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On Dahlberg's Lusin area integral theorem

Author: Marius Mitrea
Journal: Proc. Amer. Math. Soc. 123 (1995), 1449-1455
MSC: Primary 42B20; Secondary 31B10, 31B35
MathSciNet review: 1239801
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Abstract: We give new proofs to the Lusin area integral theorem of Dahlberg. Our techniques rely on the theory of elliptic boundary value problems on nonsmooth domains and are shown to extend to other important cases, including systems of equations.

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  • [Br] R. Brown, The method of layer potentials for the heat equation in Lipschitz cylinders, Amer. J. Math. 111 (1989), 339-379. MR 987761 (90d:35118)
  • [BS] R. Brown and Z. Shen, The initial-Dirichlet problem for a fourth-order parabolic equation in Lipschitz cylinders, Indiana Univ. Math. J. 39 (1990), 1313-1353. MR 1087194 (92b:35075)
  • [BG] D. L. Burkholder and R. F. Gundy, Distribution function inequalities for the area integral, Studia Math. 44 (1972), 527-544. MR 0340557 (49:5309)
  • [BGS] D. L. Burkholder, R. F. Gundy, and M. L. Silverstein, A maximal function characterization of the class $ {H^p}$, Trans. Amer. Math. Soc. 157 (1971), 137-153. MR 0274767 (43:527)
  • [Ca] A. P. Calderón, Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1092-1099. MR 0177312 (31:1575)
  • [CJS] R. R. Coifman, P. Jones, and S. Semme, Two elementary proofs of the $ {L^2}$ boundedness of the Cauchy integrals on Lipschitz curves, J. Amer. Math. Soc. 2 (1989), 553-564. MR 986825 (90k:42017)
  • [D1] B. E. J. Dahlberg, Weighted norm inequalities for the Lusin area integral and the nontangential maximal function for functions harmonic in a Lipschitz domain, Studia Math. 67 (1980), 297-314. MR 592391 (82f:31003)
  • [D2] -, Poisson semigroups and singular integrals, Proc. Amer. Math. Soc. 97 (1986), 41-48. MR 831384 (87g:42035)
  • [DK] B. E. J. Dahlberg and C. E. Kenig, Hardy spaces and the $ {L^p}$-Neumann problem for Laplace's equation in a Lipschitz domain, Ann. of Math. (2) 125 (1987), 437-465. MR 890159 (88d:35044)
  • [DKV1] B. E. J. Dahlberg, C. E. Kenig, and G. Verchota, The Dirichlet problem for the biharmonic equation in a Lipschitz domain, Ann. Inst. Fourier (Grenoble) 36 (1986), 109-135. MR 865663 (88a:35070)
  • [DKV2] -, Boundary value problems for the systems of elastostatics in Lipschitz domains, Duke Math. J. 57 (1988), 795-818. MR 975122 (90d:35259)
  • [DJS] G. David, J.-L. Journé, and S. Semmes, Operatéurs de Calderó-Zygmund, fonctiones paraaccretives et interpolation, Rev. Mat. Iberoamericana 1 (1985), 1-56. MR 850408 (88f:47024)
  • [Fa] E. Fabes, Layer potential methods for boundary value problems on Lipschitz domains, Potential Theory, Surveys and Problems (J. Král et al., eds.), Lecture Notes in Math., vol. 1344, Springer-Verlag, New York, 1988, pp. 55-80. MR 973881
  • [FS] C. Fefferman and E. M. Stein, $ {H^p}$ spaces of several variables, Acta Math. 129 (1972), 137-193. MR 0447953 (56:6263)
  • [JK] D. S. Jerison and C. E. Kenig, Boundary value problems on Lipschitz domains, Studies in Partial Differential Equations (W. Littman, ed.), Stud. Math., vol. 23, Math. Assoc. America, Washington, D. C., 1982, pp. 1-68. MR 716504 (85f:35057)
  • [K1] C.E. Kenig, Weighted $ {H^p}$ spaces on Lipschitz domains, Amer. J. Math. 102 (1980), 129-163. MR 556889 (81d:30060)
  • [K2] -, Square function estimates for solutions to elliptic equations and systems on Lipschitz domain, Proc. of the 17th Spring Lecture Series in Mathematics (Fayetteville, 1993) (to appear).
  • [LMS] C. Li, A. McIntosh, and S. Semmes, Convolution singular integrals on Lipschitz surfaces, J. Amer. Math. Soc. 5 (1992), 455-481. MR 1157291 (93b:42029)
  • [Mc] A. McIntosh, Clifford algebras and the higher dimensional Cauchy integral, Approximation Theory and Function Spaces, Banach Center Publ., vol. 22, PWN, Warsaw, 1989, pp. 253-267. MR 1097197 (92d:30032)
  • [MM] A. McIntosh and Y. Meyer, Algebres d'opérateurs definis par des integrales singulieres, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), 395-397. MR 808636 (87b:47053)
  • [Me] Y. Meyer, Ondelettes et opérateurs, Hermann, Paris, 1990. MR 1085487 (93i:42002)
  • [M1] M. Mitrea, Singular integrals, Hardy spaces and Clifford wavelets, Lecture Notes in Math., vol. 1575, Springer-Verlag, Berlin and New York, 1994. MR 1295843 (96e:31005)
  • [M2] -, Clifford algebras and boundary estimates for harmonic functions, Proc. of the Third Internat. Conf. on "Clifford Algebras and Their Applications in Mathematical Physics", Kluwer Acad. Publ, New York, 1993. MR 1266864 (95c:30062)
  • [M3] -, Boundary value problems and Hardy spaces associated to the Helmholtz equation on Lipschitz domains, submitted.
  • [MTW] M. Mitrea, R. Torres, and G. Welland, Layer potential techniques in electromagnetism on $ {C^1}$ and Lipschitz domains, preprint.
  • [Mu] M. A. M. Murray, The Cauchy integral, Calderón commutators and conjugations of singular integrals in $ {\mathbb{R}^m}$, Trans. Amer. Math. Soc. 289 (1985), 497-518. MR 784001 (86f:42009)
  • [Ne] J. Nečas, Les méthodes directes en théorie des équations élliptique, Academia, Prague, 1967.
  • [S] C. Segovia, On the area function of Lusin, Studia Math. 33 (1969), 312-343. MR 0288299 (44:5497)
  • [Se] S. Semmes, Square function estimates and the $ T(b)$ theorem, Proc. Amer. Math. Soc. 110 (1990), 721-726. MR 1028049 (91h:42018)
  • [St] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, NJ, 1970. MR 0290095 (44:7280)
  • [Ta] T. Tao, Convolution operators generated by right-monogenic and harmonic kernels, preprint.
  • [TW] R. Torres and G. Welland, The Helmholtz equation and transmission problems with Lipschitz interfaces, preprint. MR 1266102 (95b:35043)
  • [Ve] G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, J. Funct. Anal. 59 (1984), 572-611. MR 769382 (86e:35038)

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