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For a particular class of pseudo manifolds, we show that the intersectioncohomology groups for any perversity may be naturally represented by extendedweighted $L^2$ harmonic forms for a complete metric on the regular stratum withrespect to some weight determined by the perversity. Extended weighted $L^2$harmonic forms are harmonic forms that are almost in the given weighted $L^2$space for the metric in question, but not quite. This result is akin to therepresentation of absolute and relative cohomology groups for a manifold withboundary by extended harmonic forms on the associated manifold with cylindricalends. As in that setting, in the unweighted $L^2$ case, the boundary values ofthe extended harmonic forms define a Lagrangian splitting of the boundary spacein the long exact sequence relating upper and lower middle perversityintersection cohomology groups.

Extended Hodge Theory for Fibred Cusp Manifolds