Manuscript XV-B@2 · XV-B.tex · corollary · 4d3f12e8

Corollary: Finite normal order alone does not ensure extension

Nonextendability and flat multipliers · lines 1590–1608 · proof included or linked

XV-B@2#cor:normalnotextension

A punctured supported distribution of normal order zero need not have a supported normal-order-preserving extension.

Proof. Take

U = (- 1, 1) \times V_{q}, S = (- 1, 1) \times {0}, A = {0} \times V_{q},

and set $t^{\circ} = E_{0} b_{1}^{\circ}$ on $U ∖ A$ . Theorem thm:jetextension would turn an extension of the required support and normal order into an extension of $b_{1}^{\circ}$ , contradicting Proposition prop:nonextendable.

Displayed formulas in this passage

unlabeled · line 1598
$U = (- 1, 1) \times V_{q}, S = (- 1, 1) \times {0}, A = {0} \times V_{q},$

Nearby in this section

proposition: Proposition: An explicit nonextendable coefficient
proposition: Proposition: Flat multipliers can fail to repair extension
remark: Remark: Flatness is not a general impossibility theorem
proposition: Proposition: Bounded continuous coefficients extend distributionally
section: Nonextendability and flat multipliers