It is a well known fact that given a vector field $F$ in $\mathbb{R}^3$, this can be decomposed as
$$ F= \nabla V+ \nabla \times R$$
with $V$ a potential and $R$ another vector field. These components are respectively curl free and divergence free. I am interested in particular in the properties of the vector field $R$, and in the regularity it inherits from $F$. For example, $F$ bounded implies $R$ bounded? Or $F$ Sobolev implies $R$ Sobolev? Where can I find some info on this topic, everything I find online is just basically saying that $R$ exists without much insight on its properties. If in the reference there are also properties for $V$, even better.
