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Phase space localizing operators

Marco Fraccaroli, Olli Saari, Christoph Thiele

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.

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.
Paper Structure (8 sections, 13 theorems, 122 equations)

This paper contains 8 sections, 13 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Construction of the phase space projection
  3. Cone decomposition
  4. Construction
  5. Proof of inequality (\ref{['e:unifgnorm']})
  6. Proof of Inequality (\ref{['e:unif-gtheo']})
  7. Proof of Inequality (\ref{['e:unigtheo-variant']})
  8. Proof of Proposition \ref{['t:spq']}

Key Result

Theorem 1.1

Let $1 \le p \le \infty$ and $1/p+1/p'=1$. Let $\alpha>d$ be real. There exists $C=C_{p,d,\alpha}>0$ such that for all $m \in {\mathbb N}$ the following holds. Let $i_0\in {\mathbb Z}$ and $U \in {\mathcal{D}}_{i_0}$. Let $M$ be a finite non-empty collection of pairwise disjoint cubes contained in $ Then and for every $j\le i_0$ and every $J\in {\mathcal{D}}_j$ it holds and if no $J'\in {\mathca

Theorems & Definitions (24)

  • Theorem 1.1: Main theorem
  • Proposition 1.2
  • Lemma 2.1
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 14 more