2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/50049We give an explicit $L^2$-representation of chiral charged fermions using the Hardy-Lebesgue octant decomposition. In the " pure" case such a representation was already used by M. Sato in holonomic field theory. We study both "pure" and " mixed" cases. In the compact case we rigorously define unsmeared chiral charged fermion operators inside the unit circle. Using chiral fermions we orient our findings towards a functional analytic study of vertex algebras as one dimensional quantum field theory.to appear Lett.Math.PhysHigh Energy Physics - Theorytext