Exercise 19, § 6

Let $f$ be a numerical function $\geq 0$ defined in the space $\mathbf{R}_+^* = ]0, +\infty[$ and $p$-th power integrable for Lebesgue measure ($1 < p < +\infty$). Set $F(x) = \int_0^x f(t) \, dt$ for all $x > 0$.

a) Show that as $x$ tends to 0 or to $+\infty$, $F(x) = o(x^{(p-1)/p})$ (use Hölder’s inequality).

b) Show that the function $F(x)/x$ is $p$-th power integrable in $\mathbf{R}_+^*$ and that

(Hardy’s inequality). (Consider first the case that $f \in \mathscr{K}(\mathbf{R}_+^*)$; for every compact interval $[a, b] \subset \mathbf{R}_+^*$, obtain an upper bound for the integral

by integrating by parts and making use of Hölder’s inequality.)