[FORTHCOMING] Opérateurs de composition des espaces de Bergman harmonique et de Hardy vers les espaces de Bloch harmonique à poids

We characterize the boundedness and compactness of the composition operator $$$ C_{\psi}$$$, induced by an analytic self-map $$$\psi$$$ of the unit disk, acting from the weighted harmonic Bergman space $$$\mathcal{A}^{p}_{\beta}$$$, $$$1 ≤ p ≤ \infty$$$, into the harmonic Bloch spaces $$$\mathcal{B}^{\mu}_H$$$ and $$$\mathcal{B}^{\mu}_{H,0}$$$, where $$$\mu$$$ is a normal weight. The criteria are formulated both in terms of the action of $$$C_{\psi}$$$ on a family of normalized kernel-type test functions and in terms of explicit conditions on the symbol $$$\psi$$$. Analogous characterizations, with independent proofs, are obtained for $$$C_{\psi}$$$ acting from the harmonic Hardy space $$$\mathcal{H}^{q}$$$ into $$$\mathcal{B}^{\mu}_H$$$ and $$$\mathcal{B}^{\mu}_{H,0}$$$. The approach works directly with harmonic mappings and therefore also covers the endpoint cases $$$p=1$$$ and $$$q=1$$$, in which the analytic and co-analytic parts of a mapping in the space need not belong to the corresponding analytic space.

We characterize the boundedness and compactness of the composition operator $$$ C_{\psi}$$$, induced by an analytic self-map $$$\psi$$$ of the unit disk, acting from the weighted harmonic Bergman space $$$\mathcal{A}^{p}_{\beta}$$$, $$$1 ≤ p ≤ \infty$$$, into the harmonic Bloch spaces $$$\mathcal{B}^{\mu}_H$$$ and $$$\mathcal{B}^{\mu}_{H,0}$$$, where $$$\mu$$$ is a normal weight. The criteria are formulated both in terms of the action of $$$C_{\psi}$$$ on a family of normalized kernel-type test functions and in terms of explicit conditions on the symbol $$$\psi$$$. Analogous characterizations, with independent proofs, are obtained for $$$C_{\psi}$$$ acting from the harmonic Hardy space $$$\mathcal{H}^{q}$$$ into $$$\mathcal{B}^{\mu}_H$$$ and $$$\mathcal{B}^{\mu}_{H,0}$$$. The approach works directly with harmonic mappings and therefore also covers the endpoint cases $$$p=1$$$ and $$$q=1$$$, in which the analytic and co-analytic parts of a mapping in the space need not belong to the corresponding analytic space.

Composition operators harmonic Bloch space weighted harmonic Bergman space harmonic Hardy space normal weight

Composition operators harmonic Bloch space weighted harmonic Bergman space harmonic Hardy space normal weight