On perturbation of a surjective convolution operator

TL;DR

This paper investigates conditions under which a perturbed convolution operator remains surjective, focusing on operators defined by distributions with convex support and providing criteria for the surjectivity of the sum of such an operator and a linear perturbation.

Contribution

It establishes a specific condition on a linear operator B that ensures the surjectivity of the perturbed convolution operator A+B, extending the understanding of convolution operator stability under perturbations.

Findings

Provided a criterion for the surjectivity of A+B

Extended the theory of convolution operators with convex support distributions

Analyzed stability of surjectivity under linear perturbations

Abstract

Let $\mu \in {\cal E}'({\mathbb R}^n)$ be a compactly supported distribution such that its support is a convex set with non-empty interior. Let $X_2$ be a convex domain in ${\mathbb R}^n$, $X_1 = X_2 + supp \ \mu $. Assuming that a convolution operator $A: {\cal E}(X_1) \to {\cal E}(X_2)$ acting by the rule $(Af)(x) = (\mu * f)(x)$ is surjective we provide a condition on a linear continuous operator $B: {\cal E}(X_1) \to {\cal E}(X_2)$ that guarantees surjectivity of the operator $A+B$.