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Boundary Behavior of Solutions of the Helmholtz Equation

  • Kentaro Hirata (a1)
Abstract

This paper is concerned with the boundary behavior of solutions of the Helmholtz equation in ℝ n . In particular, we give a Littlewood-type theorem to show that the approach region introduced by Korányi and Taylor (1983) is best possible.

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References
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[1] Aikawa, H., Harmonic functions having no tangential limits. Proc. Amer. Math. Soc. 108(1990), no. 2, 457464.
[2] Aikawa, H., Harmonic functions and Green potentials having no tangential limits. J. London Math. Soc. (2) 43(1991), no. 1, 125136.
[3] Berman, R. and Singman, D., Boundary behavior of positive solutions of the Helmholtz equation and associated potentials. Michigan Math. J. 38(1991), no. 3, 381393.
[4] Brawn, F. T., The Martin boundary of Rn ×]0, 1[. J. London Math. Soc. (2) 5(1972), 5966.
[5] Fatou, P., Séries trigonométriques et séries de Taylor. Acta Math. 30(1906), no. 1, 335400.
[6] Gowrisankaran, K. and Singman, D., Thin sets and boundary behavior of solutions of the Helmholtz equation. Potential Anal. 9(1998), no. 4, 383398.
[7] Hirata, K., Sharpness of the Korányi approach region. Proc. Amer.Math. Soc. 133(2005), no. 8, 23092317.
[8] Korányi, A., Harmonic functions on Hermitian hyperbolic space. Trans. Amer. Math. Soc. 135(1969), 507516.
[9] Korányi, A. and Taylor, J. C., Fine convergence and parabolic convergence for the Helmholtz equation and the heat equation. Illinois J. Math. 27(1983), no. 1, 7793.
[10] Littlewood, J. E., On a theorem of Fatou. J. London Math. Soc. 2(1927), 172176.
[11] Nagel, A. and Stein, E. M., On certain maximal functions and approach regions. Adv. in Math. 54(1984), no. 1, 83106.
[12] Sueiro, J., On maximal functions and Poisson–Szegö integrals. Trans. Amer. Math. Soc. 298(1986), no. 2, 653669.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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