The augmented operator of a surjective partial differential operator with constant coefficients need not be surjective

Abstract

For d⩾3 we give an example of a constant coefficient surjective differential operator P(D):풟′(X)→풟′(X) over some open subset X⊂ℝd such that P+(D):풟′(X×ℝ)→풟′(X×ℝ) is not surjective, where P+(x1, …, xd+1)≔P(x1, …, xd). This answers Problem 9.1 posed by Bonet and Domański (‘Parameter dependence of solutions of differential equations on spaces of distributions and the splitting of short exact sequences’, J. Funct. Anal. 230 (2006) 329–381) in the negative.

Surjectivity of differential operators and linear topological invariants for spaces of zero solutions

“…This problem was posed by Bonet and Domański in [1]. Although in general this problem has a negative solution as shown in [13], in section 3 we show that this problem has always a positive solution for certain semi-elliptic differential operators including the heat operator, and for operators acting along a subspace of R n and being elliptic there.…”

“…, x n ). It was shown in [12], that in case of n = 2 the augmented operator of a surjective partial differential operator P (D) is always surjective while in [13] for n ≥ 3 a hypoelliptic differential operator P (D) was constructed for which there is some open X ⊆ R n such that P (D) is surjective on D ′ (X) while the augmented operator P + (D) is not surjective on D ′ (X × R). Thus, although not true in general, for certain differential operators, including the heat operator, the problem of parameter dependence for solutions of partial differential equations [1,Problem 9.1] has a positive solution, as will be shown as a consequence of the results from this section in section 4.…”

“…However, only recently, this conjecture has been settled in the negative. In [13] for any n ≥ 3 an example of a hypoelliptic operator P (D) and a P -convex open set X ⊆ R n has been given such that C ∞ P (X) does not have (Ω). (That n ≥ 3 is essential here follows from results obtained in [12].)…”

Linear topological invariants for kernels of convolution and differential operators

“…We remark that in all the above cases also a geometric characterization of the sets X for which [14,15,18]. On the negative side, in [16] this method was used to provide a concrete example of a (hypoelliptic) surjective operator P (D) on D ′ (X) such that D ′ P (X) does not satisfy (P Ω). There does not seem to exist an analogue of the above approach to study (Ω) for E P (X).…”

“…These results were obtained previously by Petzsche [24] and Vogt [29], respectively, via different more direct methods. Note that the example from [16] also shows that for surjective operators P (D) on E (X) the smooth kernel E P (X) in general does not satisfy (Ω).…”