Elliptic corner operators in spaces with continuous radial asymptotics. I (Q1206395)

The Mellin transform (multidimensional) has been used to analyse radial regularity of solutions to elliptic corner operators when these solutions belong to the space \(M(\Omega,\rho)\) of distributions with continuous radial asymptotics. This paper continues investigations published in: \textit{Z. Szmydt} and \textit{B. Ziemian} [J. Differ. Equations 83, No. 1, 1- 25 (1990; Zbl 0702.35105)]. For a linear partial differential operator \(R\) on an open set \(U\), \(0\in U\subset\mathbb{R}^ n\) of the order \(m\) with smooth coefficients, \(R=R(x,x{\partial \over {\partial x}})=P(x{\partial \over {\partial x}})- Q(x,x {\partial \over {\partial x}})\) it is assumed the ellipticity condition: For every \(a\in\mathbb{R}^ n\) there exist \(C_ 1>0\) and \(C_ 2>0\) such that \(| P(a+i\beta)|>C_ 2(1+\|\beta\|)^ m\) for \(\|\beta\|>C_ 1\). The main result is the following: Let \(\delta x_ 0\in \mathbb{R}_ +^ n\). Suppose \(w\in M(\widetilde\Omega,\widetilde\rho)\) and \(u\in M(\Omega,\rho)\) 2-locally at \((0;\delta x_ 0)\) and \(Ru=w\) in a local conical neighbourhood of \((0,\delta x_ 0)\). Then: (1) for any \(\varepsilon>0\), \(u\in{\mathcal M}(\Omega\cap\widetilde\Omega; \rho(-\infty)+\varepsilon)\) 2-locally at \((0;\delta x_ 0)\) if \(\rho- m\leq\rho(-\infty) \overset{\text{def}}= \lim_{a_ 1\to-\infty} \rho(a_ 1)\); (2) \(u\in M(\Omega\cap\widetilde\Omega; \widetilde\rho-m)\) 2-locally at \((0;\delta x_ 0)\) if \(Q\equiv 0\).