Composition operators on generalized Hardy spaces

Juliette Leblond (1) , Elodie Pozzi (2) , Emmanuel Russ (3)

1 APICS - Analysis and Problems of Inverse type in Control and Signal processing
2 IMB - Institut de Mathématiques de Bordeaux
3 IF - Institut Fourier

Let $\Omega_1,\Omega_2\subset {\mathbb C}$ be bounded domains. Let $\phi:\Omega_1\rightarrow \Omega_2$ holomorphic in $\Omega_1$ and belonging to $W^{1,\infty}_{\Omega_2}(\Omega_1)$. We study the composition operators $f\mapsto f\circ\phi$ on generalized Hardy spaces on $\Omega_2$, recently considered in \cite{bfl, BLRR}. In particular, we provide necessary and/or sufficient conditions on $\phi$, depending on the geometry of the domains, ensuring that these operators are bounded, invertible, isometric or compact. Some of our results are new even for Hardy spaces of analytic functions.

Mots clés: composition operators; conjugate Beltrami equation; Generalized Hardy spaces

Domaines: Analyse fonctionnelle [math.FA]

Juliette Leblond, Elodie Pozzi, Emmanuel Russ. Composition operators on generalized Hardy spaces. Complex Analysis and Operator Theory, 2015, 119, pp.354-381. ⟨hal-01242032v2⟩