$$(p,\sigma )$$ -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.