Consider $f\in B_{p,q}^s(\Omega)$, $\Omega$ compact in $\mathbb{R}^d$, with $p,q\geq 1$ and $s>d/p$ (so the elements in the space are regular enough to be continuous functions), such that $\|f\|_{B_{p,q}^s}\leq R$.

Since $f$ is continuous and $\Omega$ is bounded, $e^f$ is a continuous mapping (with $\|\cdot\|_\infty$).

Do we have more general results with Besov norms, such as $\|e^f\| \leq K \|f\|^*$, where $K$ is a positive constant possibly depending on $R$ and $\Omega$, and $\|\cdot\|,\|\cdot\|^*$ are some Besov norms?