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Hardy spaces, BMO, and boundary value problems for the Laplacian on a smooth domain in $\mathbf{R}^n$

Authors: Der-Chen Chang, Galia Dafni and Elias M. Stein
Journal: Trans. Amer. Math. Soc. 351 (1999), 1605-1661
MSC (1991): Primary 35J25, 42B25; Secondary 46E15, 42B30
MathSciNet review: 1458319
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Abstract: We study two different local $H^p$ spaces, $0 < p \leq 1$, on a smooth domain in $\mathbf{R}^n$, by means of maximal functions and atomic decomposition. We prove the regularity in these spaces, as well as in the corresponding dual BMO spaces, of the Dirichlet and Neumann problems for the Laplacian.

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  • [ADN] 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
  • [C] Der-Chen Chang, The dual of Hardy spaces on a bounded domain in 𝑅ⁿ, Forum Math. 6 (1994), no. 1, 65–81. MR 1253178, 10.1515/form.1994.6.65
  • [CKS] Der-Chen Chang, Steven G. Krantz, and Elias M. Stein, 𝐻^{𝑝} theory on a smooth domain in 𝑅^{𝑁} and elliptic boundary value problems, J. Funct. Anal. 114 (1993), no. 2, 286–347. MR 1223705, 10.1006/jfan.1993.1069
  • [D] G. Dafni, Hardy Spaces on Strongly Pseudoconvex Domains in $\mathbf C^n$ and Domains of Finite Type in $\mathbf C^2$, Ph.D. Thesis, Princeton University, 1993.
  • [D2] G. Dafni, Distributions supported in a hypersurface and local $h^p$, Proc. Amer. Math. Soc. 126 (1998), 2933-2943. CMP 98:16
  • [F] Gerald B. Folland, Introduction to partial differential equations, Princeton University Press, Princeton, N.J., 1976. Preliminary informal notes of university courses and seminars in mathematics; Mathematical Notes. MR 0599578
  • [FS] C. Fefferman and E. M. Stein, 𝐻^{𝑝} spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 0447953
  • [G] David Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979), no. 1, 27–42. MR 523600
  • [GS] P. C. Greiner and E. M. Stein, Estimates for the \overline∂-Neumann problem, Princeton University Press, Princeton, N.J., 1977. Mathematical Notes, No. 19. MR 0499319
  • [J] Peter W. Jones, Extension theorems for BMO, Indiana Univ. Math. J. 29 (1980), no. 1, 41–66. MR 554817, 10.1512/iumj.1980.29.29005
  • [JSW] Alf Jonsson, Peter Sjögren, and Hans Wallin, Hardy and Lipschitz spaces on subsets of 𝑅ⁿ, Studia Math. 80 (1984), no. 2, 141–166. MR 781332
  • [KL] S. G. Krantz, S. Y. Li, Elliptic boundary value problems for the inhomogeneous Laplace equation on bounded domains, preprint.
  • [M] Akihiko Miyachi, 𝐻^{𝑝} spaces over open subsets of 𝑅ⁿ, Studia Math. 95 (1990), no. 3, 205–228. MR 1060724
  • [R] Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
  • [Ry] V. S. Rychkov, Intrinsic characterizations of distribution spaces on domains, Studia Math. 127 (1998), 227-298. CMP 98:06
  • [S1] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • [S2] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
  • [SW] Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972

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Additional Information

Der-Chen Chang
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Address at time of publication: Department of Mathematics, Georgetown University, Washingon, DC 20057

Galia Dafni
Affiliation: Department of Mathematics & Statistics, Concordia University, Montreal, Quebec H3G-1M8, Canada

Elias M. Stein
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544

Received by editor(s): September 5, 1996
Received by editor(s) in revised form: March 20, 1997
Article copyright: © Copyright 1999 American Mathematical Society