A vector fields approach to smoothing and decaying estimates for equations in anisotropic media

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.