Lipschitz continuity of the dilation of Bloch functions on the unit ball of a Hilbert space and applications

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2024年2月13日

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Alejandro Miralles

摘要

Let $B_E$ $B_{E}$ be the open unit ball of a complex finite- or infinite-dimensional Hilbert space. If f belongs to the space $\mathcal {B}(B_E)$ $B (B_{E})$ of Bloch functions on $B_E$ $B_{E}$ , we prove that the dilation map given by $x \mapsto (1-\Vert x\Vert ^2) \mathcal {R}f(x)$ $x \mapsto (1 - ‖ x ‖^{2}) R f (x)$ for $x \in B_E$ $x \in B_{E}$ , where $\mathcal {R}f$ $R f$ denotes the radial derivative of f, is Lipschitz continuous with respect to the pseudohyperbolic distance $\rho _E$ $ρ_{E}$ in $B_E$ $B_{E}$ , which extends to the finite- and infinite-dimensional setting the result given for the classical Bloch space $\mathcal {B}$ $B$ . To provide this result, we will need to prove that $\rho _E(zx,zy) \le |z| \rho _E(x,y)$ $ρ_{E} (z x, z y) \leq | z | ρ_{E} (x, y)$ for $x,y \in B_E$ $x, y \in B_{E}$ under some conditions on $z \in \mathbb {C}$ $z \in C$ . Lipschitz continuity of $x \mapsto (1-\Vert x\Vert ^2) \mathcal {R}f(x)$ $x \mapsto (1 - ‖ x ‖^{2}) R f (x)$ will yield some applications on interpolating sequences for $\mathcal {B}(B_E)$ $B (B_{E})$ which also extends classical results from $\mathcal {B}$ $B$ to $\mathcal {B}(B_E)$ $B (B_{E})$ . Indeed, we show that it is necessary for a sequence in $B_E$ $B_{E}$ to be separated to be interpolating for $\mathcal {B}(B_E)$ $B (B_{E})$ and we also prove that any interpolating sequence for $\mathcal {B}(B_E)$ $B (B_{E})$ can be slightly perturbed and it remains interpolating.