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

@article{Kakizawa2018,
abstract = {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 $\mathbb \{R\}^\{n\}$, $n \in \mathbb \{Z\}$, $n\ge 2$, $1<p<\infty $ and Muckenhoupt $A_\{p\}$-weight $w \in A_\{p\}$. Set $p^\{\prime \}:=\{p\}/\{(p-1)\}$ and $w^\{\prime \}:=w^\{-\{1\}/\{(p-1)\}\}$. Then the Helmholtz decomposition of $(L^\{p\}_\{w\}(\Omega ))^\{n\}$ and $(L^\{p^\{\prime \}\}_\{w^\{\prime \}\}(\Omega ))^\{n\}$ and the variational estimate of $L^\{p\}_\{w,\pi \}(\Omega )$ and $L^\{p^\{\prime \}\}_\{w^\{\prime \},\pi \}(\Omega )$ are equivalent. Furthermore, we can replace $L^\{p\}_\{w,\pi \}(\Omega )$ and $L^\{p^\{\prime \}\}_\{w^\{\prime \},\pi \}(\Omega )$ by $L^\{p\}_\{w,\sigma \}(\Omega )$ and $L^\{p^\{\prime \}\}_\{w^\{\prime \},\sigma \}(\Omega )$, 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 $(L^\{p\}_\{w\}(\Omega ))^\{n\}$.},
author = {Kakizawa, Ryôhei},
journal = {Czechoslovak Mathematical Journal},
keywords = {Helmholtz decomposition; Muckenhoupt $A_\{p\}$-weighted $L^\{p\}$-spaces; variational estimate},
language = {eng},
number = {3},
pages = {771-789},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Equivalent conditions for the validity of the Helmholtz decomposition of Muckenhoupt $A_\{p\}$-weighted $L^\{p\}$-spaces},
url = {http://eudml.org/doc/294364},
volume = {68},
year = {2018},
}

TY - JOUR
AU - Kakizawa, Ryôhei
TI - Equivalent conditions for the validity of the Helmholtz decomposition of Muckenhoupt $A_{p}$-weighted $L^{p}$-spaces
JO - Czechoslovak Mathematical Journal
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 3
SP - 771
EP - 789
AB - 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 $\mathbb {R}^{n}$, $n \in \mathbb {Z}$, $n\ge 2$, $1<p<\infty $ and Muckenhoupt $A_{p}$-weight $w \in A_{p}$. Set $p^{\prime }:={p}/{(p-1)}$ and $w^{\prime }:=w^{-{1}/{(p-1)}}$. Then the Helmholtz decomposition of $(L^{p}_{w}(\Omega ))^{n}$ and $(L^{p^{\prime }}_{w^{\prime }}(\Omega ))^{n}$ and the variational estimate of $L^{p}_{w,\pi }(\Omega )$ and $L^{p^{\prime }}_{w^{\prime },\pi }(\Omega )$ are equivalent. Furthermore, we can replace $L^{p}_{w,\pi }(\Omega )$ and $L^{p^{\prime }}_{w^{\prime },\pi }(\Omega )$ by $L^{p}_{w,\sigma }(\Omega )$ and $L^{p^{\prime }}_{w^{\prime },\sigma }(\Omega )$, 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 $(L^{p}_{w}(\Omega ))^{n}$.
LA - eng
KW - Helmholtz decomposition; Muckenhoupt $A_{p}$-weighted $L^{p}$-spaces; variational estimate
UR - http://eudml.org/doc/294364
ER -