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| Consider a general domain $\Omega\subseteq \mathbb{R}^n$, $n\ge 2$, and let $1 < q < \infty$.
Our first result is based on the estimate for the gradient $\nabla p \in G^q(\Omega)$ in the form
$$\|\nabla p\|_q \le C \,\sup |\langle\nabla p,\nabla v\rangle_{\Omega}|/\|\nabla v\|_{q'},\;\;\; \nabla v
\in G^{q'}(\Omega), \;\;\;q' = \frac{q}{q-1},$$ with some constant $C=C(\Omega,q)>0$.
This estimate was introduced by Simader and Sohr for smooth bounded and exterior domains.
We show for general domains that the validity of this gradient estimate in $G^q(\Omega)$ and in $G^{q'}(\Omega)$ is necessary and sufficient for the validity of the Helmholtz decomposition in $L^q(\Omega)$ and in $L^{q'}(\Omega)$.
Another new aspect concerns the estimate for divergence free functions $f_0 \in L^q_{\sigma}(\Omega)$ in the form
$$\|f_0\|_q \le C \sup |\langle f_0,w\rangle_{\Omega}|/\|w\|_{q'},\;\;\; w\in L^{q'}_{\sigma}(\Omega),$$
for the second part of the Helmholtz decomposition.
We show again for general domains that the validity of this estimate in$L^q_{\sigma}(\Omega)$ and in $L^{q'}_{\sigma}(\Omega)$ is necessary and sufficient for the validity of the Helmholtz decomposition in $L^q(\Omega)$ and in $L^{q'}(\Omega)$. |
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