= Proof outline of Hardy's integral inequality
{c}
For a nonnegative $f$, first set $g_n(x)=\min(f(x),n)\mathbf1_{(1/n,n)}(x)$ and $F_n(x)=\int_0^xg_n(t)\,dt$. These bounded <functions> vanish near zero and have <compact support>. Their averages have finite <Lp norms>: they vanish near zero and decay as $1/x$ beyond $n$. <Integration by parts> gives
$$
I_n:=\int_0^\infty(F_n(x)/x)^p\,dx=\frac{p}{p-1}\int_0^\infty(F_n(x)/x)^{p-1}g_n(x)\,dx\leq\frac{p}{p-1}I_n^{(p-1)/p}\|g_n\|_p.
$$
The boundary term $x^{1-p}F_n(x)^p$ vanishes at zero because $F_n$ is zero there, and at infinity because $F_n$ is eventually constant and $p>1$. The inequality is <Hölder's inequality> with <conjugate exponents> $p$ and $p/(p-1)$. Divide when $I_n>0$; the zero case is immediate. Since $g_n\uparrow f$ and $Hg_n\uparrow Hf$, the <monotone convergence theorem> proves the full bound without presupposing that $Hf$ has finite <Lp norm>. Apply it to $|f|$ for the signed or complex case.<Inequalities by Godfrey Harold Hardy and others>[Inequalities, second edition, §9.8(ii), truncation and integration by parts, printed pp. 242–243]{ref}
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