Boundary value problems for elliptic functional-differential equations with dilatations and compressions of the arguments (Q2761510)

An elliptic functional-differential equation of the form NEWLINE\[NEWLINEAu(x)\equiv \sum^{\ell_2}_{k= -\ell_1} \sum_{|\alpha |\leq 2m}a_{k \alpha} (x)D^\alpha \bigl(u(q^{-k}x)\bigr)= f(x),\;x\in Q\subseteq \mathbb{R}^n,NEWLINE\]NEWLINE NEWLINE\[NEWLINET_j(x,D)u(x)= g_j(x)\quad (j=1,\dots, m;\;x\in\partial Q),NEWLINE\]NEWLINE \((q>1)\) is said to have dilations \((\ell_1>0)\) and compressions \((\ell_2>0)\) in the arguments. In the case of compressions, the paper uses techniques of pseudodifferential operators and the `symbol' NEWLINE\[NEWLINEa(x,\xi,\lambda)= \sum^\ell_{k=0} \sum_{|\alpha |=2m}a_{k\alpha} (x) \xi^\alpha \lambda^kNEWLINE\]NEWLINE to prove that the map NEWLINE\[NEWLINE[A,T_1, \dots,T_m]: H^{s+2m} (Q)\to H^s(Q)\times \prod^m_{j=1} H^{s+2m-m_j-1/2} (\partial Q)NEWLINE\]NEWLINE is Fredholm.