On possible deterioration of smoothness under the operation of convolution

Let $\mu$ be a completely finite Borel non-negative measure on the real line $\mathbf R$. We give condition on measure $\mu$ which is necessary and sufficient for the existence of a non-negative, integrable on the real line, and entire function $p$ such that
\begin{equation} \operatorname{ess\,sup}\{(p\ast\mu)(x):x\in I\}=\infty \text{ для любого интервала } I\subset\mathbf R. \tag{1} \end{equation}
We give also conditions on measure $\mu$ which are sufficient for the existence of an entire function $p$ with prescribed growth in complex plane (for example, of finite order $\varrho>1$) that is non-negative and integrable on the real line and satisfies condition (1).