(q, p)-Mixing Bloch maps

Antonio Jiménez-Vargas; David Ruiz-Casternado

doi:10.1515/dema-2025-0134

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(q, p)-Mixing Bloch maps

Antonio Jiménez-Vargas and David Ruiz-Casternado

Published/Copyright: June 13, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

As a Bloch variant of ( q , p ) -mixing linear operators, we introduce the notion of ( q , p ) -mixing Bloch maps. We prove Pietsch’s domination theorem and Maurey’s splitting theorem in this Bloch context, following the corresponding results from the linear case.

Keywords: vector-valued holomorphic map; Bloch function; p-summing operator; (q, p)-mixing operator

MSC 2010: 30H30; 47B10; 46E15; 46E40; 47L20

1 Introduction

A reading of the known monographs authored by Defant and Floret [1], Diestel et al. [2], and Pietsch [3] shows that the concept of p -summing linear operators plays an important role in the theory of Banach spaces.

Given Banach spaces X and Y , let ℒ ( X , Y ) be the Banach space of all bounded linear operators from X into Y , with the operator canonical norm. As usual, X * stands for ℒ ( X , K ) and B X for the closed unit ball of X .

For any 1 ≤ p ≤ ∞ , an operator T ∈ ℒ ( X , Y ) is p -summing if there is a constant C ≥ 0 such that

∑ i = 1 n ∥ T ( x i ) ∥ p 1 p ≤ C sup x * ∈ B X * ∑ i = 1 n ∣ x * ( x i ) ∣ p 1 p , if 1 ≤ p < ∞ , max 1 ≤ i ≤ n ∥ T ( x i ) ∥ ≤ C sup x * ∈ B X * ( max 1 ≤ i ≤ n ∣ x * ( x i ) ∣ ) , if p = ∞ ,

for all n ∈ N and all x 1 , … , x n ∈ X . The linear space of all p -summing operators from X into Y is denoted by Π p ( X , Y ) , and the p -summing norm of T , denoted by π p ( T ) , is the infimum of such constants C . By [3, 17.3.1], ( Π ∞ ( X , Y ) , π ∞ ) = ( ℒ ( X , Y ) , ‖ ⋅ ‖ ) , where ‖ ⋅ ‖ is the operator canonical norm.

The theory of ideals of p -summing operators was first developed by Pietsch [3]. He also sketched a multilinear approach to this theory in [4]. In the survey [5] by Pellegrino et al., the reader will find a complete information on summability for homogeneous polynomials and multilinear operators.

In the holomorphic context, some articles by Cabrera-Padilla and the two authors of the present article [6], Matos [7] and Pellegrino [8] have followed this direction, where holomorphic variants of some known results of the linear theory were established.

Closely related to the class of p -summing operators, Pietsch introduced the class of ( q , p ) -mixing operators in [3, Chapter IV, 20], although he remarked that such operators appear implicitly in the thesis of Maurey [9].

Following [1, 32.1], an operator T ∈ ℒ ( X , Y ) is called ( q , p ) -mixing for 1 ≤ p , q ≤ ∞ if for all Banach spaces Z and all q -summing operators S ∈ Π q ( Y , Z ) , the composition S ∘ T ∈ Π p ( X , Z ) . In this case,

m ( q , p ) ( T ) = sup { π p ( S ∘ T ) : π q ( S ) ≤ 1 } ,

where the supremum is taken over all Banach spaces Z and all operators S ∈ Π q ( Y , Z ) such that S ∘ T ∈ Π p ( X , Z ) . The linear space of all ( q , p ) -mixing operators from X into Y , denoted by ℳ ( q , p ) ( X , Y ) and equipped with the norm m ( q , p ) , is a Banach space.

Some generalizations of the concept of ( q , p ) -mixing linear operator to the multilinear case have been obtained by Achour et al. [10] and Popa [11–13]. Among the extensions to non-linear settings, the Lipschitz context was considered by Achour et al. [10] and Chávez-Domínguez [14].

In this article, we introduce and analyse the notion of ( q , p ) -mixing Bloch maps from the open unit disc D ⊆ C into a complex Banach space X for any 1 ≤ p , q ≤ ∞ . Following the theory of the linear case, as presented by Pietsch [3, Chapter IV, 20] or Defant and Floret [1, Chapter III, 32], we extend some known results on the class of ( q , p ) -mixing linear operators between Banach spaces, to the class of ( q , p ) -mixing Bloch maps from D into Banach spaces.

Towards this end, we now are going to introduce some concepts and notations in the Bloch setting. Let us recall that a map f : D → X is holomorphic if for each a ∈ D , there exists

f ′ ( a ) ≔ lim z → a f ( z ) − f ( a ) z − a ∈ X .

If we denote by ℋ ( D , X ) the space of all holomorphic maps from D into X , a map f ∈ ℋ ( D , X ) is called Bloch if we can find C ≥ 0 so that ( 1 − ∣ z ∣ 2 ) ∥ f ′ ( z ) ∥ ≤ C for all z ∈ D .

The Bloch space ℬ ( D , X ) is the linear space of all maps f ∈ ℋ ( D , X ) so that

ρ ℬ ( f ) ≔ sup { ( 1 − ∣ z ∣ 2 ) ∥ f ′ ( z ) ∥ : z ∈ D } < ∞ ,

with the Bloch seminorm ρ ℬ . The normalized Bloch space ℬ ^ ( D , X ) is the Banach space of all Bloch maps f : D → X so that f ( 0 ) = 0 , with the Bloch norm ρ ℬ . It is usual to denote ℬ ^ ( D , C ) by ℬ ^ ( D ) . Also, ℋ ^ ( D , D ) stands for the set of all holomorphic functions h from D into itself for which h ( 0 ) = 0 . For a complete study on spaces of Bloch maps, we recommend the book [15] by Zhu for the scalar-valued case and the article [16] by Arregui and Blasco for the vector-valued case.

We recently introduced in [6] the property of p -summability for Bloch maps and obtained the Bloch analogue of the Pietsch’s celebrated domination/factorization theorem for p -summing linear operators. A map f ∈ ℋ ( D , X ) is said to be p -summing Bloch with 1 ≤ p ≤ ∞ if there is a constant C ≥ 0 such that

∑ i = 1 n ∣ λ i ∣ p ∥ f ′ ( z i ) ∥ p 1 p ≤ C sup g ∈ B ℬ ^ ( D ) ∑ i = 1 n ∣ λ i ∣ p ∣ g ′ ( z i ) ∣ p 1 p , if 1 ≤ p < ∞ , max 1 ≤ i ≤ n ∣ λ i ∣ ∥ f ′ ( z i ) ∥ ≤ C sup g ∈ B ℬ ^ ( D ) ( max 1 ≤ i ≤ n ∣ λ i ∣ ∣ g ′ ( z i ) ∣ ) , if p = ∞ ,

for any n ∈ N , λ 1 , … , λ n ∈ C and z 1 , … , z n ∈ D . The infimum of the constants C such that the aforementioned inequality holds, denoted π p ℬ ( f ) , defines a seminorm on Π p ℬ ( D , X ) (the linear space of all p -summing Bloch maps f : D → X ), and this seminorm is a norm on Π p ℬ ^ ( D , X ) (the linear subspace of all f ∈ Π p ℬ ( D , X ) so that f ( 0 ) = 0 ). For p = ∞ , ( Π ∞ ℬ ^ ( D , X ) , π ∞ ℬ ) = ( ℬ ^ ( D , X ) , ρ ℬ ) by [6, Proposition 1.1].

The Bloch variant of the concept of ( q , p ) -mixing linear operator can be introduced as follows.

Definition 1.1

Let 1 ≤ p , q ≤ ∞ , and let X be a complex Banach space. A map f ∈ ℋ ( D , X ) is said to be ( q , p ) -mixing Bloch if T ∘ f ∈ Π p ℬ ( D , Y ) for all complex Banach spaces Y and all operators T ∈ Π q ( X , Y ) . We set

m ( q , p ) ℬ ( f ) = sup { π p ℬ ( T ∘ f ) : π q ( T ) ≤ 1 } ,

where the supremum is taken over all Banach spaces Y and all operators T ∈ Π q ( X , Y ) with T ∘ f ∈ Π p ℬ ( D , Y ) . The linear space of all ( q , p ) -mixing Bloch maps from D into X is denoted by ℳ ( q , p ) ℬ ( D , X ) , and its subspace formed by of all those maps f such that f ( 0 ) = 0 by ℳ ( q , p ) ℬ ^ ( D , X ) .

We now describe the content of this article. We start by studying the structure of the space of ( q , p ) -mixing Bloch maps as an injective Banach normalized Bloch ideal. Then, two different characterizations of ( q , p ) -mixing Bloch maps are presented. The first one consists in an integral inequality inspired by Pietsch’s domination theorem, while the second one involves in ( q , p ) -mixed sequences influenced by Maurey’s splitting theorem. The invariance of ( q , p ) -mixing Bloch maps on D under Möbius group of D is also established. For an introduction to Möbius invariant function spaces, see [17] by Arazy et al.

2 Results

Certain spaces of ( q , p ) -mixing Bloch maps can be identified with some distinguished spaces of Bloch maps. Given two semi-normed spaces ( X , ρ X ) and ( Y , ρ Y ) , the inequality ( X , ρ X ) ≤ ( Y , ρ Y ) will indicate that X ⊆ Y and ρ Y ( x ) ≤ ρ X ( x ) for all x ∈ X .

Let us recall (see [6]) that a Banach ideal of normalized Bloch maps (or, in short, a Banach normalized Bloch ideal) is an assignment [ ℐ ℬ ^ , ‖ ⋅ ‖ ℐ ℬ ^ ] , which associates with every complex Banach space X , a subset ℐ ℬ ^ ( D , X ) of ℬ ^ ( D , X ) , and a function ‖ ⋅ ‖ ℐ ℬ ^ : ℐ ℬ ^ → R satisfying:

( ℐ ℬ ^ ( D , X ) , ‖ ⋅ ‖ ℐ ℬ ^ ) is a Banach space and ρ ℬ ( f ) ≤ ‖ f ‖ ℐ ℬ ^ for all f ∈ ℐ ℬ ^ ( D , X ) ,
For each g ∈ ℬ ^ ( D ) and x ∈ X , the map g ⋅ x : z ∈ D ↦ g ( z ) x ∈ X belongs to ℐ ℬ ^ ( D , X ) and ‖ g ⋅ x ‖ ℐ ℬ ^ = ρ ℬ ( g ) ∥ x ∥ ,
The ideal property: if f ∈ ℐ ℬ ^ ( D , X ) , h ∈ ℋ ^ ( D , D ) and T ∈ ℒ ( X , Y ) where Y is a complex Banach space, then T ∘ f ∘ h is in ℐ ℬ ^ ( D , Y ) and ‖ T ∘ f ∘ h ‖ ℐ ℬ ^ ≤ ‖ T ‖ ‖ f ‖ ℐ ℬ ^ .

A Banach normalized Bloch ideal [ ℐ ℬ ^ , ‖ ⋅ ‖ ℐ ℬ ^ ] is called:

injective if for any map f ∈ ℬ ^ ( D , X ) , any complex Banach space Y and any into linear isometry ι : X → Y , it holds that f ∈ ℐ ℬ ^ ( D , X ) and ∥ f ∥ ℐ ℬ ^ = ∥ ι ∘ f ∥ ℐ ℬ ^ whenever ι ∘ f ∈ ℐ ℬ ^ ( D , Y ) .

For each z ∈ D , the function f z : D → C defined by

f z ( w ) = ( 1 − ∣ z ∣ 2 ) w 1 − z ¯ w , ( w ∈ D ) ,

belongs to ℬ ^ ( D ) with p ℬ ( f z ) = 1 = ( 1 − ∣ z ∣ 2 ) f z ′ ( z ) . We will use these Bloch functions in the proof of the following result.

Proposition 2.1

Let 1 ≤ p , q ≤ ∞ , and let X be a complex Banach space. Then,

( ℳ ( q , p ) ℬ ( D , X ) , m ( q , p ) ℬ ) ≤ ( ℬ ( D , X ) , ρ ℬ ) ,
( ℳ ( q , p ) ℬ ^ ( D , X ) , m ( q , p ) ℬ ) = ( ℬ ^ ( D , X ) , ρ ℬ ) whenever q ≤ p ,
( ℳ ( ∞ , p ) ℬ ^ ( D , X ) , m ( ∞ , p ) ℬ ) = ( Π p ℬ ^ ( D , X ) , π p ℬ ) .

Proof

(i) If f ∈ ℳ ( q , p ) ℬ ( D , X ) , then T ∘ f ∈ Π p ℬ ( D , Y ) for all complex Banach spaces Y and all operators T ∈ Π q ( X , Y ) . Given a point z ∈ D fixed, Hahn–Banach theorem provides x z * ∈ B X * so that ∥ f ′ ( z ) ∥ = ∣ x z * ( f ′ ( z ) ) ∣ . Since [ Π q , π q ] is a Banach operator ideal by [3, 17.1.2, 17.3.1], the functional x z * ⊗ 1 : X → C , given by ( x z * ⊗ 1 ) ( x ) = x z * ( x ) if x ∈ X , belongs to Π q ( X , C ) with π q ( x z * ⊗ 1 ) = ∥ x z * ∥ ≤ 1 (see, for example, [2, p. 131]). Hence, ( x z * ⊗ 1 ) ∘ f ∈ Π p ℬ ( D , C ) and

( 1 − ∣ z ∣ 2 ) ∥ f ′ ( z ) ∥ = ( 1 − ∣ z ∣ 2 ) ∣ ( ( x z * ⊗ 1 ) ∘ f ) ′ ( z ) ∣ ≤ ρ ℬ ( ( x z * ⊗ 1 ) ∘ f ) ≤ π p ℬ ( ( x z * ⊗ 1 ) ∘ f ) ≤ m ( q , p ) ℬ ( f ) .

Therefore, f ∈ ℬ ( D , X ) and ρ ℬ ( f ) ≤ m ( q , p ) ℬ ( f ) .

(ii) Assume q ≤ p and let f ∈ ℬ ^ ( D , X ) . Let Y be a complex Banach space and T ∈ Π q ( X , Y ) . Then, T ∈ Π p ( X , Y ) with π p ( T ) ≤ π q ( T ) by [3, 17.3.9]. Given any n ∈ N , λ i ∈ C and z i ∈ D for i = 1 , … , n , we can take a constant C ≥ 0 such that

∑ i = 1 n ∣ λ i ∣ p ∥ T ( f ′ ( z i ) ) ∥ p 1 p ≤ C sup x * ∈ B X * ∑ i = 1 n ∣ λ i ∣ p ∣ x * ( f ′ ( z i ) ) ∣ p 1 p ,

and taking the infimum over all such constants C , we deduce that

∑ i = 1 n ∣ λ i ∣ p ∥ T ( f ′ ( z i ) ) ∥ p 1 p ≤ π p ( T ) sup x * ∈ B X * ∑ i = 1 n ∣ λ i ∣ p ∣ x * ( f ′ ( z i ) ) ∣ p 1 p .

On the other hand, given any x * ∈ B X * , we have

∑ i = 1 n ∣ λ i ∣ p ∣ x * ( f ′ ( z i ) ) ∣ p 1 p = ∑ i = 1 n ∣ λ i ∣ p 1 1 − ∣ z i ∣ 2 p ∣ x * ( ( 1 − ∣ z i ∣ 2 ) f ′ ( z i ) ) ∣ p 1 p ≤ ρ ℬ ( f ) ∑ i = 1 n ∣ λ i ∣ p ∣ f z i ′ ( z i ) ∣ p 1 p ,

where f z i ∈ B ℬ ^ ( D ) for i = 1 , … , n . It follows that

∑ i = 1 n ∣ λ i ∣ p ∥ ( T ∘ f ) ′ ( z i ) ∥ p 1 p = ∑ i = 1 n ∣ λ i ∣ p ∥ T ( f ′ ( z i ) ) ∥ p 1 p ≤ π p ( T ) sup x * ∈ B X * ∑ i = 1 n ∣ λ i ∣ p ∣ x * ( f ′ ( z i ) ) ∣ p 1 p ≤ π q ( T ) ρ ℬ ( f ) sup g ∈ B ℬ ^ ( D ) ∑ i = 1 n ∣ λ i ∣ p ∣ g ′ ( z i ) ∣ p 1 p ,

whenever p < ∞ . Similarly, we obtain

max 1 ≤ i ≤ n ∣ λ i ∣ ∥ ( T ∘ f ) ′ ( z i ) ∥ ≤ π q ( T ) ρ ℬ ( f ) sup g ∈ B ℬ ^ ( D ) ( max 1 ≤ i ≤ n ∣ λ i ∣ ∣ g ′ ( z i ) ∣ ) .

Hence, T ∘ f ∈ Π p ℬ ^ ( D , Y ) and π p ℬ ( T ∘ f ) ≤ π q ( T ) ρ ℬ ( f ) . Thus, f ∈ ℳ ( q , p ) ℬ ^ ( D , X ) and m ( q , p ) ℬ ( f ) ≤ ρ ℬ ( f ) .

(iii) Let f ∈ ℳ ( ∞ , p ) ℬ ^ ( D , X ) . Then, T ∘ f ∈ Π p ℬ ( D , Y ) for any complex Banach space Y and operator T ∈ Π ∞ ( X , Y ) = ℒ ( X , Y ) . Let n ∈ N , λ i ∈ C and z i ∈ D for i = 1 , … , n . For p < ∞ , as mentioned earlier, we can write

∑ i = 1 n ∣ λ i ∣ p ∥ f ′ ( z i ) ∥ p 1 p ≤ ρ ℬ ( f ) ∑ i = 1 n ∣ λ i ∣ p ∣ f z i ′ ( z i ) ∣ p 1 p ≤ π p ℬ ( Id X ∘ f ) sup g ∈ B ℬ ^ ( D ) ∑ i = 1 n ∣ λ i ∣ p ∣ g ′ ( z i ) ∣ p 1 p ≤ m ( ∞ , p ) ℬ ( f ) sup g ∈ B ℬ ^ ( D ) ∑ i = 1 n ∣ λ i ∣ p ∣ g ′ ( z i ) ∣ p 1 p .

We also deduce

max 1 ≤ i ≤ n ∣ λ i ∣ ∥ f ′ ( z i ) ∥ ≤ m ( ∞ , ∞ ) ℬ ( f ) sup g ∈ B ℬ ^ ( D ) ( max 1 ≤ i ≤ n ∣ λ i ∣ ∣ g ′ ( z i ) ∣ ) .

Thus, f ∈ Π p ℬ ^ ( D , X ) , with π p ℬ ( f ) ≤ m ( ∞ , p ) ℬ ( f ) .

Conversely, let f ∈ Π p ℬ ^ ( D , X ) . Let Y be a complex Banach space and T ∈ Π ∞ ( X , Y ) . Since [ Π p ℬ ^ , π p ℬ ] is a Banach-normalized Bloch ideal by [6, Proposition 1.2], it follows that T ∘ f ∈ Π p ℬ ^ ( D , Y ) and π p ℬ ( T ∘ f ) ≤ ∥ T ∥ π p ℬ ( f ) = π ∞ ( T ) π p ℬ ( f ) . Hence, f ∈ ℳ ( ∞ , p ) ℬ ^ ( D , X ) and m ( ∞ , p ) ℬ ( f ) ≤ π p ℬ ( f ) .□

From now on, unless otherwise, X will denote a complex Banach space and 1 ≤ p , q ≤ ∞ but, in view of Proposition 2.1, we will only prove the results whenever 1 ≤ p < q < ∞ .

Next result is a Bloch variant of a Pietsch’s result on ( q , p ) -mixing linear operators (see [3, 20.1.2, 20.1.7]). In light of Proposition 2.1 (iii), it is an extension of [6, Proposition 1.2].

Proposition 2.2

[ ℳ ( q , p ) ℬ ^ , m ( q , p ) ℬ ] is a Banach normalized Bloch ideal.

Proof

Let X be a complex Banach space. For the proofs of (P1) and (P2), let Y be a complex Banach space and T ∈ Π q ( X , Y ) .

(P1) ( ℳ ( q , p ) ℬ ^ ( D , X ) , m ( q , p ) ℬ ) is a Banach space and ρ ℬ ( f ) ≤ m ( q , p ) ℬ ( f ) for all f ∈ ℳ ( q , p ) ℬ ^ ( D , X ) .

This inequality has been proven in Proposition 2.1 (i). If f ∈ ℳ ( q , p ) ℬ ^ ( D , X ) and m ( q , p ) ℬ ( f ) = 0 , then ρ ℬ ( f ) = 0 , and so f = 0 .

Let f , g ∈ ℳ ( q , p ) ℬ ^ ( D , X ) . Then, T ∘ f , T ∘ g ∈ Π p ℬ ^ ( D , Y ) , and since ( Π p ℬ ^ ( D , Y ) , π p ℬ ) is a normed space, it follows that T ∘ ( f + g ) = T ∘ f + T ∘ g ∈ Π p ℬ ^ ( D , Y ) and

π p ℬ ( T ∘ ( f + g ) ) ≤ π p ℬ ( T ∘ f ) + π p ℬ ( T ∘ g ) ≤ π q ( T ) m ( q , p ) ℬ ( f ) + π q ( T ) m ( q , p ) ℬ ( g ) .

Therefore, f + g ∈ ℳ ( q , p ) ℬ ^ ( D , X ) , and taking supremum over all T ∈ Π q ( X , Y ) with π q ( T ) ≤ 1 yields m ( q , p ) ℬ ( f + g ) ≤ m ( q , p ) ℬ ( f ) + m ( q , p ) ℬ ( g ) .

Let λ ∈ C . Then, T ∘ ( λ f ) = λ ( T ∘ f ) ∈ Π p ℬ ^ ( D , Y ) and

π p ℬ ( T ∘ ( λ f ) ) = ∣ λ ∣ π p ℬ ( T ∘ f ) ≤ ∣ λ ∣ π q ( T ) m ( q , p ) ℬ ( f ) .

Hence, λ f ∈ ℳ ( q , p ) ℬ ^ ( D , X ) and m ( q , p ) ℬ ( λ f ) ≤ ∣ λ ∣ m ( q , p ) ℬ ( f ) . This implies that m ( q , p ) ℬ ( λ f ) = 0 = ∣ λ ∣ m ( q , p ) ℬ ( f ) if λ = 0 . For λ ≠ 0 , we have m ( q , p ) ℬ ( f ) = m ( q , p ) ℬ ( λ − 1 ( λ f ) ) ≤ ∣ λ ∣ − 1 m ( q , p ) ℬ ( λ f ) ; hence, ∣ λ ∣ m ( q , p ) ℬ ( f ) ≤ m ( q , p ) ℬ ( λ f ) , and so m ( q , p ) ℬ ( λ f ) = ∣ λ ∣ m ( q , p ) ℬ ( f ) . This proves that m ( q , p ) ℬ is a norm on ℳ ( q , p ) ℬ ^ ( D , X ) .

To show that this norm is complete, let ( f n ) n ≥ 1 be a Cauchy sequence in ( ℳ ( q , p ) ℬ ^ ( D , X ) , m ( q , p ) ℬ ) . Since ρ ℬ ( f n ) ≤ m ( q , p ) ℬ ( f n ) for all n ∈ N and ( ℬ ^ ( D , X ) , ρ ℬ ) is a Banach space, there exists a map f ∈ ℬ ^ ( D , X ) such that ρ ℬ ( f n − f ) → 0 as n → + ∞ , and clearly, ρ ℬ ( T ∘ f n − T ∘ f ) → 0 as n → + ∞ . On the other hand, the inequality

π p ℬ ( T ∘ f r − T ∘ f s ) = π p ℬ ( T ∘ ( f r − f s ) ) ≤ π q ( T ) m ( q , p ) ℬ ( f r − f s ) , ( r , s ∈ N ) ,

yields that ( T ∘ f n ) n ≥ 1 is a Cauchy sequence in ( Π p ℬ ^ ( D , Y ) , π p ℬ ) . Hence there is a g ∈ Π p ℬ ^ ( D , Y ) so that π p ℬ ( T ∘ f n − g ) → 0 as n → + ∞ . Since ρ ℬ ≤ π p ℬ on Π p ℬ ^ ( D , Y ) , we obtain that T ∘ f = g , and thus, f ∈ ℳ ( q , p ) ℬ ^ ( D , X ) and π p ℬ ( T ∘ f n − T ∘ f ) → 0 as n → + ∞ .

To prove that ( f n ) n ≥ 1 converges to f in ( ℳ ( q , p ) ℬ ^ ( D , X ) , m ( q , p ) ℬ ) , let ε > 0 . Hence, there exists m ∈ N such that m ( q , p ) ℬ ( f r − f s ) < ε ⁄ 2 for all r , s ≥ m . Consequently, π p ℬ ( T ∘ f r − T ∘ f r + n ) < π q ( T ) ε ⁄ 2 for all r ≥ m , n ∈ N and T ∈ Π q ( X , Y ) . Taking limits with n → + ∞ , it follows that π p ℬ ( T ∘ f r − T ∘ f ) ≤ π q ( T ) ε ⁄ 2 for all r ≥ m and T ∈ Π q ( X , Y ) . Taking supremum over all T ∈ Π q ( X , Y ) such that π q ( T ) ≤ 1 , we obtain that m ( q , p ) ℬ ( f r − f ) < ε for all r ≥ m .

(P2) g ⋅ x ∈ ℳ ( q , p ) ℬ ^ ( D , X ) with m ( q , p ) ℬ ( g ⋅ x ) = ρ ℬ ( g ) ∥ x ∥ for any g ∈ ℬ ^ ( D ) and x ∈ X .

Note that g ⋅ x ∈ Π p ℬ ^ ( D , X ) with π p ℬ ( g ⋅ x ) = ρ ℬ ( g ) ‖ x ‖ since [ Π p ℬ ^ , π p ℬ ] is a normed normalized Bloch ideal. Hence, T ∘ ( g ⋅ x ) = g ⋅ T ( x ) ∈ Π p ℬ ^ ( D , Y ) , with π p ℬ ( T ∘ ( g ⋅ x ) ) = ρ ℬ ( g ) ∥ T ( x ) ∥ , and thus, g ⋅ x ∈ ℳ ( q , p ) ℬ ^ ( D , X ) with

m ( q , p ) ℬ ( g ⋅ x ) = ρ ℬ ( g ) sup { ‖ T ( x ) ‖ : T ∈ Π q ( X , Y ) , π q ( T ) ≤ 1 } ≤ ρ ℬ ( g ) sup { π q ( T ) ∥ x ∥ : π q ( T ) ≤ 1 } = ρ ℬ ( g ) ‖ x ‖ .

Conversely, we have ρ ℬ ( g ) ‖ x ‖ = π p ℬ ( g ⋅ x ) ≤ m ( q , p ) ℬ ( g ⋅ x ) by Proposition 2.1 (i).

(P3) T ∘ f ∘ h ∈ ℳ ( q , p ) B ^ ( D , Y ) with m ( q , p ) ℬ ( T ∘ f ∘ h ) ≤ ∥ T ∥ m ( q , p ) ℬ ( f ) for all f ∈ ℳ ( q , p ) ℬ ^ ( D , X ) , T ∈ ℒ ( X , Y ) , and h ∈ ℋ ^ ( D , D ) .

Let Z be a complex Banach space and S ∈ Π q ( Y , Z ) . Then, S ∘ T ∈ Π q ( X , Z ) with π q ( S ∘ T ) ≤ π q ( S ) ∥ T ∥ by the ideal property of [ Π q , π q ] . Hence, S ∘ T ∘ f ∈ Π p ℬ ^ ( D , Z ) , with π p ℬ ( S ∘ T ∘ f ) ≤ π p ( S ∘ T ) m ( q , p ) ℬ ( f ) by the definition of [ ℳ ( q , p ) ℬ ^ , m ( q , p ) ℬ ] . Therefore, S ∘ T ∘ f ∘ h ∈ Π p ℬ ^ ( D , Z ) with π p ℬ ( S ∘ T ∘ f ∘ h ) ≤ π p ℬ ( S ∘ T ∘ f ) by the ideal property of [ Π p ℬ ^ , π p ℬ ] . Consequently, T ∘ f ∘ h ∈ ℳ ( q , p ) B ^ ( D , Y ) , and from π p ℬ ( S ∘ T ∘ f ∘ h ) ≤ ∥ T ∥ m ( q , p ) ℬ ( f ) for all S ∈ Π q ( Y , Z ) with π q ( S ) ≤ 1 , we infer that m ( q , p ) ℬ ^ ( T ∘ f ∘ h ) ≤ ∥ T ∥ m ( q , p ) ℬ ( f ) .□

Next, we will establish two different characterizations of ( q , p ) -mixing Bloch maps. The first one consists in an integral inequality influenced by Pietsch domination theorem for p -summing linear operators [3, 17.3.2].

Since ℬ ^ ( D ) is a dual Banach space (see [15]), let P ( B ℬ ^ ( D ) ) be the set of all Borel regular probability measures μ on ( B ℬ ^ ( D ) , w * ) , where w * denotes the weak* topology.

Theorem 2.3

Let f ∈ ℬ ^ ( D , X ) and C ≥ 0 . The following statements are equivalent:

f is ( q , p ) -mixing Bloch with m ( q , p ) ℬ ( f ) ≤ C .
For each measure μ ∈ P ( B X * ) , there is a measure ν ∈ P ( B ℬ ^ ( D ) ) such that
∫ B X * ∣ x * ( f ′ ( z ) ) ∣ q d μ ( x * ) 1 q ≤ C ∫ B ℬ ^ ( D ) ∣ g ′ ( z ) ∣ p d ν ( g ) 1 p ,
for all z ∈ D .
For any m , n ∈ N , λ 1 , … , λ m ∈ C , z 1 , … , z m ∈ D and x 1 * , … , x n * ∈ X * ,
∑ j = 1 m ∑ k = 1 n ∣ λ j ∣ q ∣ x k * ( f ′ ( z j ) ) ∣ q p q 1 p ≤ C ∑ k = 1 n ∥ x k * ∥ q 1 q sup g ∈ B ℬ ^ ( D ) ∑ j = 1 m ∣ λ j ∣ p ∣ g ′ ( z j ) ∣ p 1 p .
For any m ∈ N , λ 1 , … , λ m ∈ C , z 1 , … , z m ∈ D and measure μ ∈ P ( B X * ) ,
∑ j = 1 m ∫ B X * ∣ λ j ∣ q ∣ x * ( f ′ ( z j ) ) ∣ q d μ ( x * ) p q 1 p ≤ C sup g ∈ B ℬ ^ ( D ) ∑ j = 1 m ∣ λ j ∣ p ∣ g ′ ( z j ) ∣ p 1 p .

In this case, m ( q , p ) ℬ ( f ) is the minimum of such constants C in either (ii), (iii), or (iv).

Proof

( i ) ⇒ ( i i ) : Let f ∈ ℳ ( q , p ) ℬ ^ ( D , X ) , and let μ ∈ P ( B X * ) . Consider the map ι X : X → L q ( μ ) defined by ι X ( x ) ( x * ) = x * ( x ) for all x * ∈ B X * and x ∈ X . Clearly, ι X ∈ Π q ( X , L q ( μ ) ) and π q ( ι X ) ≤ 1 . Hence, ι X ∘ f ∈ Π p ℬ ^ ( D , L q ( μ ) ) . By the Pietsch domination theorem for p -summing Bloch maps [6, Theorem 1.4], there exists a measure ν ∈ P ( B ℬ ^ ( D ) ) such that

∥ ( ι X ∘ f ) ′ ( z ) ∥ ≤ π p ℬ ( ι X ∘ f ) ∫ B ℬ ^ ( D ) ∣ g ′ ( z ) ∣ p d ν ( g ) 1 p ,

for all z ∈ D , i.e.,

∫ B X * ∣ x * ( f ′ ( z ) ) ∣ q d μ ( x * ) 1 q ≤ π p ℬ ( ι X ∘ f ) ∫ B ℬ ^ ( D ) ∣ g ′ ( z ) ∣ p d ν ( g ) 1 p ,

for all z ∈ D , and thus, (ii) follows with C = π p ℬ ( ι X ∘ f ) ≤ m ( q , p ) ℬ ( f ) π q ( ι X ) ≤ m ( q , p ) ℬ ( f ) .

( i i ) ⇒ ( i i i ) : Let m , n ∈ N , λ 1 , … , λ m ∈ C , z 1 , … , z m ∈ D and x 1 * , … , x n * ∈ X * . We can suppose x k * ≠ 0 for some k ∈ { 1 , … , n } and define μ = ∑ k = 1 n α k δ k , where α k = ∣ ∣ x k * ∣ ∣ q ⁄ ( ∑ k = 1 n ∣ ∣ x k * ∣ ∣ q ) and δ k is the Dirac measure at x k * ⁄ ∣ ∣ x k * ∣ ∣ . Clearly, μ ∈ P ( B X * ) . Hence, there is a measure ν ∈ P ( B ℬ ^ ( D ) ) satisfying the inequality in (ii). Therefore,

∑ j = 1 m ∑ k = 1 n ∣ λ j ∣ q ∣ x k * ( f ′ ( z j ) ) ∣ q p q 1 p = ∑ k = 1 n ∥ x k * ∥ q 1 q ∑ j = 1 m ∫ B X * ∣ λ j ∣ q ∣ x * ( f ′ ( z j ) ) ∣ q d μ ( x * ) p q 1 p ≤ C ∑ k = 1 n ∥ x k * ∥ q 1 q ∑ j = 1 m ∫ B ℬ ^ ( D ) ∣ λ j ∣ p ∣ g ′ ( z j ) ∣ p d ν ( g ) 1 p ≤ C ∑ k = 1 n ∥ x k * ∥ q 1 q ∑ j = 1 m ∫ B ℬ ^ ( D ) ∣ λ j ∣ p 1 1 − ∣ z j ∣ 2 p d ν ( g ) 1 p = C ∑ k = 1 n ∥ x k * ∥ q 1 q ∑ j = 1 m ∣ λ j ∣ p ∣ f z j ′ ( z j ) ∣ p 1 p ≤ C ∑ k = 1 n ∥ x k * ∥ q 1 q sup g ∈ B ℬ ^ ( D ) ∑ j = 1 m ∣ λ j ∣ p ∣ g ′ ( z j ) ∣ p 1 p .

( i i i ) ⇒ ( i v ) : Condition (iii) means that all finitely supported measures μ ∈ P ( B X * ) satisfy the inequality in (iii). Since the set of all such measures is dense in P ( B X * ) for the topology σ ( C ( B X * ) * , C ( B X * ) ) , it follows that the inequality in (iv) holds for all μ ∈ P ( B X * ) .

( i v ) ⇒ ( i ) : Let Y be a complex Banach space and T ∈ Π q ( X , Y ) . By the Pietsch domination theorem for p -summing operators [3, 17.3.2], there exists a measure μ ∈ P ( B X * ) so that

∥ T ( x ) ∥ p ≤ π q ( T ) p ∫ B X * ∣ x * ( x ) ∣ q d μ ( x * ) p q ,

for all x ∈ X . Take m ∈ N , λ 1 , … , λ m ∈ C , and z 1 , … , z m ∈ D . The inequality above yields

∑ j = 1 m ∣ λ j ∣ p ∥ ( T ∘ f ) ′ ( z j ) ∥ p 1 p ≤ π q ( T ) ∑ j = 1 m ∫ B X * ∣ λ j ∣ q ∣ x * ( f ′ ( z j ) ) ∣ q d μ ( x * ) p q 1 p ,

which combined with the inequality in (iv) gives

∑ j = 1 m ∣ λ j ∣ p ∥ ( T ∘ f ) ′ ( z j ) ∥ p 1 p ≤ C π q ( T ) sup g ∈ B ℬ ^ ( D ) ∑ j = 1 m ∣ λ j ∣ p ∣ g ′ ( z j ) ∣ p 1 p .

Hence, T ∘ f ∈ Π p ℬ ^ ( D , Y ) and π p ℬ ( T ∘ f ) ≤ C π q ( T ) . Therefore, f ∈ ℳ ( q , p ) ℬ ^ ( D , X ) and m ( q , p ) ℬ ( f ) ≤ C .□

Now, the injectivity of the ideal of ( q , p ) -mixing Bloch mappings can be easily obtained.

Corollary 2.4

The Banach normalized Bloch ideal [ ℳ ( q , p ) ℬ ^ , m ( q , p ) ℬ ] is injective.

Proof

Let f ∈ ℬ ^ ( D , X ) , let Y be a complex Banach space, and let ι : X → Y be an into linear isometry such that ι ∘ f ∈ ℳ ( q , p ) B ^ ( D , Y ) . Let z j ∈ D and λ j ∈ C be for j = 1 , … , m , and x k * ∈ X * for k = 1 , … , n . By the Hahn–Banach theorem, for each k = 1 , … , n there exists z k * ∈ Y * such that z k * ∘ ι = x k * and ∥ z k * ∥ = ∥ x k * ∥ . An application of Theorem 2.3 (iii) yields that

∑ j = 1 m ∑ k = 1 n ∣ λ j ∣ q ∣ x k * ( f ( z j ) ) ∣ q p q 1 p = ∑ j = 1 m ∑ k = 1 n ∣ λ j ∣ q ∣ z k * ( ( ι ∘ f ) ( z j ) ) ∣ q p q 1 p ≤ m ( q , p ) ℬ ( ι ∘ f ) ∑ k = 1 n ∥ z k * ∥ q 1 q sup g ∈ B B ^ ( D ) ∑ j = 1 m ∣ λ j ∣ p ∣ g ( z j ) ∣ p 1 p = m ( q , p ) ℬ ( ι ∘ f ) ∑ k = 1 n ∥ x k * ∥ q 1 q sup g ∈ B B ^ ( D ) ∑ j = 1 m ∣ λ j ∣ p ∣ g ( z j ) ∣ p 1 p .

It follows that f ∈ ℳ ( q , p ) ℬ ^ ( D , X ) and m ( q , p ) ℬ ( f ) ≤ m ( q , p ) ℬ ( ι ∘ f ) again by Theorem 2.3 (iii). The converse inequality follows from (P3) in Proposition 2.2.□

An easy application of Theorem 2.3 and [3, 20.1.7] yields the following composition formula.

Corollary 2.5

Let 1 ≤ p < q < r ≤ ∞ and f ∈ ℬ ^ ( D , X ) . If f ∈ ℳ ( q , p ) ℬ ^ ( D , X ) and S ∈ ℳ ( r , q ) ( X , Y ) , then S ∘ f ∈ ℳ ( r , p ) ℬ ^ ( D , Y ) and m ( r , p ) ℬ ( S ∘ f ) ≤ m ( q , p ) ℬ ( f ) m ( r , q ) ( S ) .

Using Theorem 2.3 and the monotonicity of the p -norms for p ≥ 1 , we deduce the inclusions:

Corollary 2.6

Let 1 ≤ p 2 ≤ p 1 < q 1 < q 2 ≤ ∞ . Then,

□ ( ℳ ( q 2 , p 2 ) ℬ ^ ( D , X ) , m ( q 2 , p 2 ) ℬ ) ≤ ( ℳ ( q 1 , p 1 ) ℬ ^ ( D , X ) , m ( q 1 , p 1 ) ℬ ) .

In the sequel, we will prove a result with a certain relation to Pietsch’s theorem for ( q , p ) -mixing operators (see [3, 20.1.10]).

Theorem 2.7

Let 1 ≤ p , q , r ≤ ∞ , with 1 ⁄ p = 1 ⁄ q + 1 ⁄ r . Then, Π p ℬ ^ ( D , X ) ⊆ ℳ ( q , p ) ℬ ^ ( D , X ) and m ( q , p ) ℬ ( f ) ≤ ρ ℬ ( f ) p q π p ℬ ( f ) p r for all f ∈ Π p ℬ ^ ( D , X ) .

Proof

Let f ∈ Π p ℬ ^ ( D , X ) . Given a complex Banach space Y and T ∈ Π q ( X , Y ) , we have that T ∘ f ∈ Π p ℬ ^ ( D , Y ) , with π p ℬ ( T ∘ f ) ≤ ∥ T ∥ π p ℬ ( f ) ≤ π q ( T ) π p ℬ ( f ) by [6, Proposition 1.2]. Hence, f ∈ ℳ ( q , p ) ℬ ^ ( D , X ) and m ( q , p ) ℬ ( f ) ≤ π p ℬ ( f ) . In order to obtain the inequality of the statement, let m ∈ N , λ 1 , … , λ m ∈ C and z 1 , … , z m ∈ D . Given μ ∈ P ( B X * ) , we obtain

∑ j = 1 m ∫ B X * ∣ λ j ∣ q ∣ x * ( f ′ ( z j ) ) ∣ q d μ ( x * ) p q 1 p ≤ ∑ j = 1 m ∫ B X * ( ∣ λ j ∣ ∣ x * ( f ′ ( z j ) ) ∣ ) p d μ ( x * ) p q ( ∣ λ j ∣ ∥ f ′ ( z j ) ∥ ) ( q − p ) p q 1 p ≤ ∑ j = 1 m ∫ B X * ( ∣ λ j ∣ ∣ x * ( f ′ ( z j ) ) ∣ ) p d μ ( x * ) 1 q ∑ j = 1 m ( ∣ λ j ∣ ∥ f ′ ( z j ) ∥ ) p 1 r ≤ ρ ℬ ( f ) p q π p ℬ ( f ) p r sup g ∈ B ℬ ^ ( D ) ∑ j = 1 m ∣ λ j ∣ p ∣ g ′ ( z j ) ∣ p 1 p ,

where we have applied Hölder’s inequality using that ( q − p ) r ⁄ q = p , and the inequalities

∑ j = 1 m ∫ B X * ∣ λ j ∣ p ∣ x * ( f ′ ( z j ) ) ∣ p d μ ( x * ) 1 q ≤ ∑ j = 1 m ∫ B X * ∣ λ j ∣ p ρ ℬ ( f ) 1 − ∣ z j ∣ 2 p d μ ( x * ) 1 q = ρ ℬ ( f ) p q ∑ j = 1 m ∫ B X * ∣ λ j ∣ p ∣ f z j ′ ( z j ) ∣ p d μ ( x * ) 1 q ≤ ρ ℬ ( f ) p q sup g ∈ B ℬ ^ ( D ) ∑ j = 1 m ∣ λ j ∣ p ∣ g ′ ( z j ) ∣ p 1 q

and

∑ j = 1 m ∣ λ j ∣ p ∥ f ′ ( z j ) ∥ p 1 r ≤ ∑ j = 1 m ∣ λ j ∣ p ρ ℬ ( f ) 1 − ∣ z j ∣ 2 p 1 r ≤ ∑ j = 1 m ∣ λ j ∣ p π p ℬ ( f ) p ∣ f z j ′ ( z j ) ∣ p 1 r ≤ π p ℬ ( f ) p r sup g ∈ B ℬ ^ ( D ) ∑ j = 1 m ∣ λ j ∣ p ∣ g ′ ( z j ) ∣ p 1 r .

Then, m ( q , p ) ℬ ( f ) ≤ ρ ℬ ( f ) p q π p ℬ ( f ) p r follows applying Theorem 2.3.□

Following the proof of [14, Proposition 4.2], one can show that if 1 ≤ p < q < ∞ and 1 ⁄ p = 1 ⁄ q + 1 ⁄ r , then

sup ∑ j = 1 n ∫ B X * ∣ λ j ∣ q ∣ x * ( x j ) ∣ q d μ ( x * ) p q 1 p : μ ∈ P ( B X * ) = inf ∑ j = 1 n ∣ λ j ∣ r 1 r sup x * ∈ B X * ∑ j = 1 n ∣ x * ( x j ) ∣ q 1 q : λ 1 , … , λ n ∈ C ,

for all n ∈ N and x 1 , … , x n ∈ X . Using this equality and Theorem 2.3, we next characterize ( q , p ) -mixing Bloch maps in terms of a Maurey splitting property [9]. Compare with [14, Corollary 4.3].

Theorem 2.8

Let 1 ≤ p , q , r ≤ ∞ with 1 ⁄ p = 1 ⁄ q + 1 ⁄ r . For f ∈ ℬ ^ ( D , X ) , the following are equivalent:

f is ( q , p ) -mixing Bloch.
There exists a constant C ≥ 0 such that
inf ∑ j = 1 n ∣ λ j ∣ r 1 r sup x * ∈ B X * ∑ j = 1 n ∣ x * ( f ′ ( z j ) ) ∣ q 1 q : λ 1 , … , λ n ∈ C ≤ C sup g ∈ B ℬ ^ ( D ) ∑ j = 1 n ∣ λ j ∣ p ∣ g ′ ( z j ) ∣ p 1 p ,
for all n ∈ N and z 1 , … , z n ∈ D .

In this case, m ( q , p ) ℬ ( f ) is the infimum of such constants C.□

We now present a result of interpolation type for ( q , p ) -mixing Bloch maps whose proof is based on [3, 20.1.13] (see also [14, Theorem 5.5]).

Theorem 2.9

Let 0 < θ < 1 , 1 ≤ p ≤ q 0 , q 1 ≤ ∞ , 1 ⁄ q = ( 1 − θ ) ⁄ q 0 + θ ⁄ q 1 , and f ∈ ℳ ( q 0 , p ) ℬ ^ ( D , X ) ∩ ℳ ( q 1 , p ) ℬ ^ ( D , X ) . Then, f ∈ ℳ ( q , p ) ℬ ^ ( D , X ) and m ( q , p ) ℬ ( f ) ≤ m ( q 0 , p ) ℬ ( f ) 1 − θ m ( q 1 , p ) ℬ ( f ) θ .

Proof

Define 1 ⁄ r = 1 ⁄ p − 1 ⁄ q , 1 ⁄ r 0 = 1 ⁄ p − 1 ⁄ q 0 , and 1 ⁄ r 1 = 1 ⁄ p − 1 ⁄ q 1 . Note that 1 ⁄ r = ( 1 − θ ) ⁄ r 0 + θ ⁄ r 1 . Let n ∈ N and z 1 , … , z n ∈ D . Given ε > 0 , for k = 0,1 , the necessary condition in Theorem 2.8 provides a finite set ( λ j , k ) j = 1 n in C * such that

∑ j = 1 n ∣ λ j , k ∣ r k 1 r k sup x * ∈ B X * ∑ j = 1 n ∣ x * ( f ′ ( z j ) ) ∣ q k 1 q k ≤ ( 1 + ε ) m ( q k , p ) ℬ ( f ) sup g ∈ B ℬ ^ ( D ) ∑ j = 1 n ∣ λ j ∣ p ∣ g ′ ( z j ) ∣ p 1 p .

Dividing by an appropriate constant, we can suppose that

∑ j = 1 n ∣ λ j , k ∣ r k 1 r k ≤ ( 1 + ε ) m ( q k , p ) ℬ ( f ) sup g ∈ B ℬ ^ ( D ) ∑ j = 1 n ∣ λ j ∣ p ∣ g ′ ( z j ) ∣ p 1 p

and

sup x * ∈ B X * ∑ j = 1 n ∣ x * ( f ′ ( z j ) ) ∣ q k 1 q k ≤ 1 .

Define λ j = λ j , 0 1 − θ λ j , 1 θ for j = 1 , … , n . Now, Hölder’s inequality yields

∑ j = 1 n ∣ λ j ∣ r 1 r ≤ ∑ j = 1 n ∣ λ j , 0 ∣ r 0 1 − θ r 0 ∑ j = 1 n ∣ λ j , 1 ∣ r 1 θ r 1 ≤ ( 1 + ε ) m ( q 0 , p ) ℬ ( f ) 1 − θ m ( q 1 , p ) ℬ ( f ) θ sup g ∈ B ℬ ^ ( D ) ∑ j = 1 n ∣ λ j ∣ p ∣ g ′ ( z j ) ∣ p 1 p .

On the other hand, writing ∣ x * ( f ′ ( z j ) ) ∣ q = ∣ x * ( f ′ ( z j ) ) ∣ ( 1 − θ ) q ∣ x * ( f ′ ( z j ) ) ∣ θ q , Hölder’s inequality gives

sup x * ∈ B X * ∑ j = 1 n ∣ x * ( f ′ ( z j ) ) ∣ q 1 q ≤ sup x * ∈ B X * ∑ j = 1 n ∣ x * ( f ′ ( z j ) ) ∣ q 0 1 − θ q 0 sup x * ∈ B X * ∑ j = 1 n ∣ x * ( f ′ ( z j ) ) ∣ q 1 θ q 1 ≤ 1 .

Therefore, we have

∑ j = 1 n ∣ λ j ∣ r 1 r sup x * ∈ B X * ∑ j = 1 n ∣ x * ( f ′ ( z j ) ) ∣ q 1 q ≤ ( 1 + ε ) m ( q 0 , p ) ℬ ( f ) 1 − θ m ( q 1 , p ) ℬ ( f ) θ sup g ∈ B ℬ ^ ( D ) ∑ j = 1 n ∣ λ j ∣ p ∣ g ′ ( z j ) ∣ p 1 p .

The sufficient condition in Theorem 2.8 shows now that f ∈ ℳ ( q , p ) ℬ ^ ( D , X ) with

m ( q , p ) ℬ ( f ) ≤ ( 1 + ε ) m ( q 0 , p ) ℬ ( f ) 1 − θ m ( q 1 , p ) ℬ ( f ) θ ,

and the result follows by the arbitrariness of ε .□

Let Aut ( D ) be the group of Möbius transformations of D . A linear space A ( D , X ) ⊆ ℬ ( D , X ) with a seminorm ρ A is called Möbius-invariant if it satisfies the two conditions: (i) there is a constant C > 0 so that ρ ℬ ( f ) ≤ C ρ A ( f ) for all f ∈ A ( D , X ) ; and (ii) f ∘ ϕ ∈ A ( D , X ) with ρ A ( f ∘ ϕ ) = ρ A ( f ) for all ϕ ∈ Aut ( D ) and f ∈ A ( D , X ) .

Invariance of ( q , p ) -mixing Bloch maps by Möbius transformations over D can be stated as follows.

Proposition 2.10

( ℳ ( q , p ) ℬ ( D , X ) , m ( q , p ) ℬ ) is Möbius-invariant.

Proof

(i) ( ℳ ( q , p ) ℬ ( D , X ) , m ( q , p ) ℬ ) ≤ ( ℬ ( D , X ) , ρ ℬ ) follows from Proposition 2.1 (i).

(ii) Let f ∈ ℳ ( q , p ) ℬ ( D , X ) and ϕ ∈ Aut ( D ) . Let Y be a complex Banach space and T ∈ Π q ( X , Y ) . Then, T ∘ f ∈ Π p ℬ ( D , Y ) , and since [ Π p ℬ ( D , Y ) , π p ℬ ] is Möbius-invariant by [6, Proposition 1.3], we have that T ∘ f ∘ ϕ ∈ Π p ℬ ( D , Y ) and π p ℬ ( T ∘ f ∘ ϕ ) = π p ℬ ( T ∘ f ) . Hence, f ∘ ϕ ∈ ℳ ( q , p ) ℬ ( D , X ) and m ( q , p ) ℬ ( f ∘ ϕ ) = m ( q , p ) ℬ ( f ) .□

We finish the article with some open problems. For 1 ≤ p < q < ∞ and r defined by 1 ⁄ p = 1 ⁄ q + 1 ⁄ r , the following relations are well known for such operator ideals (see [3, 20.1, 20.2]):

[ Π r , π r ] ≤ [ ℳ ( q , p ) , m ( q , p ) ]

and

[ Π q , π q ] ∘ [ ℳ ( q , p ) , m ( q , p ) ] ≤ [ Π p , π p ] ,

and, as a consequence, the Pietsch’s composition formula is deduced:

[ Π q , π q ] ∘ [ Π r , π r ] ≤ [ Π p , π p ] .

Theorem 2.7 provides a type of inclusion distinct to the first above. The following questions are raised in the setting of Bloch maps:

If f ∈ Π r ℬ ^ ( D , X ) , then f ∈ ℳ ( q , p ) ℬ ^ ( D , X ) with m ( q , p ) ℬ ( f ) ≤ π r ℬ ( f ) .
If T ∈ Π q ( X , Y ) and f ∈ Π r ℬ ^ ( D , X ) , then T ∘ f ∈ Π p ℬ ^ ( D , Y ) with π p ℬ ( T ∘ f ) ≤ π q ( T ) π r ℬ ( f ) .

Acknowledgments

Part of the research of the topic was developed during the participation of the authors to the congress “Approximation Theory and Special Functions” (ATSF 2024 Conference – 8th Series) which took place at Ankara (Turkey) from September 4–7, 2024. They are very grateful to the organizers for their kind hospitality.

Funding information: This research was partially supported by Ministerio de Ciencia e Innovación Grant PID2021-122126NB-C31 funded by MICIU/AEI/10.13039/501100011033 and by ERDF/EU; by Junta de Andalucía Grant FQM194; and by P_FORT_GRUPOS_2023/76, PPIT-UAL, Junta de Andalucía- ERDF 2021-2027. Programme: 54.A
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. A.J-V prepared the manuscript with contributions from the co-author.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: No new data were created during the study.

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Received: 2024-11-05

Revised: 2025-04-01

Accepted: 2025-04-29

Published Online: 2025-06-13

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/dema-2025-0134

Keywords for this article

vector-valued holomorphic map; Bloch function; p-summing operator; (q, p)-mixing operator

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