comparable. c) If $\mu$ is Lebesgue measure on the interval $[0, 1]$, show similarly that if $p < q$, the topology of convergence in mean of order $q$ is strictly finer than the topology of convergence in mean of order $p$ (on $\mathscr{L}^q$).