Weighted isoperimetric inequalities at infinity for a weighted Dirichlet normalized Laplacian associated with 2-simplexes

In this paper, we take a simplicial complex of dimension [Formula: see text] and we introduce on it oriented [Formula: see text]-simplexes and oriented [Formula: see text]-simplexes. As a result, we create a weighted simplicial set of [Formula: see text]-simplexes and [Formula: see text]-simplexes [Formula: see text], on which we introduce the cochains of dimension [Formula: see text] and we use them to construct a weighted normalized Laplacian associated with [Formula: see text]-simplexes [Formula: see text] and its weighted Dirichlet normalized Laplacian associated with [Formula: see text]-simplexes [Formula: see text]. Next, we study the spectrum of [Formula: see text] and the spectrum of [Formula: see text]. Finally, we develop the weighted isoperimetric inequalities at infinity associated to [Formula: see text] and we use them to analyze the essential spectrum of [Formula: see text] and the essential spectrum of [Formula: see text].