The EMS Publishing House is now **EMS Press** and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

# Zeitschrift für Analysis und ihre Anwendungen

Full-Text PDF (309 KB) | Abstract as PDF | Metadata | Table of Contents | ZAA summary

**Volume 22, Issue 2, 2003, pp. 315–338**

**DOI: 10.4171/ZAA/1148**

Published online: 2003-06-30

Pseudodifferential Operators on R^n with Variable Shifts

Vladimir S. Rabinovich^{[1]}(1) Escuelo Superior de Mat y Fis del IPN, México, D.f., Mexico

The aim of the paper is the study of pseudodifferential operators with shifts of the form $$ Au(x) = \sum_{j=1}^N a_j(x,D)V_{h_j} + \sum_{j=1}^N b_j(x,D)T_{g_j} $$ where $a_j(x,D) \in OPS_{1,0}^m$ and $b_j(x,D) \in OPS_{1,0}^{m-\epsilon} \ \ (\epsilon > 0)$ are pseudodifferential operators in the H\"ormander classes, and $V_{h_j}$ and $T_{g_j}$ are shift operators of the form $$ V_{h_j}u(x) = u(x - h_j), \qquad T_{g_j}u(x) = u(x - g_j(x)), \qquad x \in \mathbb R^n $$ 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.

*Keywords: *Pseudodifferential operator, shift, limit operators method

Rabinovich Vladimir S.: Pseudodifferential Operators on R^n with Variable Shifts. *Z. Anal. Anwend.* 22 (2003), 315-338. doi: 10.4171/ZAA/1148