Proposition: Support of nonsummable coefficients

Suppose Theorem thm:main applies and the same family has a raw limiting reference on an open set $V\subset U$. Then every nonsummable coefficient restricts to zero on $V$, and

Moreover $T^{\rho }{|}_V$ equals that reference.

Proof. Apply the fixed-test dominance argument to tests supported in $V$. A nonzero restricted nonsummable coefficient would contradict convergence there. After those coefficients vanish, the remainder estimate identifies the limit with $T^{\rho }{|}_V$.