The EMS Publishing House is now EMS Press and has its new home at

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