General perversities and L2 de Rham and Hodge theorems for stratified pseudomanifolds

Given a compact stratified pseudomanifold X with a Thom-Mather stratification and a class of riemannian metrics over its regular part, we study the relationships between the L2 de Rham and Hodge cohomology of the regular part of X and the intersection cohomology of X associated to some perversities. More precisely, to a kind of metric which we call quasi edge with weights, we associate two general perversities in the sense of G. Friedman, pg and its dual qg. We then show that: