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

Chang, Der-Chen; Dafni, Galia; Stein, Elias

doi:10.1090/S0002-9947-99-02111-X

Hardy spaces, BMO, and boundary value problems for the Laplacian on a smooth domain in $\mathbf R^N$
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by Der-Chen Chang, Galia Dafni and Elias M. Stein

Trans. Amer. Math. Soc. 351 (1999), 1605-1661

DOI: https://doi.org/10.1090/S0002-9947-99-02111-X

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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.

References

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 125307, DOI 10.1002/cpa.3160120405
Der-Chen Chang, The dual of Hardy spaces on a bounded domain in $\textbf {R}^n$, Forum Math. 6 (1994), no. 1, 65–81. MR 1253178, DOI 10.1515/form.1994.6.65
Der-Chen Chang, Steven G. Krantz, and Elias M. Stein, $H^p$ theory on a smooth domain in $\textbf {R}^N$ and elliptic boundary value problems, J. Funct. Anal. 114 (1993), no. 2, 286–347. MR 1223705, DOI 10.1006/jfan.1993.1069
G. Dafni, Hardy Spaces on Strongly Pseudoconvex Domains in $\mathbb {C}^n$ and Domains of Finite Type in $\mathbb {C}^2$, Ph.D. Thesis, Princeton University, 1993.
G. Dafni, Distributions supported in a hypersurface and local $h^p$, Proc. Amer. Math. Soc. 126 (1998), 2933–2943.
Gerald B. Folland, Introduction to partial differential equations, Mathematical Notes, Princeton University Press, Princeton, N.J., 1976. Preliminary informal notes of university courses and seminars in mathematics. MR 0599578, DOI 10.1515/9780691213033
C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
David Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979), no. 1, 27–42. MR 523600
P. C. Greiner and E. M. Stein, Estimates for the $\overline \partial$-Neumann problem, Mathematical Notes, No. 19, Princeton University Press, Princeton, N.J., 1977. MR 0499319
Peter W. Jones, Extension theorems for BMO, Indiana Univ. Math. J. 29 (1980), no. 1, 41–66. MR 554817, DOI 10.1512/iumj.1980.29.29005
Alf Jonsson, Peter Sjögren, and Hans Wallin, Hardy and Lipschitz spaces on subsets of $\textbf {R}^n$, Studia Math. 80 (1984), no. 2, 141–166. MR 781332, DOI 10.4064/sm-80-2-141-166
S. G. Krantz, S. Y. Li, Elliptic boundary value problems for the inhomogeneous Laplace equation on bounded domains, preprint.
Akihiko Miyachi, $H^p$ spaces over open subsets of $\textbf {R}^n$, Studia Math. 95 (1990), no. 3, 205–228. MR 1060724, DOI 10.4064/sm-95-3-205-228
Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
V. S. Rychkov, Intrinsic characterizations of distribution spaces on domains, Studia Math. 127 (1998), 227–298.
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
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
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972