JournalszaaVol. 22 , No. 2DOI 10.4171/zaa/1148

Pseudodifferential Operators on R^n with Variable Shifts

  • Vladimir S. Rabinovich

    Escuelo Superior de Mat y Fis del IPN, México, D.f., Mexico
Pseudodifferential Operators on R^n with Variable Shifts cover

Abstract

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

Au(x)=j=1Naj(x,D)Vhj+j=1Nbj(x,D)TgjAu(x) = \sum_{j=1}^N a_j(x,D)V_{h_j} + \sum_{j=1}^N b_j(x,D)T_{g_j}

where aj(x,D)OPS1,0ma_j(x,D) \in OPS_{1,0}^m and bj(x,D)OPS1,0mϵ  (ϵ>0)b_j(x,D) \in OPS_{1,0}^{m-\epsilon} \ \ (\epsilon > 0) are pseudodifferential operators in the H\"ormander classes, and VhjV_{h_j} and TgjT_{g_j} are shift operators of the form

Vhju(x)=u(xhj),Tgju(x)=u(xgj(x)),xRnV_{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 hjRnh_j \in \mathbb R^n and the mappings gj:RnRng_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 AA acting from Hs(Rn)H^s(\mathbb R^n) into Hsm(Rn)H^{s-m}(\mathbb R^n) applying the limit operators method.