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VOL. 64 | 2015 A vector fields approach to smoothing and decaying estimates for equations in anisotropic media
Mitsuru Sugimoto

Editor(s) Shin-Ichiro Ei, Shuichi Kawashima, Masato Kimura, Tetsu Mizumachi

## Abstract

It is well known that the vector fields $$\Omega = x\wedge D = (\Omega_{ij})_{i<j},\qquad \Omega_{ij} = x_i D_j - x_j D_i$$ commute with the Laplacian $-\Delta$. Hence we have $$Pu = f \quad\Rightarrow\quad P(\Omega u) = \Omega f,$$ where $P$ is a function of $-\Delta$, and in this way we can control the growth/decaying order of solution $u$ to the equation $Pu = f$. This fact was actually used to induce some decaying estimates for the wave equation ([3]) in a context of nonlinear analysis, and smoothing estimates for the Scrödinger equation ([6]) in a critical case. In this article, we will discuss how to trace this idea for equations with the Laplacian $-\Delta$ replaced by general elliptic (pseudo-)differential operators.

## Information

Published: 1 January 2015
First available in Project Euclid: 30 October 2018

zbMATH: 1342.35062
MathSciNet: MR3381218

Digital Object Identifier: 10.2969/aspm/06410319

Subjects:
Primary: 35E15 , 35L45 , 35S10

Keywords: Anisotropic media , Elastic equations , Maxwell equation , Scrödinger equation