Composition operators on Hardy-Orlicz spaces

Abstract

We investigate composition operators on Hardy-Orlicz spaces when the Orlicz function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Psi"> <mml:semantics> <mml:mi mathvariant="normal"> Ψ </mml:mi> <mml:annotation encoding="application/x-tex">\Psi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> grows rapidly: compactness, weak compactness, to be <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -summing, order bounded, …, and show how these notions behave according to the growth of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Psi"> <mml:semantics> <mml:mi mathvariant="normal"> Ψ </mml:mi> <mml:annotation encoding="application/x-tex">\Psi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We introduce an adapted version of Carleson measure. We construct various examples showing that our results are essentially sharp. In the last part, we study the case of Bergman-Orlicz spaces.