(p, σ)-Absolute continuity of Bloch maps

Abstract

Motivated by new progress in the theory of ideals of Bloch maps, we introduce (p, σ)-absolutely continuous Bloch maps with p ∈ [1,∞) and σ ∈ [0, 1) from the complex unit open disc 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, σ)-absolutely continuous Bloch zero-preserving maps from D into X* under a suitable norm π^B_(p,σ) with the duals of the spaces of X-valued Bloch molecules on D equipped with the Bloch version of the (p*, σ)-Chevet–Saphar tensor norms.