The “qualitative” extension theorem of Demailly guarantees the existence of holomorphic extensions of holomorphic sections on some subvariety under certain positive-curvature assumption, but that comes without any estimate of the extensions, especially when the singular locus of the subvariety is non-empty and the holomorphic section to be extended does not vanish identically there. Residue functions are analytic functions which connect the \(L^2\) norms on the subvarieties (or their singular loci) to \(L^2\) norms with specific weights on the ambient space. Motivated by the conjectural “dlt extension”, this note discusses the possibility of retrieving the \(L^2\) estimates for the extensions in the general situation via the use of the residue functions. It is also shown in this note that the 1-lc-measure defined via the residue function of index 1 is indeed equal to the Ohsawa measure in the Ohsawa–Takegoshi \(L^2\) extension theorem.