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The integral weight and superposition operators between Bloch-type spaces. (English) Zbl 1506.47093

Given a weight \(\mu\) (a continuous, positive, bounded, radial (\(\nu(z)=\mu (|z|)\)) function) on the unit disc, \(\Delta\), the authors use \(\mathcal{B}^\mu\) to denote the space of all holomorphic functions \(f\colon\Delta\to\mathbb{C}\) such that \(\sup_{z\in\Delta}\mu(z)|f'(z)|<\infty\). Endowed with the norm \(\|f\|:=|f(0)| +\sup_{z\in\Delta}\mu(z)|f'(z)|\), \(\mathcal{B}^\mu\) becomes a Banach space. A weight \(\mu\) is said to be typical if \(\lim_{|z|\to 1^-}\mu(z)=0\), while it is said to satisfy the growth condition if there is a holomorphic function \(H_\mu\colon\Delta\to\mathbb{C}\) such that (i) \(\mu(z)=H_\mu(|z^2|)\) for all \(z\) in \(\Delta\); (ii) there is \(L_\mu>0\) such that \(\mu(z)\le L_\mu|H_\mu({\bar a}z)|\) for all \(a,z\) in \(\Delta\). When \(\mu\) satifies the growth condition, the function \(\mu_I\colon\Delta\to\mathbb{R}\) defined by \(\mu_I(z)=\int_0^1\frac{dt}{H_\mu(t|z|^2)}\) is called the integral weight associated with \(\mu\). Given a function \(\phi \colon\mathbb{C}\to\mathbb{C}\) and a set of holomorphic functions, \(\mathcal{H}\), on the unit disc, the superpostion operator \(S_\phi\) is defined on \(\mathcal{H}\) by \(S_\phi(f)=\phi\circ f\). The authors assume that if \(\mu_1\) and \(\mu_2\) are weights on the unit disc such that \(\mu_1\) safisfies the growth condition. They show that (i) if \(\lim_{|z|\to 1^-}\frac{\mu_1(z)}{\mu_2(z)}=0\), then \(S_\phi\colon\mathcal{B}^{\mu_1}\to\mathcal{B}^{\mu_2}\) is bounded if and only if \(\phi\) is a constant function; (ii) if \(\lim_{|z|\to 1^-}\frac{\mu_1(z)}{\mu_2(z)}=L>0\), then \(S_\phi\colon\mathcal{B}^{\mu_1}\to\mathcal{B}^{\mu_2}\) is bounded if and only if \(\phi\) is an affine linear function; (iii) if the associated integral weight \(\mu_{1I}\) is typical and there is an increasing unbounded continuous function \(\varphi\colon (0,\infty)\to(0,\infty)\) such that \(L=\lim_{|z|\to 1^-}\frac{\mu_2(z)}{\mu_1(z)}\varphi\left(\frac{1}{\mu_{I1}(z)}\right)>0\), then \(S_\phi\colon\mathcal{B}^{\mu_1}\to\mathcal{B}^{\mu_2}\) is bounded if and only if for each \(\lambda\) in \((0,1)\) there are positive constants \(\delta\) and \(R\) such that \(|\phi'(w)|\le\delta\varphi(\lambda|w|)\) for \(|w|>R\).

MSC:

47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
30H30 Bloch spaces
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