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Smoothing effect for third order operators with variable coefficients

Serena Federico, Davide Tramontana

TL;DR

This work develops a robust framework for local smoothing of third-order variable-coefficient dispersive operators by introducing verifiable symbol conditions on the real part and a smallness assumption on the imaginary part. A new Doi-type lemma for order $m\ge 2$ yields the notion of admissible symbols, enabling smoothing estimates (homogeneous and inhomogeneous) and linear/nonlinear well-posedness results for NLIVPs, including derivative nonlinearities in KdV-type settings. The analysis covers both constant and variable coefficient dispersive operators, with extensive examples (KdV-type, ultrahyperbolic, and higher-dimensional generalizations) and a detailed study of non-trapping for strongly elliptic points. The results extend classical smoothing theory for Schrödinger-type equations to third-order variable-coefficient operators, offering tools for applications in KdV-type dynamics and related dispersive systems. The work thus unifies and broadens smoothing, well-posedness, and geometric considerations in a variable-coefficient, higher-order dispersive context.

Abstract

In this work we study the smoothing effect of some variable coefficient operators of the form $D_t-A$, where $A$ is a Weyl-quantized pseudo-differential operator of order $m=2,3$. The class under consideration includes, among others, KdV-type and ultrahyperbolic Schrödinger operators. We prove homogeneous and inhomogeneous smoothing estimates and use them to get well-posedness results for some NLIVPs with derivative nonlinearities. Finally, we investigate the so called non-trapping property of the bicharacteristic curves of the principal symbol of our operators.

Smoothing effect for third order operators with variable coefficients

TL;DR

This work develops a robust framework for local smoothing of third-order variable-coefficient dispersive operators by introducing verifiable symbol conditions on the real part and a smallness assumption on the imaginary part. A new Doi-type lemma for order yields the notion of admissible symbols, enabling smoothing estimates (homogeneous and inhomogeneous) and linear/nonlinear well-posedness results for NLIVPs, including derivative nonlinearities in KdV-type settings. The analysis covers both constant and variable coefficient dispersive operators, with extensive examples (KdV-type, ultrahyperbolic, and higher-dimensional generalizations) and a detailed study of non-trapping for strongly elliptic points. The results extend classical smoothing theory for Schrödinger-type equations to third-order variable-coefficient operators, offering tools for applications in KdV-type dynamics and related dispersive systems. The work thus unifies and broadens smoothing, well-posedness, and geometric considerations in a variable-coefficient, higher-order dispersive context.

Abstract

In this work we study the smoothing effect of some variable coefficient operators of the form , where is a Weyl-quantized pseudo-differential operator of order . The class under consideration includes, among others, KdV-type and ultrahyperbolic Schrödinger operators. We prove homogeneous and inhomogeneous smoothing estimates and use them to get well-posedness results for some NLIVPs with derivative nonlinearities. Finally, we investigate the so called non-trapping property of the bicharacteristic curves of the principal symbol of our operators.

Paper Structure

This paper contains 10 sections, 17 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Admissible operators
  4. A new version of Doi's lemma
  5. Admissible operators and sufficient conditions for admissibility
  6. Examples of admissible operators and dispersive operators with variable coefficients
  7. Smoothing estimates
  8. Well-posedness for the NLIVP for KdV-type operators with variable coefficients
  9. Non-trapping for some admissible operators
  10. Technical lemmas

Key Result

Theorem 1.1

Let $a\in S^m(\mathbb{R}^n)$, with $m=2,3$ be such that cond.a and cond.ima hold with $\lambda(\lvert x\rvert)=\langle x\rangle^{-N}$, where $N\in\mathbb{N}$ and $N>1$. Assume also that $\mathrm{Re}(a)$ satisfies ellderivatives and smallcoeffa2a3 with the same $\lambda(\lvert x\rvert)$. Then, given

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6: Sharp Gå rding and Fefferman-Phong inequality
  • Definition 2.7
  • Definition 2.8
  • ...and 49 more