It is well known that there exists a constant $C$ such that $$\forall f\in C^\infty_c(\mathbb R^3), \quad \Vert f\Vert_{L^6(\mathbb R^3)}\le C\Vert \nabla f\Vert_{L^2(\mathbb R^3)}. \tag{$\ast$}$$ Now let us assume that $f$ is locally integrable on $\mathbb R^3$ (or more generally that $f$ is a distribution on $\mathbb R^3$) and that $\nabla f\in L^2(\mathbb R^3)$. Does that imply that $$ f=f_0+\alpha, \quad\text{where $\alpha$ is a constant and $f_0$ belongs to $L^6(\mathbb R^3)$?} \tag{$\ast\ast$}$$