Abstract
Let Ω ⊂ Rn be a strongly Lipschitz domain. In this article, the authors study Hardy spaces, Hpr (Ω)and Hpz (Ω), and Hardy-Sobolev spaces, H1,pr (Ω) and H1,pz,0 (Ω) on , for p ∈ ( n/n+1, 1]. The authors establish grand maximal function characterizations of these spaces. As applications, the authors obtain some div-curl lemmas in these settings and, when is a bounded Lipschitz domain, the authors prove that the divergence equation div u = f for f ∈ Hpz (Ω) is solvable in H1,pz,0 (Ω) with suitable regularity estimates.
References
[1] G. Acosta, R. G. Durán, M. A. Muschietti, Solution of the divergence operator on John domains, Adv. Math. 206 (2006), 373-401. 10.1016/j.aim.2005.09.004Search in Google Scholar
[2] P. Auscher, E. Russ, Hardy spaces and divergence operators on strongly Lipschitz domains of Rn, J. Funct. Anal. 201 (2003), 148-184. 10.1016/S0022-1236(03)00059-4Search in Google Scholar
[3] P. Auscher, E. Russ, P. Tchamitchian, Hardy Sobolev spaces on strongly Lipschitz domains of Rn, J. Funct. Anal. 218 (2005), 54-109. 10.1016/j.jfa.2004.06.005Search in Google Scholar
[4] P. Auscher, E. Russ, P. Tchamitchian, Corrigendum to “Hardy Sobolev spaces on strongly Lipschitz domains of Rn", J. Funct. Anal. 253 (2007), 782-785. 10.1016/j.jfa.2007.02.003Search in Google Scholar
[5] J. Bourgain, H. Brezis, On the equation div Y = f and application to control of phases, J. Amer.Math. Soc. 16 (2003), 393-426. 10.1090/S0894-0347-02-00411-3Search in Google Scholar
[6] J. Cao, D.-C. Chang, D. Yang, S. Yang, Weighted local Orlicz-Hardy spaces on domains and their applications in inhomogeneous Dirichlet and Neumann problems, Trans. Amer. Math. Soc. 365 (2013), 4729-4809. 10.1090/S0002-9947-2013-05832-1Search in Google Scholar
[7] D.-C. Chang, G. Dafni, E. M. Stein, Hardy spaces, BMO, and boundary value problems for the Laplacian on a smooth domain in RN, Trans. Amer. Math. Soc. 351 (1999), 1605-1661. 10.1090/S0002-9947-99-02111-XSearch in Google Scholar
[8] D.-C. Chang, S. G. Krantz, E. M. Stein, Hp theory on a smooth domain in Rn and elliptic boundary value problems, J. Funct. Anal. 114 (1993), 286-347. 10.1006/jfan.1993.1069Search in Google Scholar
[9] L. Diening, M. Ružicka, K. Schumacher, A decomposition technique for John domains, Ann. Acad. Sci. Fenn.Math. 35 (2010), 87-114. 10.5186/aasfm.2010.3506Search in Google Scholar
[10] X. T. Duong, S. Hofmann, D. Mitrea, M. Mitrea, L. X. Yan, Hardy spaces and regularity for the inhomogeneous Dirichlet and Neumann problems, Rev. Mat. Iberoam. 29 (2013), 183-236. 10.4171/RMI/718Search in Google Scholar
[11] R. G. Durán, M. A.Muschietti, E. Russ, P. Tchamitchian, Divergence operator and Poincaré inequalities on arbitrary bounded domains, Complex Var. Elliptic Equ. 55 (2010), 795-816. 10.1080/17476931003786659Search in Google Scholar
[12] L. C. Evans, R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. viii+268 pp. Search in Google Scholar
[13] R. Gong, J. Li, Hardy-Sobolev spaces on product domains and application, J. Math. Anal. Appl. 377 (2011), 296-302. 10.1016/j.jmaa.2010.10.057Search in Google Scholar
[14] R. Jiang, A. Kauranen, P. Koskela, Solvability of the divergence equation implies John via Poincaré inequality, Nonlinear Anal. 101 (2014), 80-88. 10.1016/j.na.2014.01.021Search in Google Scholar
[15] A. Jonsson, P. Sjögren, H. Wallin, Hardy and Lipschitz spaces on subsets of Rn, Studia Math. 80 (1984), 141-166. 10.4064/sm-80-2-141-166Search in Google Scholar
[16] P. Koskela, E. Saksman, Pointwise Characterizations of Hardy-Sobolev functions, Math. Res. Lett. 15(2008), 727-744. 10.4310/MRL.2008.v15.n4.a11Search in Google Scholar
[17] Z. J. Lou, S. Z. Yang, An atomic decomposition for the Hardy-Sobolev space, Taiwanese J. Math. 11 (2007), 1167-1176. 10.11650/twjm/1500404810Search in Google Scholar
[18] Z. J. Lou, A. McIntosh, Hardy spaces of exact forms on Lipschitz domains in Rn, Indiana Univ. Math. J. 53 (2004), 583-611. 10.1512/iumj.2004.53.2395Search in Google Scholar
[19] D. Mitrea, I. Mitrea, M. Mitrea, L. X. Yan, Coercive energy estimates for differential forms in semi-convex domains, Commun. Pure Appl. Anal. 9 (2010), 987-1010. 10.3934/cpaa.2010.9.987Search in Google Scholar
[20] D. Mitrea, M. Mitrea, L. X. Yan, Boundary value problems for the Laplacian in convex and semiconvex domains, J. Funct. Anal. 258 (2010), 2507-2585. 10.1016/j.jfa.2010.01.012Search in Google Scholar
[21] A. Miyachi, Hp spaces over open subsets of Rn, Studia Math., 95 (3) (1990), 205-228. 10.4064/sm-95-3-205-228Search in Google Scholar
[22] A. Miyachi, Hardy-Sobolev spaces and maximal functions, J. Math. Soc. Japan 42 (1990), 73-90. 10.2969/jmsj/04210073Search in Google Scholar
[23] A. Miyachi, Extension theorems for real variable Hardy and Hardy-Sobolev spaces, in: Harmonic Analysis (Sendai, 1990), 170-182, ICM-90 Satell. Conf. Proc., Springer, Tokyo, 1991. 10.1007/978-4-431-68168-7_15Search in Google Scholar
[24] J. Orobitg, Spectral synthesis in spaces of functions with derivatives in H1, in: Harmonic Analysis and Partial Differential Equations (El Escorial, 1987), 202-206, Lecture Notes in Math., 1384, Springer, Berlin, 1989. 10.1007/BFb0086804Search in Google Scholar
[25] R. S. Strichartz, Hp Sobolev spaces, Colloq. Math. 60/61 (1990), 129-139. 10.4064/cm-60-61-1-129-139Search in Google Scholar
[26] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993. xiv+695 pp. 10.1515/9781400883929Search in Google Scholar
[27] D. Yang, S. Yang, Orlicz-Hardy spaces associated with divergence operators on unbounded strongly Lipschitz domains of Rn, Indiana Univ. Math. J. 61 (2012), 81-129. 10.1512/iumj.2012.61.4535Search in Google Scholar
[28] D. Yang, S. Yang, Real-variable characterizations of Orlicz-Hardy spaces on strongly Lipschitz domains of Rn, Rev. Mat. Iberoam. 29 (2013), 237-292. 10.4171/RMI/719Search in Google Scholar
© 2016 Xiaming Chen et al.
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.
