| Title:
|
Equivalent conditions for the validity of the Helmholtz decomposition of Muckenhoupt $A_{p}$-weighted $L^{p}$-spaces (English) |
| Author:
|
Kakizawa, Ryôhei |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
68 |
| Issue:
|
3 |
| Year:
|
2018 |
| Pages:
|
771-789 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $(L^{p}_{w}(\Omega ))^{n}$. (English) |
| Keyword:
|
Helmholtz decomposition |
| Keyword:
|
Muckenhoupt $A_{p}$-weighted $L^{p}$-spaces |
| Keyword:
|
variational estimate |
| MSC:
|
35Q30 |
| MSC:
|
46E30 |
| MSC:
|
76D05 |
| idZBL:
|
Zbl 06986971 |
| idMR:
|
MR3851890 |
| DOI:
|
10.21136/CMJ.2018.0646-16 |
| . |
| Date available:
|
2018-08-09T13:13:52Z |
| Last updated:
|
2020-10-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147367 |
| . |
| Reference:
|
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