Let \(M\) be a \(C^\infty\) submanifold of \(\mathbb R^n\) endowed by a natural structure of Riemannian manifold, and let \((\Omega^\bullet(M),d)\) be the de Rham complex on \(M\). Suppose that for a differential form \(\omega\in \Omega^p(M)\) there is a constant \(C\) such that for any \(x\in M\) one has \(| \omega(x)| \leq C,\) where \(| \omega(x)|\) denotes the norm of \(\omega(x).\) Then \(\omega\) is called the \(L^{\infty }\) differential \(p\)-form. The real vector space generated by all \(\omega\in\Omega^p(M)\) such that \(\omega\) and \(d \omega\) are both \(L^{\infty }\) differential forms is denoted by \(\Omega^p_{\infty}(M)\). Thus, one obtains an increasing subcomplex \((\Omega^\bullet_{\infty}(M),d)\) of the de Rham complex. The cohomology groups of this cochain complex are called the \(L^{\infty }\)-cohomology groups of \(M;\) they are denoted by \(H^\bullet_\infty(M).\)
Given a subanalytic set \(X,\) the set of its points at which \(X\) is locally a \(C^\infty\) manifold is denoted by \(X_{\text{reg}},\) while the complement \(X\setminus X_{\text{reg}}\) is denoted by \(X_{\text{sing}}\). By definition, an \(\ell\)-dimensional pseudomanifold is a globally subanalytic locally closed set \(X\subset \mathbb R^n\) such that \(X_{\text{reg}}\) is a manifold of dimension \(\ell\) and \(\dim X_{\text{sing}} \leq \ell-2.\) The aim of the paper is to prove the following variant of de Rham theorem for compact subanalyitic \(\ell\)-dimensional pseudomanifolds: \(L^{\infty }\)-cohomology groups of \(X_{\text{reg}}\) and intersection cohomology groups of maximal perversity of \(X\) are naturally isomorphic, that is, \(H^q_{\infty}(X_{\text{reg}}) \cong I^\ell H^q(X)\) for all \(q\geq 0\). The proof is a result of detail analysis of the metric type of subanalytic singular sets; it is based essentially on properties of Lipschitz functions and homemomorphisms, on a version of the classical Poincaré lemma in the context of \(L^{\infty }\) differential forms, on the theory of integration on subanalytic singular simplices, etc. In fact, the author exploits techniques developed in his earlier papers [Ill. J. Math. 49, No. 3, 953–979 (2005; Zbl 1154.14323); J. Symb. Log. 73, No. 2, 439–447 (2008; Zbl 1145.03017)].