Pseudodifferential Operators on R^n with Variable Shifts

Abstract

The aim of the paper is the study of pseudodifferential operators with shifts of the form

where $a_j(x,D)\in OP{S}_{1,0}^m$ and $b_j(x,D)\in OP{S}_{1,0}^{m-ϵ}(ϵ>0)$ are pseudodifferential operators in the Hörmander classes, and $V_{h_j}$ and $T_{g_j}$ are shift operators of the form

where $h_j\in {\mathbb{R}}^n$ and the mappings $g_j:{\mathbb{R}}^n\to {\mathbb{R}}^n$ have infinitely differentiable coordinate functions bounded with all their derivatives. We will investigate the Fredholm and semi-Fredholm properties of the operator $A$ acting from $H^s({\mathbb{R}}^n)$ into $H^{s-m}({\mathbb{R}}^n)$ applying the limit operators method.