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Zeitschrift für Analysis und ihre Anwendungen


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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