Proceedings of the American Mathematical Society

An approximation property of harmonic functions in Lipschitz domains and some of its consequences

Author(s): Jorge Rivera-Noriega
Journal: Proc. Amer. Math. Soc. 132 (2004), 1321-1331.
MSC (2000): Primary 42B25, 35J67
Posted: December 18, 2003
Retrieve article in: PDF DVI PostScript

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.: 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

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 42B25, 35J67

Jorge Rivera-Noriega
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email: rnoriega@math.uiuc.edu

DOI: 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
Copyright of article: Copyright 2003, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS