On perturbation of a surjective convolution operator

Let $\mu \in {\mathcal{E}}^{\mathrm{\prime }}({\mathbb{R}}^n)$ 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 $A:\mathcal{E}(X_1)\to \mathcal{E}(X_2)$ acting by the rule $(Af)(x)=(\mu \ast f)(x)$ is surjective we provide a condition on a linear continuous operator $B:\mathcal{E}(X_1)\to \mathcal{E}(X_2)$ that guarantees surjectivity of the operator $A+B$.