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

DOI:
https://doi.org/10.1090/S0002-9939-03-07293-9

Published electronically:
December 18, 2003

MathSciNet review:
2053336

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

Retrieve articles in all journals with MSC (2000): 42B25, 35J67

Additional Information

**Jorge Rivera-Noriega**

Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801

Email:
rnoriega@math.uiuc.edu

DOI:
https://doi.org/10.1090/S0002-9939-03-07293-9

Keywords:
Approximation of harmonic functions,
exponential square class,
area integral estimates

Received by editor(s):
February 26, 2002

Published electronically:
December 18, 2003

Communicated by:
Andreas Seeger

Article copyright:
© Copyright 2003
American Mathematical Society