Czechoslovak Mathematical Journal, Vol. 68, No. 3, pp. 771-789, 2018
Equivalent conditions for the validity of the Helmholtz decomposition of Muckenhoupt $A_p$-weighted $L^p$-spaces
Ryôhei Kakizawa
Received December 23, 2016. Published online April 16, 2018.
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 $\mathbb{R}^n$, $n \in\mathbb{Z}$, $n\geq2$, $1<p<\infty$ 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 $(L^p_w(\Omega))^n$.
References: [1] G. de Rham: Variétés différentiables. Formes, courants, formes harmoniques. Publications de l'Institut de Mathématique de l'Université de Nancago III. Actualités Scientifiques et Industrielles 1222 b, Hermann, Paris (1973), French. MR 0346830 | Zbl 0284.58001
[2] E. Fabes, O. Mendez, M. Mitrea: Boundary layers on Sobolev-Besov spaces and Poisson's equation for the Laplacian in Lipschitz domains. J. Funct. Anal. 159 (1998), 323-368. DOI 10.1006/jfan.1998.3316 | MR 1658089 | Zbl 0930.35045
[3] R. Farwig: Weighted $L^q$-Helmholtz decompositions in infinite cylinders and in infinite layers. Adv. Differ. Equ. 8 (2003), 357-384. MR 1948530 | Zbl 1038.35068
[4] R. Farwig, H. Kozono, H. Sohr: An $L^q$-approach to Stokes and Navier-Stokes equations in general domains. Acta Math. 195 (2005), 21-53. DOI 10.1007/BF02588049 | MR 2233684 | Zbl 1111.35033
[5] R. Farwig, H. Kozono, H. Sohr: On the Helmholtz decomposition in general unbounded domains. Arch. Math. 88 (2007), 239-248. DOI 10.1007/s00013-006-1910-8 | MR 2305602 | Zbl 1121.35097
[6] R. Farwig, H. Sohr: Weighted $L^q$-theory for the Stokes resolvent in exterior domains. J. Math. Soc. Japan 49 (1997), 251-288. DOI 10.2969/jmsj/04920251 | MR 1601373 | Zbl 0918.35106
[7] A. Fröhlich: The Helmholtz decomposition of weighted $L^q$-spaces for Muckenhoupt weights. Ann. Univ. Ferrara, Nuova Ser., Sez. VII 46 (2000), 11-19. MR 1896920 | Zbl 1034.35089
[8] A. Fröhlich: Maximal regularity for the non-stationary Stokes system in an aperture domain. J. Evol. Equ. 2 (2002), 471-493. DOI 10.1007/PL00012601 | MR 1941038 | Zbl 1040.35059
[9] A. Fröhlich: The Stokes operator in weighted $L^q$-spaces I. Weighted estimates for the Stokes resolvent problem in a half space. J. Math. Fluid Mech. 5 (2003), 166-199. DOI 10.1007/s00021-003-0080-8 | MR 1982327 | Zbl 1040.35060
[10] A. Fröhlich: The Stokes operator in weighted $L^q$-spaces II. Weighted resolvent estimates and maximal $L^p$-regularity. Math. Ann. 339 (2007), 287-316. DOI 10.1007/s00208-007-0114-2 | MR 2324721 | Zbl 1126.35041
[11] J. García-Cuerva, J. L. Rubio de Francia: Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies 116, North-Holland, Amsterdam (1985). DOI 10.1016/s0304-0208(08)x7154-3 | MR 0807149 | Zbl 0578.46046
[12] J. Geng, Z. Shen: The Neumann problem and Helmholtz decomposition in convex domains. J. Funct. Anal. 259 (2010), 2147-2164. DOI 10.1016/j.jfa.2010.07.005 | MR 2671125 | Zbl 1195.35128
[13] A. S. Kim, Z. Shen: The Neumann problem in $L^p$ on Lipschitz and convex domains. J. Funct. Anal. 255 (2008), 1817-1830. DOI 10.1016/j.jfa.2008.06.032 | MR 2442084 | Zbl 1180.35202
[14] T. Kobayashi, T. Kubo: Weighted $L^p$-$L^q$ estimates of the Stokes semigroup in some unbounded domains. Tsukuba J. Math. 37 (2013), 179-205. DOI 10.21099/tkbjm/1389972027 | MR 3161575 | Zbl 1282.35103
[15] J. Lang, O. Méndez: Potential techniques and regularity of boundary value problems in exterior non-smooth domains. Regularity in exterior domains. Potential Anal. 24 (2006), 385-406. DOI 10.1007/s11118-006-9008-2 | MR 2224756 | Zbl 1220.35120
[16] Y. Maekawa, H. Miura: Remark on the Helmholtz decomposition in domains with noncompact boundary. Math. Ann. 359 (2014), 1077-1095. DOI 10.1007/s00208-014-1033-7 | MR 3231025 | Zbl 1295.35184
[17] C. G. Simader, H. Sohr, W. Varnhorn: Necessary and sufficient conditions for the existence of Helmholtz decompositions in general domains. Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 60 (2014), 245-262. DOI 10.1007/s11565-013-0193-9 | MR 3208796 | Zbl 1304.35506
Affiliations: Ryôhei Kakizawa, Department of Fundamental Mathematics Education, Shimane University, 1060 Nishikawatsu-cho, Matsue-shi, Shimane 690-8504, Japan, e-mail: kakizawa@edu.shimane-u.ac.jp
