Hardy operator ($Au(x)=x^{-1}{\int }_0^{x}u(t)dt$)

The Hardy operator is the averaging operatoron functions over the positive half-line or an interval beginning at zero.

For $1<p<\infty$, the Hardy operator is bounded on $L^p(0,1)$ with operator norm at most $p/(p-1)$.

For $1<p<\infty$, if $u\in W^{1,p}(0,1)$ has zero trace at zero, thenIndeed, the one-dimensional Sobolev representative satisfies $u(x)={\int }_0^x{u}^{\prime }(t)dt$, so the claim is the Hardy averaging inequality applied to $u^{\prime }$.