Order bounded composition operators on the Hardy spaces and the Nevanlinna class

We study the order boundedness of composition operators induced by holomorphic self-maps of the open unit disc D. We consider these operators first on the Hardy spaces ${\displaystyle H^p}$ 0 < p < ∞ and then on the Nevanlinna class N. Given a non-negative increasing function h on [0,∞[, a composition operator is said to be X,L_h-order bounded (we write (X,L_h)-ob) with ${\displaystyle X=H^p}$ or X = N if its composition with the map f ↦ f*, where f* denotes the radial limit of f, is order bounded from X into ${\displaystyle L_h}$. We give a complete characterization and a family of examples in both cases. On the other hand, we show that the (${\displaystyle N,{\log }^{+}L}$)-ob composition operators are exactly those which are Hilbert-Schmidt on ${\displaystyle H^2}$. We also prove that the (${\displaystyle N,L^q}$)-ob composition operators are exactly those which are compact from N into ${\displaystyle H^q}$.