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Equality of norms for a class of Bloch and symmetrically weighted Lipschitz spaces of vector valued functions and derivation inequalities for Pick functions. (English) Zbl 1452.32008

Summary: If \(\mathcal{X}\) and \(\mathcal{Y}\) are Banach spaces and \(f : \mathbb{B}_{\mathcal{X}} \rightarrow \mathcal{Y}\) is Fréchet differentiable on the open unit ball \(\mathbb{B}_{\mathcal{X}}\) of \(\mathcal{X}\), then for every operator monotone function \(\varphi :(- 1, 1) \rightarrow \mathbb{R}\), which satisfies \(\varphi^{\prime \prime} \geqslant 0\) on \([a, b)\), \[\sup_{a, b \in \mathbb{B}_{\mathcal{X}}, a \neq b} \frac{\| f(a) - f(b) \|}{\sqrt{\varphi^\prime(\| a \|)} \| a - b \| \sqrt{\varphi^\prime(\| b \|)}} = \sup_{a \in \mathbb{B}_{\mathcal{X}}} \frac{\| D f(a) \|}{\varphi^\prime(\| a \|)} . \tag{1}\] This generalizes Holland-Walsh-Pavlović criterium for the membership in Bloch type spaces for functions defined in the unit ball of a Banach space and taking values in another Banach space. We also established relations of the induced Bloch and Lipschitz spaces with other spaces of vector valued functions.

MSC:

32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
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