Sub-exponential decay and uniform holomorphic extensions for semilinear pseudodifferential equations

The goal of the present paper is to derive a simultaneous description of the decay and the regularity properties for elliptic equations in $\R^n$ with coefficients admitting irregular decay at infinity of the type $O(|x|^{-\sigma}), \sigma >0$, filling the gap between the case of Cordes globally elliptic operators and the case of regular/Fuchs behaviour at infinity. Representative examples in $\R^n$ are the equations $$-\Delta u+\frac{\omega(x)}{\px^{\sigma}}u =f+F[u], \qquad x \in \R^{n},$$ where $0<\sigma <2, \px= (1+|x|^2)^{1/2}, \omega(x)$ a bounded smooth function, $f$ given and $F[u]$ a polynomial in $u$, and similar Schr\"odinger equations at the endpoint of the spectrum. Other relevant examples are given by linear and nonlinear ordinary differential equations with irregular type of singularity for $x \rightarrow \infty$, admitting solutions $y(x)$ with holomorphic extension in a strip and sub-exponential decay of type $|y(x)|\leq Ce^{-\veps|x|^{r}}, 0<r<1.$ Sobolev estimates for the linear case are proved in the frame of a suitable pseudodifferential calculus; decay and uniform holomorphic extensions are then obtained in terms of Gelfand-Shilov spaces by an inductive technique. The same technique allows to extend the results to the semilinear case.