Multilinear variants of the Maurey factorization theorem. Issue 19 (16th December 2022)

• Record Type: Journal Article
• Authors: Popa, Dumitru
• Is Part Of: Linear & multilinear algebra. Volume 70:Issue 19(2022)
• Page Start: 3722
• Page End: 3733
• Publication Date: 2022-12-16
• Subjects: Maurey factorization theory -- strong-type factorization theorems for multilinear operators -- vector valued norm inequalities -- spaces of operators; Primary 47B38 -- 46E30 -- Secondary46B28; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2020.1853020
• Languages: English
• ISSNs: 0308-1087

Abstract

Abstract : We prove multilinear variants of the Maurey factorization theorem. Let n be a natural number, π = { A 1, …, A k } a partition of the set {1, …, n }, 0 < q 1, …, q k < ∞, 1/ v k = 1/ q 1 + · · · + 1/ q k and 1 ≤ p, r < ∞ such that 1/ p = 1/ v k + 1/ r . Let U : X 1 × · · · × X n → L p ( μ, Y ) be a positive homogeneous operator in each variable and V 1 : ∏ i ∈ A 1 X i → 0, ∞, …, V k : ∏ i ∈ A k X i → 0, ∞ be all positive homogeneous operators in each variable. We give the necessary and sufficient conditions that there exist g ∈ L r ( μ ), g ≥ 0, g L r μ ≤ 1 and T : X 1 × ⋯ × X n → L v k μ, Y a positive homogeneous operator in each variable such that U = M g ∘ T and for all ( x 1, …, x n ) ∈ X 1 × · · · × X n we have 1 T x 1, ⋯, x n L v k μ, Y ≤ V 1 x i i ∈ A 1 ⋯ V k x i i ∈ A k .