calculus - Support of convolution

Assume $u \in L^1(\mathbb{R}^n)$ and $\mathrm{ess\,supp}(u) \subset U,$ where $U$ is a bounded open set. Now we compute the convolution of $u$ with a function $\eta \in C(\mathbb{R}^n)$ with $\mathrm{ess\,supp}(\eta) \subset B(0,h).$ Does this mean then that $\forall x \in \mathrm{ess\,supp}(u * \eta): dist(x,U)\le h$? Somehow this convolution confuses me completely, so I am not sure if this holds, although it sounds natural

By support I mean for an $L^p$ function that $u$ is zero outside $\mathrm{ess\,supp}(u)$ a.e.