Perturbation of Differential Operators Admitting a Continuous Linear Right Inverse on Ultradistributions

Let Pm be a homogeneous polynomial of degree m in n ≥ 2 variables for which the associated partial differential operator Pm(D) admits a continuous linear right inverse on C∞(Rn). Examples suggest that then for each polynomial Q of degree less than m there exists a number 0 < β < 1 such that the operator (Pm+Q)(D) admits a continuous linear right inverse on the space of all ωβ-ultradifferentiable functions on R, where ωβ(t) = (1 + t). The main result of the present paper is to determine the optimal value of β for which the above holds for all perturbations Q of a given degree in the case n = 3. When n > 3 sufficient conditions as well as necessary conditions of this type are presented, but there is a gap between them. The results are illustrated by several examples.