[1] M. E. Bogovski˘ı, Decomposition of Lp (Ω, Rn ) into a direct sum of subspaces of solenoidal
and potential vector fields, Dokl. Akad. Nauk SSSR, 286 (1986), no. 4, 781–786
[2] M. Bolkart and Y. Giga, On L∞ -BM O estimates for derivatives of the Stokes semigroup,
Math. Z., 284 (2016), no. 3–4, 1163–1183
[3] M. Bolkart, Y. Giga, T.-H. Miura, T. Suzuki, Y. Tsutsui, On analyticity of the Lp -Stokes
semigroup for some non-Helmholtz domains, Math. Nachr., 290 (2017), no. 16, 2524–2546
[4] M. Bolkart, Y. Giga and T. Suzuki, Analyticity of the Stokes semigroup in BM O-type
spaces, J. Math. Soc. Japan, 70 (2018), no. 1, 153–177
[5] M. Bolkart, Y. Giga, T. Suzuki, Y. Tsutsui, Equivalence of BM O-type norms with applications to the heat and Stokes semigroups, Potential Anal., 49 (2018), no. 1, 105–130
[6] J. Boman, Lp -estimates for very strongly elliptic systems, Report 29, Department of Mathematics, University of Stockholm (1982)
[7] G. E. Bredon, Topology and geometry, Grad. Texts in Math., 139, Springer-Verlag, New
York (1997)
[8] L. C. Evans, Partial differential equations, Grad. Stud. Math., 19, American Mathematical
Society, Providence, RI (2010)
[9] R. Farwig, Weighted Lq -Helmholtz decompositions in infinite cylinders and in infinite layers,
Adv. Differential Equation, 8 (2003), no. 3, 357–384
[10] R. Farwig and H. Sohr, Helmholtz decomposition and Stokes resolvent system for aperture
domains in Lq -spaces, Analysis, 16 (1996), no. 1, 1–26
39
[11] R. Farwig, H. Kozono and H. Sohr, An Lq -approach to Stokes and Navier-Stokes equations
in general domains, Acta Math., 195 (2005), 21–53
[12] R. Farwig, H. Kozono and H. Sohr, On the Helmholtz decomposition in general unbounded
domains, Arch. Math. (Basel), 88 (2007), no. 3, 239–248
[13] H. Federer, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), 418–491
[14] D. Fujiwara and H. Morimoto, An Lr -theorem of the Helmholtz decomposition of vector
fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), no. 3, 685–700
[15] G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations.
Steady-state problems. Second edition, Springer Monogr. Math., Springer, New York (2011)
[16] F. W. Gehring and B. G. Osgood, Uniform domains and the quasihyperbolic metric, J.
Analyse Math., 36 (1979), 50–74
[17] Y. Giga and Z. Gu, On the Helmholtz decompositions of vector fields of bounded mean
oscillation and in real Hardy spaces over the half space, Adv. Math. Sci. Appl., 29 (2020),
no. 1, 87–128
[18] Y. Giga and Z. Gu, Normal trace for a vector field of bounded mean oscillation, Potential
Anal., 59 (2023), 409–434
[19] Y. Giga and Z. Gu, The Helmholtz decomposition of a space of vector fields with bounded
mean oscillation in a bounded domain, Math. Ann., 386 (2023), 673–712
[20] Y. Giga and Z. Gu, The Helmholtz decomposition of a BM O type vector field in a slightly
perturbed half space, J. Math. Fluid Mech., 25 (2023), no. 2, Paper No. 41, 46 pp
[21] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order,
Second edition, Grundlehren Math. Wiss., 224 [Fundamental Principles of Mathematical
Sciences], Springer-Verlag, Berlin (1983)
[22] L. Grafakos, Classical Fourier analysis, Third edition, Grad. Texts in Math., 249, Springer,
New York (2014)
[23] Z. Gu, Extension theorem for bmo in a domain, preprint (2023)
[24] M. Hieber, H. Kozono, A. Seyfert, S. Shimizu and T. Yanagisawa, A characterization of
harmonic Lr -vector fields in two-dimensional exterior domains, J. Geom. Anal., 30 (2020),
no. 4, 3742–3759
[25] M. Hieber, H. Kozono, A. Seyfert, S. Shimizu and T. Yanagisawa, The Helmholtz-Weyl
decomposition of Lr vector fields for two dimensional exterior domains, J. Geom. Anal., 31
(2021), no. 5, 5146–5165
[26] M. Hieber, H. Kozono, A. Seyfert, S. Shimizu and T. Yanagisawa, Lr Helmholtz-Weyl
decomposition for three dimensional exterior domains, J. Funct. Anal., 281 (2021), no. 8,
Paper No. 109144, 52 pp
40
[27] M. Hieber, H. Kozono, A. Seyfert, S. Shimizu and T. Yanagisawa, A characterization of
harmonic Lr -vector fields in three dimensional exterior domains, J. Geom. Anal., 32 (2022),
no. 7, Paper No. 206, 26 pp
[28] P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces,
Acta Math., 147 (1981), no. 1-2, 71–88
[29] P. W. Jones, Extension theorems for BM O, Indiana Univ. Math. J., 29 (1980), no. 1, 41–66
[30] H. Kozono and H. Wadade, Remarks on Gagliardo-Nirenberg type inequality with critical
Sobolev space and BM O, Math. Z., 259 (2008), no. 4, 935–950
[31] H. Kozono and T. Yanagisawa, Lr -variational inequality for vector fields and the HelmholtzWeyl decomposition in bounded domains, Indiana Univ. Math. J., 58 (2009), 1853–1920
[32] S. G. Krantz and H. R. Parks, The implicit function theorem. History, theory, and applications, Reprint of the 2003 edition, Mod. Birkh¨auser Class., Birkh¨auser/Springer, New York
(2013)
[33] J. M. Lee, Introduction to smooth manifolds, Second edition, Grad. Texts in Math., 218,
Springer, New York (2013)
[34] O. Martio and J. Sarvas, Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn.
Ser. A I Math., 4 (1979), no. 2, 383–401
[35] V. N. Maslennikova and M. E. Bogovski˘ı, Elliptic boundary value problems in unbounded
domains with noncompact and nonsmooth boundaries, Rend. Sem. Mat. Fis. Milano, 56
(1986), 125–138
[36] T. Miyakawa, The Helmholtz decomposition of vector fields in some unbounded domains,
Math. J. Toyama Univ., 17 (1994), 115–149
[37] T. Miyakawa, Hardy spaces of solenoidal vector fields, with applications to the Navier-Stokes
equations, Kyushu J. Math., 50 (1996), no. 1, 1–64
[38] J. Neˇcas, Direct methods in the theory of elliptic equations, Springer Monogr. Math.,
Springer, Heidelberg (2012)
[39] C. G. Simader and H. Sohr, A new approach to the Helmholtz decomposition and the
Neumann problem in Lq -spaces for bounded and exterior domains, Ser. Adv. Math. Appl.
Sci., 11, World Scientific Publishing Co., Inc., River Edge, NJ (1992), 1–35
[40] H. Sohr, The Navier-Stokes equations. An elementary functional analytic approach, Mod.
Birkh¨auser Class., Birkh¨
auser/Springer Basel AG, Basel (2001)
[41] V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations, Journal
of Soviet Mathematics, 8 (1977), 467–529
[42] S. G. Staples, Lp -averaging domains and the Poincar´e inequality, Ann. Acad. Sci. Fenn.
Ser. A. I Math., 14 (1989), no. 1, 103–127
41
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