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Weighted local Orlicz-Hardy spaces on domains and their applications in inhomogeneous Dirichlet and Neumann problems


Authors: Jun Cao, Der-Chen Chang, Dachun Yang and Sibei Yang
Journal: Trans. Amer. Math. Soc. 365 (2013), 4729-4809
MSC (2010): Primary 42B35; Secondary 42B30, 42B20, 42B25, 35J25, 42B37, 47B38, 46E30
DOI: https://doi.org/10.1090/S0002-9947-2013-05832-1
Published electronically: February 27, 2013
MathSciNet review: 3066770
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Abstract: Let $ \Omega $ be either $ \mathbb{R}^n$ or a strongly Lipschitz domain of $ \mathbb{R}^n$, and $ \omega \in A_{\infty }(\mathbb{R}^n)$ (the class of Muckenhoupt weights). Let $ L$ be a second-order divergence form elliptic operator on $ L^2 (\Omega )$ with the Dirichlet or Neumann boundary condition, and assume that the heat semigroup generated by $ L$ has the Gaussian property $ (G_1)$ with the regularity of their kernels measured by $ \mu \in (0,1]$. Let $ \Phi $ be a continuous, strictly increasing, subadditive, positive and concave function on $ (0,\infty )$ of critical lower type index $ p_{\Phi }^-\in (0,1]$. In this paper, the authors first introduce the ``geometrical'' weighted local Orlicz-Hardy spaces $ h^{\Phi }_{\omega ,\,r}(\Omega )$ and $ h^{\Phi }_{\omega ,\,z}(\Omega )$ via the weighted local Orlicz-Hardy spaces $ h^{\Phi }_{\omega }(\mathbb{R}^n)$, and obtain their two equivalent characterizations in terms of the nontangential maximal function and the Lusin area function associated with the heat semigroup generated by $ L$ when $ p_{\Phi }^-\in (n/(n+\mu ),1]$. Second, the authors furthermore establish three equivalent characterizations of $ h^{\Phi }_{\omega ,\,r}(\Omega )$ in terms of the grand maximal function, the radial maximal function and the atomic decomposition when the complement of $ \Omega $ is unbounded and $ p_{\Phi }^-\in (0,1]$. Third, as applications, the authors prove that the operators $ \nabla ^2{\mathbb{G}}_D$ are bounded from $ h^{\Phi }_{\omega ,\,r}(\Omega )$ to the weighted Orlicz space $ L^{\Phi }_{\omega }(\Omega )$, and from $ h^{\Phi }_{\omega ,\,r}(\Omega )$ to itself when $ \Omega $ is a bounded semiconvex domain in $ \mathbb{R}^n$ and $ p_{\Phi }^-\in (\frac {n}{n+1},1]$, and the operators $ \nabla ^2{\mathbb{G}}_N$ are bounded from $ h^{\Phi }_{\omega ,\,z}(\Omega )$ to $ L^{\Phi }_{\omega }(\Omega )$, and from $ h^{\Phi }_{\omega ,\,z}(\Omega )$ to $ h^{\Phi }_{\omega ,\,r}(\Omega )$ when $ \Omega $ is a bounded convex domain in $ \mathbb{R}^n$ and $ p_{\Phi }^-\in (\frac {n}{n+1},1]$, where $ {\mathbb{G}}_D$ and $ {\mathbb{G}}_N$ denote, respectively, the Dirichlet Green operator and the Neumann Green operator.


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  • 1. V. Adolfsson, $ L^2$-integrability of second-order derivatives for Poisson's equation in nonsmooth domains, Math. Scand. 70 (1992), 146-160. MR 1174208 (93h:35047)
  • 2. V. Adolfsson, $ L^p$-integrability of the second order derivatives of Green potentials in convex domains, Pacific J. Math. 159 (1993), 201-225. MR 1214070 (94c:35041)
  • 3. V. Adolfsson and D. Jerison, $ L^p$-integrability of the second order derivatives for the Neumann problem in convex domains, Indiana Univ. Math. J. 43 (1994), 1123-1138. MR 1322613 (96e:35031)
  • 4. S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623-727. MR 0125307 (23:A2610)
  • 5. T. Aoki, Locally bounded linear topological space, Proc. Imp. Acad. Tokyo 18 (1942), 588-594. MR 0014182 (7:250d)
  • 6. P. Auscher, X. T. Duong and A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces, Unpublished Manuscript, 2005.
  • 7. P. Auscher, A. McIntosh and E. Russ, Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal. 18 (2008), 192-248. MR 2365673 (2009d:42053)
  • 8. P. Auscher and E. Russ, Hardy spaces and divergence operators on strongly Lipschitz domains of $ \mathbb{R}^n$, J. Funct. Anal. 201 (2003), 148-184. MR 1986158 (2004c:42049)
  • 9. P. Auscher and Ph. Tchamitchian, Square root problem for divergence operators and related topics, Astérisque 249 (1998), viii+172pp. MR 1651262 (2000c:47092)
  • 10. P. Auscher and Ph. Tchamitchian, Gaussian estimates for second order elliptic divergence operators on Lipschitz and $ C^1$ domains, Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), pp. 15-32, Lecture Notes in Pure and Applied Math., 215, Dekker, New York, 2001. MR 1816433 (2001m:35074)
  • 11. Z. Birnbaum and W. Orlicz, Über die verallgemeinerung des begriffes der zueinander konjugierten potenzen, Studia Math. 3 (1931), 1-67.
  • 12. H. Bui, Weighted Hardy spaces, Math. Nachr. 103 (1981), 45-62. MR 653914 (83h:42026)
  • 13. S.-S. Byun, F. Yao and S. Zhou, Gradient estimates in Orlicz space for nonlinear elliptic equations, J. Funct. Anal. 255 (2008), 1851-1873. MR 2462578 (2010b:35153)
  • 14. A. Carbonaro, A. McIntosh and A. J. Morris, Local Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal. 23 (2013), 106-169.
  • 15. D.-C. Chang, The dual of Hardy spaces on a bounded domain in $ \mathbb{R}^n$, Forum Math. 6 (1994), 65-81. MR 1253178 (95b:42022)
  • 16. D.-C. Chang, G. Dafni and E. M. Stein, Hardy spaces, $ \mathop \mathrm {BMO}$ and boundary value problems for the Laplacian on a smooth domain in $ \mathbb{R}^n$, Trans. Amer. Math. Soc. 351 (1999), 1605-1661. MR 1458319 (99f:46031)
  • 17. D.-C. Chang, S. G. Krantz and E. M. Stein, Hardy spaces and elliptic boundary value problems, Contemporary Math. 137 (1992), 119-131. MR 1190976
  • 18. D.-C. Chang, S. G. Krantz and E. M. Stein, $ H^p$ theory on a smooth domain in $ \mathbb{R}^N$ and elliptic boundary value problems, J. Funct. Anal. 114 (1993), 286-347. MR 1223705 (94j:46032)
  • 19. R. R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9) 72 (1993), 247-286. MR 1225511 (95d:46033)
  • 20. R. R. Coifman, Y. Meyer and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), 304-335. MR 791851 (86i:46029)
  • 21. R. R. Coifman and G. Weiss, Analyse Harmonique Non-commutative sur Certains Espaces Homogènes, Lecture Notes in Math., 242, Springer, Berlin, 1971. MR 0499948 (58:17690)
  • 22. R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645. MR 0447954 (56:6264)
  • 23. D. Cruz-Uribe and C. J. Neugebauer, The structure of the reverse Hölder classes, Trans. Amer. Math. Soc. 347 (1995), 2941-2960. MR 1308005 (95m:42026)
  • 24. B. Dahlberg, $ L^{q}$-estimates for Green potentials in Lipschitz domains, Math. Scand. 44 (1979), 149-170. MR 544584 (81d:31007)
  • 25. B. Dahlberg, G. Verchota and T. Wolff, Unpublished manuscript.
  • 26. X. T. Duong, S. Hofmann, D. Mitrea, M. Mitrea and L. Yan, Hardy spaces and regularity for the inhomogeneous Dirichlet and Neumann problems, Rev. Mat. Iberoam. 29 (2013), 183-236.
  • 27. X. T. Duong and L. Yan, On the atomic decomposition for Hardy spaces on Lipschitz domains of $ {\mathbb{R}}^n$, J. Funct. Anal. 215 (2004), 476-486. MR 2151301 (2006d:42040)
  • 28. X. T. Duong and L. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc. 18 (2005), 943-973. MR 2163867 (2006d:42037)
  • 29. X. T. Duong, J. Xiao and L. Yan, Old and new Morrey spaces with heat kernel bounds, J. Fourier Anal. Appl. 13 (2007), 87-111. MR 2296729 (2007k:42027)
  • 30. C. Fefferman and E. M. Stein, $ H^p$ spaces of several variables, Acta Math. 129 (1972), 137-193. MR 0447953 (56:6263)
  • 31. S. Fromm, Potential space estimates for Green potentials in convex domains, Proc. Amer. Math. Soc. 119 (1993), 225-233. MR 1156467 (93k:35076)
  • 32. J. García-Cuerva, Weighted $ H^p$ spaces, Dissertationes Math. (Rozprawy Mat.) 162 (1979), 1-63. MR 549091 (82a:42018)
  • 33. J. García-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics, Amsterdam, North-Holland, 1985. MR 807149 (87d:42023)
  • 34. L. Grafakos, Modern Fourier Analysis, Second Edition, Graduate Texts in Math., No. 250, Springer, New York, 2008. MR 2463316 (2011d:42001)
  • 35. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683 (86m:35044)
  • 36. M. Grüter and K.-O. Widman, The Green function for uniformly elliptic equations, Manuscripta Math. 37 (1982), 303-342. MR 657523 (83h:35033)
  • 37. D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979), 27-42. MR 523600 (80h:46052)
  • 38. S. Hofmann, G. Lu, D. Mitrea, M. Mitrea and L. Yan, Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, Mem. Amer. Math. Soc. 214 (2011), no. 1007, vi+78 pp. MR 2868142
  • 39. S. Hofmann and S. Mayboroda, Hardy and BMO spaces associated to divergence form elliptic operators, Math. Ann. 344 (2009), 37-116. MR 2481054 (2009m:42038)
  • 40. S. Hofmann and S. Mayboroda, Correction to ``Hardy and BMO spaces associated to divergence form elliptic operators'', arXiv: 0907.0129v2.
  • 41. S. Hofmann, S. Mayboroda and A. McIntosh, Second order elliptic operators with complex bounded measurable coefficients in $ L^p$, Sobolev and Hardy spaces, Ann. Sci. École Norm. Sup. (4) 44 (2011), 723-800. MR 2931518
  • 42. G. Hu, D. Yang and Y. Zhou, Boundedness of singular integrals in Hardy spaces on spaces of homogeneous type, Taiwanese J. Math. 13 (2009), 91-135. MR 2489309 (2010a:42048)
  • 43. J. Huang, Hardy spaces associated to the Schrödinger operator on strongly Lipschitz domains of $ \mathbb{R}^d$, Math. Z. 266 (2010), 141-168. MR 2670676 (2011e:42046)
  • 44. J. Huang, A characterization of Hardy space on strongly Lipschitz domains of $ \mathbb{R}^n$ by Littlewood-Paley-Stein function, Commun. Contemp. Math. 12 (2010), 71-84. MR 2649228 (2011c:42034)
  • 45. T. Iwaniec and J. Onninen, $ \mathcal {H}^1$-estimates of Jacobians by subdeterminants, Math. Ann. 324 (2002), 341-358. MR 1933861 (2003i:42027)
  • 46. S. Janson, Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation, Duke Math. J. 47 (1980), 959-982. MR 596123 (83j:46037)
  • 47. D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130 (1995), 161-219. MR 1331981 (96b:35042)
  • 48. R. Jiang and D. Yang, Orlicz-Hardy spaces associated with operators satisfying Davies-Gaffney estimates, Commun. Contemp. Math. 13 (2011), 331-373. MR 2794490 (2012e:42040)
  • 49. R. Jiang and D. Yang, New Orlicz-Hardy spaces associated with divergence form elliptic operators, J. Funct. Anal. 258 (2010), 1167-1224. MR 2565837 (2011e:42047)
  • 50. R. Jiang, Da. Yang and Do. Yang, Maximal function characterizations of Hardy spaces associated with magnetic Schrödinger operators, Forum Math. 24 (2012), 471-494.
  • 51. R. Jiang, D. Yang and Y. Zhou, Orlicz-Hardy spaces associated with operators, Sci. China Ser. A 52 (2009), 1042-1080. MR 2505009 (2011b:46054)
  • 52. R. Jiang, D. Yang and Y. Zhou, Localized Hardy spaces associated with operators, Appl. Anal. 88 (2009), 1409-1427. MR 2574337 (2011b:42071)
  • 53. R. Johnson and C. J. Neugebauer, Homeomorphisms preserving $ A_p$, Rev. Mat. Ibero. 3 (1987), 249-273. MR 990859 (90d:42013)
  • 54. J. Kadlec, The regularity of the solution of the Poisson problem in a domain whose boundary is similar to that of a convex domain, Czechoslovak Math. J. 14 (1964), 386-393. MR 0170088 (30:329)
  • 55. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966. MR 0203473 (34:3324)
  • 56. K. Kurata, An estimate on the heat kernel of magnetic Schrödinger operators and uniformly elliptic operators with non-negative potentials, J. London Math. Soc. (2) 62 (2000), 885-903. MR 1794292 (2001j:35046)
  • 57. J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, 1972. MR 0350177 (50:2670)
  • 58. R. A. Macías and C. Segovia, A decomposition into atoms of distributions on spaces of homogeneous type, Adv. Math. 33 (1979), 271-309. MR 546296 (81c:32017b)
  • 59. S. Martínez and N. Wolanski, A minimum problem with free boundary in Orlicz spaces, Adv. Math. 218 (2008), 1914-1971. MR 2431665 (2009h:35456)
  • 60. S. Mayboroda and M. Mitrea, Sharp estimates for Green potentials on non-smooth domains, Math. Res. Lett. 11 (2004), 481-492. MR 2092902 (2005i:35059)
  • 61. S. Mayboroda and M. Mitrea, The solution of the Chang-Krantz-Stein conjecture, Proceedings of the conference Harmonic Analysis and its Applications at Tokyo (Zempukuji, 2007), pp. 61-154, Tokyo Woman's Christian University, Tokyo, 2007.
  • 62. A. McIntosh, Operators which have an $ H_\infty $ functional calculus, Miniconference on operator theory and partial differential equations (North Ryde, 1986), pp. 210-231, Proc. Centre Math. Anal. Austral. Nat. Univ., 14, Austral. Nat. Univ., Canberra, 1986. MR 912940 (88k:47019)
  • 63. D. Mitrea, M. Mitrea and L. Yan, Boundary value problems for the Laplacian in convex and semiconvex domains, J. Funct. Anal. 258 (2010), 2507-2585. MR 2593333 (2010m:42049)
  • 64. D. Mitrea, I. Mitrea, M. Mitrea and L. Yan, Coercive energy estimates for differential forms in semi-convex domains, Commun. Pure Appl. Anal. 9 (2010), 987-1010. MR 2610257 (2011i:35052)
  • 65. A. Miyachi, Maximal functions for distributions on open sets, Hitotsubashi J. Arts Sci. 28 (1987), 45-58. MR 939226 (90k:42038)
  • 66. A. Miyachi, $ H^p$ spaces over open subsets of $ \mathbb{R}^n$, Studia Math. 95 (1990), 205-228. MR 1060724 (91m:42022)
  • 67. S. Müller, Hardy space methods for nonlinear partial differential equations, Tatra Mt. Math. Publ. 4 (1994), 159-168. MR 1298466 (95f:35191)
  • 68. W. Orlicz, Über eine gewisse Klasse von Räumen vom Typus B, Bull. Int. Acad. Pol. Ser. A 8 (1932), 207-220.
  • 69. E. M. Ouhabaz, Analysis of Heat Equations on Domains, Princeton University Press, Princeton, N. J., 2005. MR 2124040 (2005m:35001)
  • 70. M. Rao and Z. Ren, Theory of Orlicz Spaces, Marcel Dekker, Inc., New York, 1991. MR 1113700 (92e:46059)
  • 71. M. Rao and Z. Ren, Applications of Orlicz Spaces, Marcel Dekker, Inc., New York, 2002. MR 1890178 (2003e:46041)
  • 72. S. Rolewicz, On a certain class of linear metric spaces, Bull. Acad. Polon. Sci. Cl. III. 5 (1957), 471-473. MR 0088682 (19:562d)
  • 73. E. Russ, The atomic decomposition for tent spaces on spaces of homogeneous type, CMA/AMSI Research Symposium ``Asymptotic Geometric Analysis, Harmonic Analysis, and Related Topics'', pp. 125-135, Proc. Centre Math. Appl., 42, Austral. Nat. Univ., Canberra, 2007. MR 2328517 (2008m:46066)
  • 74. V. S. Rychkov, Littlewood-Paley theory and function spaces with $ A^{\textup {loc}}_p$ weights, Math. Nachr. 224 (2001), 145-180. MR 1821243 (2002k:42045)
  • 75. S. Semmes, A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller, Comm. Partial Differential Equations 19 (1994), 277-319. MR 1257006 (94j:46038)
  • 76. E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, N. J., 1993. MR 1232192 (95c:42002)
  • 77. E. M. Stein and G. Weiss, On the theory of harmonic functions of several variables. I. The theory of $ H^p$-spaces, Acta Math. 103 (1960), 25-62. MR 0121579 (22:12315)
  • 78. L. Tang, Weighted local Hardy spaces and their applications, Illinois J. Math. (to appear) or arXiv: 1004.5294.
  • 79. H. Triebel, Theory of Function Spaces, Birkhäuser Verlag, Basel, 1983. MR 781540 (86j:46026)
  • 80. H. Triebel, Theory of Function Spaces II, Birkhäuser Verlag, Basel, 1992. MR 1163193 (93f:46029)
  • 81. H. Triebel and H. Winkelvoß, Intrinsic atomic characterizations of function spaces on domains, Math. Z. 221 (1996), 647-673. MR 1385173 (97b:46048)
  • 82. B. E. Viviani, An atomic decomposition of the predual of $ {\rm BMO}(\rho )$, Rev. Mat. Iberoamericana 3 (1987), 401-425. MR 996824 (90e:46024)
  • 83. H. Wang and H. Jia, Potential space estimates in local Hardy spaces for Green potentials in convex domains, Anal. Theory Appl. 20 (2004), 342-349. MR 2120009
  • 84. H. Wang and X. Yang, The characterization of the weighted local Hardy spaces on domains and its application, J. Zhejiang Univ. Sci. 9 (2004), 1148-1154.
  • 85. L. Yan, Classes of Hardy spaces associated with operators, duality theorem and applications, Trans. Amer. Math. Soc. 360 (2008), 4383-4408. MR 2395177 (2010f:42045)
  • 86. D. Yang and S. Yang, Weighted local Orlicz-Hardy spaces with applications to pseudo-differential operators, Dissertationes Math. (Rozprawy Mat.) 478 (2011), 1-78. MR 2848094 (2012h:46056)
  • 87. D. Yang and S. Yang, Real-variable characterizations of Orlicz-Hardy spaces on strongly Lipschitz domains of $ \mathbb{R}^n$, Rev. Mat. Iberoam. 29 (2013), 237-292.
  • 88. D. Yang and S. Yang, Orlicz-Hardy spaces associated with divergence operators on unbounded strongly Lipschitz domains of $ \mathbb{R}^n$, Indiana Univ. Math. J. 61 (2012), 81-129.
  • 89. K. Yosida, Functional Analysis, Springer-Verlag, Berlin, 1995. MR 1336382 (96a:46001)

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Additional Information

Jun Cao
Affiliation: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
Email: caojun1860@mail.bnu.edu.cn

Der-Chen Chang
Affiliation: Department of Mathematics and Department of Computer Science, Georgetown University, Washington, DC 20057
Email: chang@georgetown.edu

Dachun Yang
Affiliation: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
Email: dcyang@bnu.edu.cn

Sibei Yang
Affiliation: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
Email: yangsibei@mail.bnu.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-2013-05832-1
Keywords: Strongly Lipschitz domain, semiconvex domain, convex domain, divergence form elliptic operator, Dirichlet Green operator, Neumann Green operator, Dirichlet boundary condition, Neumann boundary condition, weight, local Orlicz-Hardy space, Gaussian property, nontangential maximal function, Lusin area function, atom.
Received by editor(s): July 25, 2011
Published electronically: February 27, 2013
Additional Notes: The second author was partially supported by an NSF grant DMS-1203845 and a Hong Kong RGC competitive earmarked research grant $#$601410.
The third (corresponding) author was supported by the National Natural Science Foundation of China (Grant No. 11171027) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20120003110003).
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