2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/66638We obtain Sobolev inequalities for the Schrodinger operator -Δ-V, where V has critical behaviour V(x)=((N-2)/2)^2|x|^{-2} near the origin. We apply these inequalities to obtain pointwise estimates on the associated heat kernel, improving upon earlier results.23 pagesAnalysis of PDEsSpectral Theory35K65 (Primary) 26D10, 35K20, 35B05 (Secondary)text