and

This is the first of a series of three devoted to the regularity of solution of elliptic problems on nonsmooth domains in IR3. The present paper introduces various weighted spaces and countably weighted spaces in neighbourhood of edges and vertices of poly-hedral domains, and it concentrates on exploring the structure of these spaces such as the imbeddings of weighted Sobolev spaces, the relation between weighted Sobolev spaces and weighted continuous function spaces, and the relations between the weighted Sobolev spaces and countably weighted Sobolev spaces in Cartesian coordinates and in the spher-ical and cylindrical coordinates. These well-defined spaces are the foundations for comprehensive study of the regular-ity theory of elliptic problem with piecewise analytic data in IR3, which are essential for the design of effective computation and the analysis of the h − p version of the finite ele-ment method for solving elliptic problems in three-dimensional nonsmooth domain arising from mechanics and engineering