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

Remark: Flatness is not a general impossibility theorem

Nonextendability and flat multipliers · lines 1636–1655

The correct conclusion is that flatness alone does not guarantee extendability of a product. It is false that a flat multiplier can never repair nonextendability. For the same $v$ ,

v b_{1}^{\circ} (x) = {e^{- 1/ x^{2}}, 0, x > 0, x < 0

has a smooth flat extension obtained by assigning zero at the origin.

Flat-annihilator relations in Corollary cor:relationclassification concern the different hypothesis $f T^{\circ} = 0$ together with an already existing extension. They must not be confused with multiplying a nonextendable coefficient by a flat function.

Displayed formulas in this passage

unlabeled · line 1640
$v b_{1}^{\circ} (x) = {e^{- 1/ x^{2}}, 0, x > 0, x < 0$

Nearby in this section

proposition: Proposition: An explicit nonextendable coefficient
corollary: Corollary: Finite normal order alone does not ensure extension
proposition: Proposition: Flat multipliers can fail to repair extension
proposition: Proposition: Bounded continuous coefficients extend distributionally
section: Nonextendability and flat multipliers