$(p,\sigma )$-Absolute continuity of Bloch maps

Abstract

Motivated by new progress in the theory of ideals of Bloch maps, we introduce $(p,\sigma )$-absolutely continuous Bloch maps with $p\in [1,\infty )$ and $\sigma \in [0,1)$ from the complex unit open disc $\mathbb {D}$ into a complex Banach space X. We prove a Pietsch domination/factorization theorem for such Bloch maps that provides a reformulation of some results on both absolutely continuous (multilinear) operators and Lipschitz operators. We also identify the spaces of $(p,\sigma )$-absolutely continuous Bloch zero-preserving maps from $\mathbb {D}$ into $X^*$ under a suitable norm $\pi ^{\mathcal {B}}_{p,\sigma }$ with the duals of the spaces of X-valued Bloch molecules on $\mathbb {D}$ equipped with the Bloch version of the $(p^*,\sigma )$-Chevet–Saphar tensor norms.