Equivalent conditions for the validity of the Helmholtz decomposition of Muckenhoupt $A_{p}$-weighted $L^{p}$-spaces

summary:We discuss the validity of the Helmholtz decomposition of the Muckenhoupt $A_{p}$-weighted $L^{p}$-space $(L^{p}_{w}(\Omega ))^{n}$ for any domain $\Omega $ in $\mathbb {R}^{n}$, $n \in \mathbb {Z}$, $n\geq 2$, $1<p<\infty $ and Muckenhoupt $A_{p}$-weight $w \in A_{p}$. Set $p':={p}/{(p-1)}$ and $w':=w^{-{1}/{(p-1)}}$. Then the Helmholtz decomposition of $(L^{p}_{w}(\Omega ))^{n}$ and $(L^{p'}_{w'}(\Omega ))^{n}$ and the variational estimate of $L^{p}_{w,\pi }(\Omega )$ and $L^{p'}_{w',\pi }(\Omega )$ are equivalent. Furthermore, we can replace $L^{p}_{w,\pi }(\Omega )$ and $L^{p'}_{w',\pi }(\Omega )$ by $L^{p}_{w,\sigma }(\Omega )$ and $L^{p'}_{w',\sigma }(\Omega )$, respectively. The proof is based on the reflexivity and orthogonality of $L^{p}_{w,\pi }(\Omega )$ and $L^{p}_{w,\sigma }(\Omega )$ and the Hahn-Banach theorem. As a corollary of our main result, we obtain the extrapolation theorem with the aid of the Helmholtz projection of $(L^{p}_{w}(\Omega ))^{n}$