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

Rivera-Noriega, Jorge

doi:10.1090/S0002-9939-03-07293-9

An approximation property of harmonic functions in Lipschitz domains and some of its consequences
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Proc. Amer. Math. Soc. 132 (2004), 1321-1331 Request permission

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.

References

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, DOI 10.1090/S0002-9947-1989-0957080-4
Rodrigo Bañuelos and Charles N. Moore, Probabilistic behavior of harmonic functions, Progress in Mathematics, vol. 175, Birkhäuser Verlag, Basel, 1999. MR 1707297, DOI 10.1007/978-3-0348-8728-1
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, DOI 10.1007/BF02567411
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
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, DOI 10.4064/sm-67-3-297-314
John B. Garnett, Two constructions in BMO, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 295–301. MR 545269
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
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 352854, DOI 10.4064/sm-49-2-107-124
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; by the American Mathematical Society, Providence, RI, 1994. MR 1282720, DOI 10.1090/cbms/083
Jürgen Moser, On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577–591. MR 159138, DOI 10.1002/cpa.3160140329
Jill Pipher, A martingale inequality related to exponential square integrability, Proc. Amer. Math. Soc. 118 (1993), no. 2, 541–546. MR 1131038, DOI 10.1090/S0002-9939-1993-1131038-3
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
Caroline Sweezy, $L$-harmonic functions and the exponential square class, Pacific J. Math. 147 (1991), no. 1, 187–200. MR 1081681