= Normalized squared Cauchy kernel
{title2=$f_a(z)=(1-|a|^2)/(1-\overline a z)^2$}
For $a\in\mathbb D$, this <holomorphic function> has <H1 space on the unit disk> norm one: its boundary modulus is the unit-mass <Poisson kernel on the circle>. The radial integrals are bounded by that boundary integral by <subharmonic> comparison and tend to it by continuity. Choosing $a$ at depth comparable to an arc's length makes $|f_a|$ bounded below by the reciprocal length throughout its <hyperbolic geodesic cap>. It tests necessity in the <Carleson embedding theorem on the disk>.
Back to article page
