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Local smoothing estimates for Schrödinger equations in modulation spaces

Kotaro Inami

TL;DR

This paper develops local smoothing estimates for the Schrödinger equation in modulation spaces, sharpening the role of modulation-space summability through a Córdoba–Fefferman type reverse square function inequality and a bilinear Strichartz estimate. It introduces a randomized Strichartz framework in modulation spaces and proves that a reverse function estimate implies modulation-space Strichartz estimates, yielding a critical-order reverse-square-function bound. The results advance the understanding of smoothing for the Schrödinger flow with data in modulation spaces and have potential applications to nonlinear problems, including local well-posedness results and random-data global behavior. Overall, the work connects reverse-square-function theory, orthogonal Strichartz, and randomization to refine space-time control of Schrödinger evolutions in fine function spaces.

Abstract

Motivated by a recent work of Schippa (2022), we consider local smoothing estimates for Schrödinger equations in modulation spaces. By using the Córdoba-Fefferman type reverse square function inequality and the bilinear Strichartz estimate, we can refine the summability exponent of modulation spaces. Next, we will also discuss a new type of randomized Strichartz estimate in modulation spaces. Finally, we will show that the reverse function estimate implies the Strichartz estimates in modulation spaces. From this implication, we obtain the reverse square function estimate of critical order.

Local smoothing estimates for Schrödinger equations in modulation spaces

TL;DR

This paper develops local smoothing estimates for the Schrödinger equation in modulation spaces, sharpening the role of modulation-space summability through a Córdoba–Fefferman type reverse square function inequality and a bilinear Strichartz estimate. It introduces a randomized Strichartz framework in modulation spaces and proves that a reverse function estimate implies modulation-space Strichartz estimates, yielding a critical-order reverse-square-function bound. The results advance the understanding of smoothing for the Schrödinger flow with data in modulation spaces and have potential applications to nonlinear problems, including local well-posedness results and random-data global behavior. Overall, the work connects reverse-square-function theory, orthogonal Strichartz, and randomization to refine space-time control of Schrödinger evolutions in fine function spaces.

Abstract

Motivated by a recent work of Schippa (2022), we consider local smoothing estimates for Schrödinger equations in modulation spaces. By using the Córdoba-Fefferman type reverse square function inequality and the bilinear Strichartz estimate, we can refine the summability exponent of modulation spaces. Next, we will also discuss a new type of randomized Strichartz estimate in modulation spaces. Finally, we will show that the reverse function estimate implies the Strichartz estimates in modulation spaces. From this implication, we obtain the reverse square function estimate of critical order.

Paper Structure

This paper contains 12 sections, 17 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Modulation spaces
  4. Wiener's randomization
  5. Strichartz estimate for orthogonal initial data.
  6. Notations
  7. Proof of Theorem \ref{['thm:loocalsmoothing']}
  8. Proof of the bilinear Strichartz estimate
  9. Proof of Theorem \ref{['thm:loocalsmoothing']}
  10. Proof of Theorem \ref{['thm:randomized']}
  11. Proof of Proposition \ref{['prop:necessary for Demeter']}
  12. A direct proof for the reverse square function estimate when $d = 1$

Key Result

Theorem 1

Suppose that $d \geq 1$, $2\leq p \leq \infty$, and $1 \leq q \leq \infty$. Then the estimate holds if either one of the following conditions holds:

Theorems & Definitions (27)

  • Theorem 1: Theorem 1.1 in schippa2022smoothing
  • Proposition 1: Schippa schippa2022smoothing
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Proposition 2: Exercise 5.30 demeter2020fourier
  • Proposition 3
  • Proposition 4
  • Proposition 5: chaichenets2020local
  • Definition 1
  • ...and 17 more