2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/79123We establish refinements of the classical Kato inequality for sections of a vector bundle which lie in the kernel of a natural injectively elliptic first-order linear differential operator. Our main result is a general expression which gives the value of the constants appearing in the refined inequalities. These constants are shown to be optimal and are computed explicitly in most practical cases.AMS-LaTeX, 36pp, 1 figure (region.eps)Differential GeometryAnalysis of PDEsFunctional Analysis53B21; 58G03text