= Endpoint kernel bound for a zero-trace Sobolev function
{title2=$|u(x)|^2\leq x(1-x)\int_0^1|u'|^2$}
For $u\in H_0^1(0,1)$, write $u(x)=(1-x)\int_0^xu'-x\int_x^1u'$. 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 $\int w|u|^2\leq[\int wx(1-x)]\int|u'|^2$. 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.
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