Phase space localizing operators

TL;DR

This paper introduces new phase space localizing operators across all dimensions, improving previous one-dimensional phase plane projections, with applications in time-frequency analysis and multilinear modulation invariant operators.

Contribution

It constructs multidimensional phase space localizing operators, enhancing prior one-dimensional phase plane projections, for use in advanced time-frequency analysis.

Findings

Operators are frequency localized variants of conditional expectation.

Construction improves upon previous phase plane projections.

Applications include uniform estimates for multilinear modulation invariant operators.

Abstract

We construct phase space localizing operators in all dimensions. These are frequency localized variants of the conditional expectation operator related to a dyadic stopping time. Our construction is an improvement over the so-called phase plane projections of Muscalu, Tao and the third author in one dimension. The motivation for such operators comes from time-frequency analysis. They are used in particular to prove uniform estimates for multilinear modulation invariant operators.