Perturbation of differential operators admitting a continuous linear right inverse on ultradistributions

Rüdiger W. Braun, Reinhold Meise and B.A. Taylor

Vol. 212 (2003), No. 1, 25–48

DOI: 10.2140/pjm.2003.212.25

Abstract

Let P_m be a homogeneous polynomial of degree m in n ≥ 2 variables for which the associated partial differential operator P_m(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 (P_m + Q)(D) admits a continuous linear right inverse on the space of all ω_β-ultradifferentiable functions on ℝⁿ, where ω_β(t) = (1 + t)^β. 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.

Milestones

Received: 7 August 2002

Published: 1 November 2003