Perturbation of a surjective convolution operator

Abstract

Let $\mu\in\mathcal E'(\mathbb R^n)$ be a compactly supported distribution such that its support is a convex set with a non-empty interior. Let $X_2$ be a convex domain in $\mathbb R^n$, $X_1=X_2+\mathrm{supp}\,\mu $. Let the convolution operator $A\colon\mathcal E(X_1)\to\mathcal E(X_2)$ acting by the rule $(Af)(x)=(\mu*f)(x)$ is surjective. We obtain a sufficient condition for a linear continuous operator $B\colon\mathcal E(X_1)\to\mathcal E(X_2)$ ensuring the surjectivity of the operator $A+B$.