On the de Rham homomorphism for $L_{\pi }$-cohomologies

We study the procedure of regularization in the context of the Lipschitz version of de Rham calculus on metric simplicial complexes with bounded geometry. It provides us with the machinery to handle the de Rham homomorphism for $L_{\pi }$-cohomologies. In this respect, we obtain the condition resolving the question of triviality of the kernel for de Rham homomorphism. In particular, we specify the non-trivial cohomology classes explicitly for a sequence of parameters $\pi =⟨p_0,p_1,\dots ,p_n⟩$ missing non-increasing monotonicity.