 Endpoint kernel bound for a zero-trace Sobolev function

For $u\in H_0^1(0,1)$, write $u(x)=(1-x){\int }_0^x{u}^{\prime }-x{\int }_x^1{u}^{\prime }$. The squared $L^2$ norm of this piecewise constant test kernel is $x(1-x)$, so the Cauchy-Schwarz inequality gives the displayed pointwise estimate. Integrating against a nonnegative weight $w$ gives $\int w|u{|}^2\le [\int wx(1-x)]\int |u^{\prime }{|}^2$. In particular $w=x^2$ gives the constant $1/20$, useful for controlling a negative quadratic potential by derivative energy. The estimate is sharp at each fixed evaluation point, whose equality profile is a tent function, although its integrated constant need not be sharp.