Localized characteristic primitive with an explicit defect

For the characteristic integrals $I^-,I^+,J=I^-+I^+$ , choose a smooth $\chi(U)$ equal to zero below a finite interval and one above it. Define

I_\chi F=I^-F-\chi JF=(1-\chi)I^-F-\chi I^+F.

This localized primitive obeys the exact identity

D_M I_\chi F=F-\chi'(U)JF.

It has compact support in $U$ , contained between the smallest and largest endpoints of the source support and the cutoff transition.

Support and averaging

Below both intervals, $I^-F=0$ and $\chi=0$ ; above both, $I^-F=JF$ and $\chi=1$ . The support can fill gaps between source pieces. Since torus translation preserves its normalized measure,

\langle JF\rangle_y=\int_{\mathbb R}\langle F(s,\cdot)\rangle_y\,ds.

Thus a zero integrated torus mean makes the defect's mean zero. It need not make the defect itself zero. If $F$ is independent of $y$ , the same zero integral makes $JF=0$ exactly.