Phase space localizing 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.

v3: added a reference to the application of the result. "Outside of the tree" item of the main theorem now has a slightly stronger form (contained in the old proof but not in the old statement)