Multilinear singular operators with fractional rank

TL;DR

This paper establishes bounds for multilinear operators with singular multipliers along subspaces of fractional rank, exploring connections with additive combinatorics and extending previous results to non-integer rank cases.

Contribution

It introduces bounds for multilinear singular operators with fractional rank multipliers and investigates their relation to true complexity in additive combinatorics.

Findings

Bounds established for operators with fractional rank multipliers

Extension of singular operator theory to non-integer rank cases

Connections made between operator bounds and additive combinatorics concepts

Abstract

We prove bounds for multilinear operators on $\R^d$ given by multipliers which are singular along a $k$ dimensional subspace. The new case of interest is when the rank $k/d$ is not an integer. Connections with the concept of {\em true complexity} from Additive Combinatorics are also investigated.