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An approximation property of harmonic functions in Lipschitz domains and some of its consequences
Author:
Jorge Rivera-Noriega
Translated by:
Journal:
Proc. Amer. Math. Soc. 132 (2004), 1321-1331
MSC (2000):
Primary 42B25, 35J67
Posted:
December 18, 2003
MathSciNet review:
2053336
Full-text PDF Free Access
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Abstract: An extension of an inequality of J. B. Garnett (1979), with improvements by B. E. J. Dahlberg (1980), on an approximation property of harmonic functions is proved. The weighted inequality proved here was suggested by the work of J. Pipher (1993) and it implies an extension of a result of S. Y. A. Chang, J. Wilson and T. Wolff (1985) and C. Sweezy (1991) on exponential square integrability of the boundary values of solutions to second-order linear differential equations in divergence form. This implies a solution of a problem left open by R. Bañuelos and C. N. Moore (1989) on sharp estimates for the area integral of harmonic functions in Lipschitz domains.
- 1.
Rodrigo
Bañuelos and Charles
N. Moore, Sharp estimates for the nontangential
maximal function and the Lusin area function in Lipschitz domains,
Trans. Amer. Math. Soc. 312 (1989),
no. 2, 641–662. MR 957080
(90i:42030), http://dx.doi.org/10.1090/S0002-9947-1989-0957080-4
- 2.
Rodrigo
Bañuelos and Charles
N. Moore, Probabilistic behavior of harmonic functions,
Progress in Mathematics, vol. 175, Birkhäuser Verlag, Basel,
1999. MR
1707297 (2001j:31003)
- 3.
S.-Y.
A. Chang, J.
M. Wilson, and T.
H. Wolff, Some weighted norm inequalities concerning the
Schrödinger operators, Comment. Math. Helv. 60
(1985), no. 2, 217–246. MR 800004
(87d:42027), http://dx.doi.org/10.1007/BF02567411
- 4.
Björn
E. J. Dahlberg, Approximation of harmonic functions, Ann.
Inst. Fourier (Grenoble) 30 (1980), no. 2, vi,
97–107 (English, with French summary). MR 584274
(82i:31010)
- 5.
Björn
E. J. Dahlberg, Weighted norm inequalities for the Lusin area
integral and the nontangential maximal functions for functions harmonic in
a Lipschitz domain, Studia Math. 67 (1980),
no. 3, 297–314. MR 592391
(82f:31003)
- 6.
John
B. Garnett, Two constructions in BMO, Harmonic analysis in
Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown,
Mass., 1978), Part 1, Proc. Sympos. Pure Math., XXXV, Part, Amer. Math.
Soc., Providence, R.I., 1979, pp. 295–301. MR 545269
(81d:30058)
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John
B. Garnett, Bounded analytic functions, Pure and Applied
Mathematics, vol. 96, Academic Press Inc. [Harcourt Brace Jovanovich
Publishers], New York, 1981. MR 628971
(83g:30037)
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R.
F. Gundy and R.
L. Wheeden, Weighted integral inequalities for the nontangential
maximal function, Lusin area integral, and Walsh-Paley series, Studia
Math. 49 (1973/74), 107–124. MR 0352854
(50 #5340)
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Carlos
E. Kenig, Harmonic analysis techniques for second order elliptic
boundary value problems, CBMS Regional Conference Series in
Mathematics, vol. 83, Published for the Conference Board of the
Mathematical Sciences, Washington, DC, 1994. MR 1282720
(96a:35040)
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Jürgen
Moser, On Harnack’s theorem for elliptic differential
equations, Comm. Pure Appl. Math. 14 (1961),
577–591. MR 0159138
(28 #2356)
- 11.
Jill
Pipher, A martingale inequality related to
exponential square integrability, Proc. Amer.
Math. Soc. 118 (1993), no. 2, 541–546. MR 1131038
(94c:42015), http://dx.doi.org/10.1090/S0002-9939-1993-1131038-3
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and oscillatory integrals, Princeton Mathematical Series,
vol. 43, Princeton University Press, Princeton, NJ, 1993. With the
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Caroline
Sweezy, 𝐿-harmonic functions and the exponential square
class, Pacific J. Math. 147 (1991), no. 1,
187–200. MR 1081681
(91k:35072)
- 1.
- R. Bañuelos and C. N. Moore, Sharp estimates for the nontangential maximal function and the Lusin area function in Lipschitz domains, Trans. Amer. Math. Soc. 312 (1989), 641-662. MR 90i:42030
- 2.
- R. Bañuelos and C. N. Moore, Probabilistic behavior of harmonic functions, Progress in Mathematics, vol. 175, Birkhäuser-Verlag, Basel-Boston-Berlin, 1999. MR 2001j:31003
- 3.
- S. Y. A. Chang, J. M. Wilson, and T. H. Wolff, Some weighted norm inequalities concerning the Schrödinger operators, Comment. Math. Helvetici 60 (1985), 217-246. MR 87d:42027
- 4.
- B. E. J. Dahlberg, Approximation of harmonic functions, Ann. Inst. Fourier Grenoble 30 (1980), 97-107. MR 82i:31010
- 5.
- B. E. J. Dahlberg, Weighted norm inequalities for the Lusin area integral and the nontangential maximal functions for functions harmonic in a Lipschitz domain, Studia Math. 67 (1980), 297-314. MR 82f:31003
- 6.
- J. B. Garnett, Two constructions in BMO, Proc. Sympos. Pure Math., Vol. 35, Amer. Math. Soc., Providence, RI, 1979, pp. 295-301. MR 81d:30058
- 7.
- J. B. Garnett, Bounded analytic functions, Pure and Applied Math., Vol. 96, Academic Press, New York, 1981. MR 83g:30037
- 8.
- R. Gundy and R. Wheeden, Weighted integral inequalities for the nontangential maximal functions, Lusin area integral, and Walsh-Paley series, Studia Math. 49 (1974), 107-124. MR 50:5340
- 9.
- C. E. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conference Series in Mathematics, no. 83, Amer. Math. Soc., Providence, RI, 1994. MR 96a:35040
- 10.
- J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure and Appl. Math. 14 (1961), 577-591. MR 28:2356
- 11.
- J. Pipher, A martingale inequality related to exponential square integrability, Proc. Amer. Math. Soc. 118 (1993), 541-546. MR 94c:42015
- 12.
- E. M. Stein, Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton Univ. Press., Princeton, NJ, 1993. MR 95c:42002
- 13.
- C. Sweezy, L-harmonic functions and the exponential square class, Pacific J. of Math. 147 (1991), 187-200. MR 91k:35072
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Additional Information
Jorge Rivera-Noriega
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email:
rnoriega@math.uiuc.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-03-07293-9
PII:
S 0002-9939(03)07293-9
Keywords:
Approximation of harmonic functions,
exponential square class,
area integral estimates
Received by editor(s):
February 26, 2002
Posted:
December 18, 2003
Communicated by:
Andreas Seeger
Article copyright:
© Copyright 2003 American Mathematical Society
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