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A projection theorem and tangential boundary behavior of potentials


Authors: Kohur GowriSankaran and David Singman
Journal: Proc. Amer. Math. Soc. 129 (2001), 397-405
MSC (2000): Primary 31B25
DOI: https://doi.org/10.1090/S0002-9939-00-05524-6
Published electronically: August 29, 2000
MathSciNet review: 1694863
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Abstract:

Let $L_k$ be the Weinstein operator on the half space, $\mathbb{R}^n_+$. Suppose there is a sequence of Borel sets $A_j \subset \mathbb{R}^n_+$ such that a certain tangential projection of $A_j$ onto $\mathbb{R}^{n-1}$ forms a pairwise disjoint subset of the boundary. Let $\nu$ be a finite test measure on the boundary for a specific non-isotropic Hausdorff measure. The measure $\nu$ is carried back to a measure $\lambda$on a subset of $\bigcup A_j$ by the projection. We give an upper bound for the Weinstein potential corresponding to the measure $d\lambda / x_n$ in terms of a universal constant and a Weinstein subharmonic function. We use this upper bound to deduce a result concerning tangential behavior of Weinstein potentials at the boundary with the exception of sets on the boundary of vanishing non-isotropic Hausdorff measure.


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  • [BCB1] Bernadette Brelot-Collin and Marcel Brelot, Représentation intégrale des solutions positives de l’équation 𝐿_{𝑘}(𝑢)=∑₁ⁿ∂²𝑢/∂𝑥₁²+𝑘/𝑥_{𝑛}∂𝑢/∂𝑥_{𝑛}=0(𝑘 constante réelle) dans le demi-espace 𝐸(𝑥_{𝑛}>0), de 𝑅ⁿ, Acad. Roy. Belg. Bull. Cl. Sci. (5) 58 (1972), 317–326 (French). MR 0318505
  • [BCB2] Bernadette Brelot-Collin and Marcel Brelot, Allure à la frontière des solutions positives de l’équation de Weinstein 𝐿_{𝑘}(𝑢)=Δ𝑢+(𝑘/𝑥_{𝑛}) ∂𝑢/∂𝑥_{𝑛}=0 dans le demi-espace 𝐸 (𝑥_{𝑛}>0) de 𝑅ⁿ (𝑛≥2), Acad. Roy. Belg. Bull. Cl. Sci. (5) 59 (1973), 1100–1117 (French). MR 0350035
  • [D] J. L. Doob, Classical potential theory and its probabilistic counterpart, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262, Springer-Verlag, New York, 1984. MR 731258
  • [GS1] Kohur GowriSankaran and David Singman, A generalized Littlewood theorem for Weinstein potentials on a halfspace, Illinois J. Math. 41 (1997), no. 4, 630–647. MR 1468871
  • [GS2] Kohur GowriSankaran and David Singman, Minimal fine limits for a class of potentials, to appear, Potential Analysis.
  • [Guz] M. Gusman, \cyr Differentsirovanie integralov v 𝑅ⁿ, \cyr Matematika: Novoe v Zarubezhnoĭ Nauke [Mathematics: Recent Publications in Foreign Science], vol. 9, “Mir”, Moscow, 1978 (Russian). Translated from the English by V. A. Skvorcov. MR 515884
  • [H] W. K. Hayman, Subharmonic functions. Vol. 2, London Mathematical Society Monographs, vol. 20, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1989. MR 1049148
  • [HK] W. K. Hayman and P. B. Kennedy, Subharmonic functions. Vol. I, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. London Mathematical Society Monographs, No. 9. MR 0460672
  • [He] R.-M. Hervé, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier (Grenoble) 12 (1962), 415–571 (French). MR 0139756
  • [K] Erwin Kreyszig, Advanced Engineering Mathematics, seventh edition, Wiley, Appendix 3. CMP 93:03
  • [L] Peter A. Loeb, Opening covering theorems of Besicovitch and Morse, to appear.
  • [LMT] T. J. Lyons, K. B. MacGibbon, and J. C. Taylor, Projection theorems for hitting probabilities and a theorem of Littlewood, J. Funct. Anal. 59 (1984), no. 3, 470–489. MR 769377, https://doi.org/10.1016/0022-1236(84)90061-2
  • [M] A. P. Morse, Perfect blankets, Trans. Amer. Math. Soc. 6 1947, 418-442. MR 8:571h
  • [P] I. I. Privalov, Sur un probleme limite des fonctions sous-harmoniques, Rec. Math. (Mat. Sbornik) N. S. 41 (1934), 3-10.
  • [R] C. A. Rogers, Hausdorff measures, Cambridge University Press, London-New York, 1970. MR 0281862
  • [St] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • [W] A. Weinstein, Generalized axially symmetric potential theory, Bull. Am. Math. Soc. 69 (1953), 20-38. MR 14:749c

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Additional Information

Kohur GowriSankaran
Affiliation: Department of Mathematics, McGill University, Montreal, Quebec, Canada H3A 2K6
Email: gowri@math.mcgill.ca

David Singman
Affiliation: Department of Mathematics, George Mason University, Fairfax, Virginia 22030
Email: dsingman@osf1.gmu.edu

DOI: https://doi.org/10.1090/S0002-9939-00-05524-6
Keywords: Weinstein equation, Littlewood theorem, Weinstein potential, non-isotropic Hausdorff measure, boundary behavior, minimal fine limit
Received by editor(s): August 27, 1998
Received by editor(s) in revised form: April 9, 1999
Published electronically: August 29, 2000
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2000 American Mathematical Society