Strongly (p,σ)-absolutely continuous Bloch maps

Abstract

Given 1<p,r<∞ and 0≤σ<1 such that 1/r+(1-σ)/p^{∗}=1, we study the Banach normalized Bloch ideal (D_{p,σ}^{B}(D,X),d_{p,σ}^{B}) formed by all strongly (p,σ)-absolutely continuous Bloch maps from the complex unit open disc D into a complex Banach space X. Characterizations of such Bloch maps are established in terms of: (i) Pietsch domination, (ii) linearisation on G(D) (the Bloch-free Banach space over D), (iii) Bloch transposition, and (iv) Pietsch factorization. The invariance of such maps under Möbius transformations of D and their relation with compact Bloch maps are also addressed. Furthermore, we show that such space can be identified with the dual of the tensor product space G(D)⊗X^{∗} equipped with a suitable Bloch reasonable crossnorm ϱ_{p,σ}^{B}.