Existence of the solutions of Lewy equation as the tempered ultrahyperfunctions (Recent development of microlocal analysis and asymptotic analysis)
Yasuyuki Oka, Kunio Yoshino · Institutional Repositories DataBase (IRDB) · 2013
The aim of this article is to show that there exist the solutions of the Lewy equation in the space of the tempered ultrahyperfunctions and give the example. \S 1. Introduction and Main resultIn the middle of $1950' s$ , B. Malgrange and L. Ehrenpreis independently obtained the result that every linear differential operator with constant coefficients has a fundamental solution (see [2] and [9]).This implies that if $L$ is a linear differential operator with constant coefficients on $\mathbb{R}^{d}$ and $f\in C_{0}^{\infty}(\mathbb{R}^{d})$ , there exists $u\in C^{\infty}(\mathbb{R}^{d})$ such that $Lu=f$ (see [3] and so on).Therefore everyone beheved that a linear differential equation with variable coefficients $P(x, \partial)u=\sum_{|\alpha|\leq m}a_{\alpha}(x)\partial^{\alpha}u=f$ can be also solved for an arbitrary right-hand side $f$ , especially $f\in C_{0}^{\infty}(\mathbb{R}^{d})$ .But in 1957, H. Lewy destroyed all hopes in the world by the following result: Theorem 1.1 ([4], [16]).There exist the functions $f\in C_{0}^{\infty}(\mathbb{R}_{x,y,t}^{3})$ so that the fol- lowing linear partial differential equation has no solution in the space $C^{1}$ in any neigh- borhood of the point $(x, y, t)=(O, 0, t_{0})$ :(1.1) $-( \frac{\partial}{\partial x}+\dot{\iota}\frac{\partial}{\partial y})u(x, y, t)+2i(x+iy)\frac{\partial}{\partial t}u(x, y, t)=f(x, y, t),$ $f\in C_{0}^{\infty}(\mathbb{R}_{x,y,t}^{3})$ .