Exponential Decay and Holomorphic Extension of Solutions

Abstract

In the preceding chapters we proved, under different assumptions of global hypoellipticity on the symbol in ℝ^d, that the solutions in $S'\left( {\mathbb{R}^d } \right)$ of the homogeneous equation belong to $S\left( {\mathbb{R}^d } \right)$. In particular, for the self-adjoint operators discussed in Chapter 4, all the eigenfunctions belong to $S\left( {\mathbb{R}^d } \right)$. In the present chapter we show that this information can be strongly improved for G and Γ operators, namely we may give precise results of exponential decay and holomorphic extension of solutions.