Multilinear variants of the Maurey factorization theorem

Abstract

We prove multilinear variants of the Maurey factorization theorem. Let n be a natural number, π = {A₁, …, A_k} a partition of the set {1, …, n}, 0 < q₁, …, q_k < ∞, 1/v_k = 1/q₁ + · · · + 1/q_k and 1 ≤ p, r < ∞ such that 1/p = 1/v_k + 1/r. Let U:X₁ × · · · × X_n → L_p (μ, Y) be a positive homogeneous operator in each variable and $V_1:\prod _{i\in A_1}{X}_i\to \left[0,\mathrm{\infty }\right),\dots$, $V_k:\prod _{i\in A_k}{X}_i\to \left[0,\mathrm{\infty }\right)$ be all positive homogeneous operators in each variable. We give the necessary and sufficient conditions that there exist g ∈ L_r (μ), g ≥ 0, ${∥g∥}_{L_r\left(\mu \right)}\le 1$ and $T:X_1\times \cdots \times X_n\to L_{v_k}\left(\mu ,Y\right)$ a positive homogeneous operator in each variable such that $U=M_g\circ T$ and for all (x₁, …, x_n) ∈ X₁ × · · · × X_n we have ${∥T\left(x_1,\cdots ,x_n\right)∥}_{L_{v_k}\left(\mu ,Y\right)}\le V_1\left({\left(x_i\right)}_{i\in A_1}\right)\cdots V_k\left({\left(x_i\right)}_{i\in A_k}\right).$

Keywords:

• Maurey factorization theory
• strong-type factorization theorems for multilinear operators
• vector valued norm inequalities
• spaces of operators

2010 Mathematics Subject Classifications:

• Primary 47B38
• 46E30
• Secondary46B28

Acknowledgments

We would like to express our gratitude to the referee for his/her careful reading of the manuscript, many valuable comments, and suggestions which have improved the final version of the paper.

Disclosure statement

No potential conflict of interest was reported by the author.