Let $\Omega\subset\mathbb{R}^{d}$ a bounded domain with $C^1$-boundary and $1<p<\infty$. The Helmholtz Decomposition Theorem states that there is a bounded projection $$ P \colon L^p(\Omega, \mathbb{R}^n)\to L^p_\sigma(\Omega,\mathbb{R}^n)$$ where $$L^p_\sigma(\Omega,\mathbb{R}^{n}) = \overline{\{u\in\mathcal{D}(\Omega,\mathbb{R}^{n})\colon \nabla\cdot u = 0\}}.$$ This projection can be constructed as $Pu=\nabla(F(\nabla\cdot u))$ where $F$ is a suitable solution operator for the Laplace operator.
There are results showing that the same construction is not possible in the case $p=\infty$. What is the situation in the case $p=1$? Is there a result on the existence or non-existence of the Helmholtz projection in this case?