Composition operators on Hardy-Orlicz spaces

Composition Operators on Hardy–Orlicz Spaces on the Ball

“…Moreover, we have the following interpolation theorem, which is not general, but which will be sufficient for our purpose. It is nothing but [9 …”

“…Proposition 4.6 of [9] ensures that this latter condition is still equivalent to the fact that, for every (or equivalently one) β > 1, there exists a constant…”

“…To finish, Proposition 4.7 (2) of [9] establishes relations between these conditions: For any 1 < p < ∞, every function x −→ x p is an Orlicz function which satisfies the uniform ∇ 0 -Condition, (so ∇ 2 and ∇ 0 -conditions too) and the ∆ 2 -Condition. At the opposite side, for any a > 0 and b ≥ 1, x −→ e ax b − 1 belongs to the ∆ 2 -Class (and then to the uniform ∇ 0 -Class), yet not to the ∆ 2 -one.…”

“…We can also find precise and useful information in [9], such as other classes of Orlicz functions and their link together with.…”

“…Since 2006, they produced some papers on this subject, e.g. [9,10,7,8], in which they gave characterizations of boundedness and compactness of C φ on these spaces, for instance. Moreover, they have been interesting in the question of the existence of some Hardy-Orlicz spaces on which the compactness of C φ is equivalent to that on H ∞ .…”