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Disentanglement, multilinear duality and factorisation for nonpositive operators

Anthony Carbery, Timo S. Hänninen and Stefán Ingi Valdimarsson

Vol. 16 (2023), No. 2, 511–543
Abstract

In a previous work we established a multilinear duality and factorisation theory for norm inequalities for pointwise weighted geometric means of positive linear operators defined on normed lattices. In this paper we extend the reach of the theory for the first time to the setting of general linear operators defined on normed spaces. The scope of this theory includes multilinear Fourier restriction-type inequalities. We also sharpen our previous theory of positive operators.

Our results all share a common theme: estimates on a weighted geometric mean of linear operators can be disentangled into quantitatively linked estimates on each operator separately. The concept of disentanglement recurs throughout the paper.

The methods we used in the previous work — principally convex optimisation — relied strongly on positivity. In contrast, in this paper we use a vector-valued reformulation of disentanglement, geometric properties (Rademacher-type) of the underlying normed spaces, and probabilistic considerations related to p-stable random variables.

Keywords
multilinear inequalities, duality, factorisation, disentanglement, $p$-convexity, Rademacher-type
Mathematical Subject Classification
Primary: 42B99, 46B99, 47H99
Milestones
Received: 18 September 2020
Revised: 9 July 2021
Accepted: 11 August 2021
Published: 3 May 2023
Authors
Anthony Carbery
School of Mathematics and Maxwell Institute for Mathematical Sciences
University of Edinburgh
Edinburgh
United Kingdom
Timo S. Hänninen
Department of Mathematics and Statistics
University of Helsinki
Helsinki
Finland
School of Mathematics and Maxwell Institute for Mathematical Sciences
University of Edinburgh
Edinburgh
United Kingdom
Stefán Ingi Valdimarsson
Arion banki
Reykjavík
Iceland
Science Institute, Mathematics Division
University of Iceland
Reykjavík
Iceland

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