Composition Operators on Generalized Hardy Spaces

Abstract

We study the composition operators $f\mapsto f\circ \phi$ on generalized analytic function spaces named generalized Hardy spaces, on bounded domains of $\mathbb {C}$, for holomorphic functions $\phi$ with uniformly bounded derivative. 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, or isometric.

Appendix: Factorization Results

We extend below [5], Thm 1] to the case of $n$-connected Dini smooth domains. Theorem 4 may be seen as a converse to the factorization result [6], Prop. 3.2], see [23] for more details. It is a straightforward generalization of a factorization result on generalized Hardy spaces on simply connected domains; however, the authors could not locate such a factorization for multiply connected Dini-smooth domains in the literature. For that reason, we give a short proof of this extension.

Theorem 4

Let $\Omega \subset \mathbb {C}$ be a $n$-connected Dini smooth domain. Let $F\in H^p(\Omega ),$ $\alpha \in L^{\infty }(\Omega ).$ There exists a function $s\in W^{1,r}(\Omega )$ for all $r\in (2,+\infty )$ such that $\hbox {tr Re}\,s=0$ on $\partial \Omega ,$ $w=e^sF$ and $\left\| s\right\| _{W^{1,r}(\Omega )}\lesssim \left\| \alpha \right\| _{L^{\infty }(\Omega )}.$

The proof is inspired by the one of [5], Thm 1]. By conformal invariance, it is enough to deal with the case where $\Omega =\mathbb {G}$ is a circular domain. We first assume that $\alpha \in W^{1,2}(\mathbb {G})\cap L^{\infty }(\mathbb {G})$. For all $\varphi \in W^{1,2}_{\mathbb {R}}(\mathbb {G})$, let $G(\varphi )\in W^{1,2}_{0,\mathbb {R}}(\mathbb {G})$ be the unique solution of

$$\begin{aligned} \Delta (G(\varphi ))=\text{ Im } \left( \partial (\alpha e^{-2i\varphi })\right) . \end{aligned}$$

We claim:

Lemma 11

The operator $G$ is bounded from $W^{1,2}_{\mathbb {R}}(\mathbb {G})$ from $W^{2,2}_{\mathbb {R}}(\mathbb {G})$ and compact from $W^{1,2}_{\mathbb {R}}(\mathbb {G})$ to $W^{1,2}_{\mathbb {R}}(\mathbb {G})$.

Proof

Let $\varphi \in W^{1,2}_{\mathbb {R}}(\mathbb {G})$. As in [5], $\partial (\alpha e^{-2i\varphi })\in L^2(\mathbb {G})$ and $\left\| \partial (\alpha e^{-2i\varphi })\right\| _{L^2(\mathbb {G})}\lesssim \left\| \varphi \right\| _{W^{1,2}(\mathbb {G})}$. It is therefore enough to show that the operator $T$, which, to any function $\psi \in L^2_{\mathbb {R}}(\mathbb {G})$, associates the solution $h\in W^{1,2}_{0,\mathbb {R}}(\mathbb {G})$ of $\Delta \psi =h$ is continuous from $L^2(\mathbb {G})$ to $W^{2,2}(\mathbb {G})$, which is nothing but the standard $W^{2,2}$ regularity estimate for second order elliptic equations (see [15], Sec. 6.3,Thm 4] and note that $\mathbb {G}$ is $C^2$). This shows that $G$ is bounded from $W^{1,2}_{\mathbb {R}}(\mathbb {G})$ from $W^{2,2}_{\mathbb {R}}(\mathbb {G})$, and its compactness on $W^{1,2}_{\mathbb {R}}(\mathbb {G})$ follows then from the Rellich–Kondrachov theorem. $\square$

Proof of Theorem 4

As in the proof of [5], Thm 1], Lemma 11 entails that $G$ has a fixed point in $W^{1,2}_{\mathbb {R}}(\Omega )$, which yields the conclusion of Theorem 4 when $\alpha \in W^{1,2}(\Omega )\cap L^{\infty }(\Omega )$, and a limiting procedure ends the proof. $\square$