On multilinear determinant functionals

TL;DR

This paper establishes $L^p$-estimates for multilinear determinant functionals, linking their boundedness to a geometric measure inequality related to Fourier restriction and averaging operators.

Contribution

It proves the equivalence between boundedness of multilinear determinant functionals and a geometric measure inequality, extending previous work in harmonic analysis.

Findings

Boundedness of functionals linked to geometric inequalities

Connection to Fourier restriction and averaging operators

General conditions for $L^p$-estimates established

Abstract

This paper considers the problem of $L^p$-estimates for a certain multilinear functional involving integration against a kernel with the structure of a determinant. Examples of such objects are ubiquitous in the study of Fourier restriction and geometric averaging operators. It is shown that, under very general circumstances, the boundedness of such functionals is equivalent to a geometric inequality for measures which has recently appeared in work by D. Oberlin (Math Proc. Cambridge. Philos. Soc., 129, 2000) and Bak, Oberlin, and Seeger (J. Aust. Math. Soc., 85, 2008).