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Schrödinger ultrahyperbolic equations with singular coefficients

Claudia Garetto, Davide Tramontana

Abstract

In this paper we investigate the Cauchy problem for Schrödinger ultrahyperbolic equations with singular (less than continuous) coefficients. We prove $H^\infty$ well-posedness in the very weak sense under suitable assumptions of the distributional structure of the coefficients and decay on the lower order terms. Consistency is proven with the classical $H^\infty$-results when the equation coefficients are smooth.

Schrödinger ultrahyperbolic equations with singular coefficients

Abstract

In this paper we investigate the Cauchy problem for Schrödinger ultrahyperbolic equations with singular (less than continuous) coefficients. We prove well-posedness in the very weak sense under suitable assumptions of the distributional structure of the coefficients and decay on the lower order terms. Consistency is proven with the classical -results when the equation coefficients are smooth.
Paper Structure (15 sections, 15 theorems, 166 equations)

This paper contains 15 sections, 15 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basics of pseudodifferential calculus and Sobolev spaces
  4. Regularisations via convolution with a mollifier
  5. Sobolev moderate and negligible nets
  6. Ultrahyperbolic operators with singular coefficients
  7. Regularisation of singular ultrahyperbolic operators
  8. Doi's Lemma for ultrahyperbolic operators with singular coefficients
  9. Statement of the problem and regularised version
  10. Statement of the problem
  11. Regularised Problem
  12. Existence and uniqueness of an $H^\infty$-very weak solution
  13. Existence
  14. Uniqueness
  15. Consistency with the classical theory

Key Result

Theorem 2.1

Let $a \in S^m$, with $m \in \mathbb R$, and let $s \in \mathbb R$. Then $\mathrm{Op}(a)$ extends to a bounded linear operator from $H^{s+m}(\mathbb R^n)$ to $H^s(\mathbb R^n)$ and there exist $k=k(n,m,s) \in \mathbb{N}$ and $C=C(n,m,s)>0$ such that

Theorems & Definitions (42)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Definition 2.8
  • ...and 32 more