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

Motivated by new progress in the theory of ideals of Bloch maps, we introduce \((p,\sigma )\) $(p,\sigma )$ -absolutely continuous Bloch maps with \(p\in [1,\infty )\) $p\in [1,\infty )$ and \(\sigma \in [0,1)\) $\sigma \in [0,1)$ from the complex unit open disc \(\mathbb {D}\) $\mathbb{D}$ 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 \((p,\sigma )\) $(p,\sigma )$ -absolutely continuous Bloch zero-preserving maps from \(\mathbb {D}\) $\mathbb{D}$ into \(X^*\) $X^{\ast }$ under a suitable norm \(\pi ^{\mathcal {B}}_{p,\sigma }\) ${\pi }_{p,\sigma }^{\mathcal{B}}$ with the duals of the spaces of X-valued Bloch molecules on \(\mathbb {D}\) $\mathbb{D}$ equipped with the Bloch version of the \((p^*,\sigma )\) $(p^{\ast },\sigma )$ -Chevet–Saphar tensor norms.