Hardy spaces, BMO, and boundary value problems for the Laplacian on a smooth domain in
Authors:
DerChen Chang, Galia Dafni and Elias M. Stein
Journal:
Trans. Amer. Math. Soc. 351 (1999), 16051661
MSC (1991):
Primary 35J25, 42B25; Secondary 46E15, 42B30
MathSciNet review:
1458319
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We study two different local spaces, , on a smooth domain in , 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.
 [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 (23 #A2610)
 [C]
DerChen
Chang, The dual of Hardy spaces on a bounded domain in
𝑅ⁿ, Forum Math. 6 (1994), no. 1,
65–81. MR
1253178 (95b:42022), http://dx.doi.org/10.1515/form.1994.6.65
 [CKS]
DerChen
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
(94j:46032), http://dx.doi.org/10.1006/jfan.1993.1069
 [D]
G. Dafni, Hardy Spaces on Strongly Pseudoconvex Domains in and Domains of Finite Type in , Ph.D. Thesis, Princeton University, 1993.
 [D2]
G. Dafni, Distributions supported in a hypersurface and local , Proc. Amer. Math. Soc. 126 (1998), 29332943. 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 (58 #29031)
 [FS]
C.
Fefferman and E.
M. Stein, 𝐻^{𝑝} spaces of several variables,
Acta Math. 129 (1972), no. 34, 137–193. MR 0447953
(56 #6263)
 [G]
David
Goldberg, A local version of real Hardy spaces, Duke Math. J.
46 (1979), no. 1, 27–42. MR 523600
(80h:46052)
 [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 (58 #17218)
 [J]
Peter
W. Jones, Extension theorems for BMO, Indiana Univ. Math. J.
29 (1980), no. 1, 41–66. MR 554817
(81b:42047), http://dx.doi.org/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
(87b:46022)
 [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
(91m:42022)
 [R]
Walter
Rudin, Functional analysis, 2nd ed., International Series in
Pure and Applied Mathematics, McGrawHill, Inc., New York, 1991. MR 1157815
(92k:46001)
 [Ry]
V. S. Rychkov, Intrinsic characterizations of distribution spaces on domains, Studia Math. 127 (1998), 227298. 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
(44 #7280)
 [S2]
Elias
M. Stein, Harmonic analysis: realvariable 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
(95c:42002)
 [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
(46 #4102)
 [ADN]
 S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Applied Math. 12 (1959), 623727. MR 23:A2610
 [C]
 D.C. Chang, The dual of Hardy spaces on a bounded domain in , Forum Math. 6 No. 1 (1994), 6581. MR 95b:42022
 [CKS]
 D.C. Chang, S. G. Krantz, and E. M. Stein, Theory on a smooth domain in and elliptic boundary value problems, J. Funct. Anal. 114, No. 2 (1993), 286347. MR 94j:46032
 [D]
 G. Dafni, Hardy Spaces on Strongly Pseudoconvex Domains in and Domains of Finite Type in , Ph.D. Thesis, Princeton University, 1993.
 [D2]
 G. Dafni, Distributions supported in a hypersurface and local , Proc. Amer. Math. Soc. 126 (1998), 29332943. CMP 98:16
 [F]
 G. B. Folland, Introduction to Partial Differential Equations, Math. Notes 17, Princeton Univ. Press, Princeton, New Jersey, 1976. MR 58:29031
 [FS]
 C. Fefferman and E. M. Stein, spaces of several variables, Acta Math. 129 (1972), 137193. MR 56:6263
 [G]
 D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979), 2742. MR 80h:46052
 [GS]
 P. C. Greiner and E. M. Stein, Estimates for the Neumann Problem, Math. Notes 19, Princeton Univ. Press, Princeton, New Jersey, 1977. MR 58:17218
 [J]
 P. W. Jones, Extension theorems for , Indiana Univ. Math. J. bf 29, No. 1 (1980), 4166. MR 81b:42047
 [JSW]
 A. Jonsson, P. Sjögren, and H. Wallin, Hardy and Lipschitz spaces on subsets of , Studia Math. 80, No. 2 (1984), 141166. MR 87b:46022
 [KL]
 S. G. Krantz, S. Y. Li, Elliptic boundary value problems for the inhomogeneous Laplace equation on bounded domains, preprint.
 [M]
 A. Miyachi, spaces over open subsets of , Studia Math. 95, No. 3 (1990), 205228. MR 91m:42022
 [R]
 W. Rudin, Functional Analysis, Second Edition, McGrawHill, New York, 1991. MR 92k:46001
 [Ry]
 V. S. Rychkov, Intrinsic characterizations of distribution spaces on domains, Studia Math. 127 (1998), 227298. CMP 98:06
 [S1]
 E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, New Jersey, 1970. MR 44:7280
 [S2]
 E. M. Stein, Harmonic Analysis: RealVariable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, New Jersey, 1993. MR 95c:42002
 [SW]
 E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, New Jersey, 1971. MR 46:4102
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (1991):
35J25,
42B25,
46E15,
42B30
Retrieve articles in all journals
with MSC (1991):
35J25,
42B25,
46E15,
42B30
Additional Information
DerChen 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
Email:
drc@math.umd.edu
Galia Dafni
Affiliation:
Department of Mathematics & Statistics, Concordia University, Montreal, Quebec H3G1M8, Canada
Email:
gdafni@discrete.concordia.ca
Elias M. Stein
Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email:
stein@math.princeton.edu
DOI:
http://dx.doi.org/10.1090/S000299479902111X
PII:
S 00029947(99)02111X
Received by editor(s):
September 5, 1996
Received by editor(s) in revised form:
March 20, 1997
Article copyright:
© Copyright 1999
American Mathematical Society
