Composition operators from the Hardy space to the nth weighted-type space on the unit disk and the half-plane

The boundedness of the composition operator Cφf(z)=f(φ(z))Cφf(z)=f(φ(z)) from the Hardy space Hp(X),p>0, where X   is the upper half-plane Π+={z∈C:Imz>0} or the unit disk D={z∈C:|z|<1}D={z∈C:|z|<1} in the complex plane CC, to the n  th weighted-type space, where φφ is an analytic self-map of X, is characterized.