Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Published electronically: August 29, 2000
MathSciNet review: 1694863
Full-text PDF

Abstract | References | Similar Articles | Additional Information


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.

References [Enhancements On Off] (What's this?)

  • [BCB1] B. Brelot-Collin and M. Brelot, Représentation intégrale des solutions positives de l'equation $L_k(f)= \sum_{i=1}^{n} \tfrac{\partial ^2 f}{\partial x_i^2} + \frac{k}{x_n} \frac{\partial f}{\partial x_n} = 0, \mbox{(k constante r\'eele)}$ dans le demi-espace E$(x_n> 0)$, de $\mathbb{R}^n$, Bull. Acd. Royale de Belg. 58 (1972), 317-326. MR 47:7052
  • [BCB2] B. Brelot-Collin and M. Brelot, Allure à la frontière des solutions positives de l'equation de Weinstein $L_k(u)=\Delta u+\frac{k}{x_n}\tfrac{\partial u}{\partial x_n}=0$dans le demi-espace E($x_n>0$) de $\mathbb{R}^n$ ($n \geq 2$), Bull Acad. Royale de Belg. 59 (1973), 1100-1117. MR 50:2528
  • [D] J. L. Doob, Classical potential theory and its probabilistic counterpart, Springer-Verlag, 1984. MR 85k:31001
  • [GS1] Kohur GowriSankaran and David Singman, A generalized Littlewood theorem for Weinstein potentials on a halfspace, Ill. J. Math., 41 no. 4 (1997), 630-647. MR 98m:31007
  • [GS2] Kohur GowriSankaran and David Singman, Minimal fine limits for a class of potentials, to appear, Potential Analysis.
  • [Guz] Miguel de Guzmán, Differentiation of Integrals in $\mathbb{R}^n$, Lecture Notes in Math. #481, Springer-Verlag. MR 80b:28005
  • [H] W. K. Hayman, Subharmonic Functions, Volume 2, London Math. Society Monograph No. 20, Academic Press, 1989. MR 91f:31001
  • [HK] W. K. Hayman and P. B. Kennedy, Subharmonic Functions, Volume 1, Academic Press, 1976. MR 57:665
  • [He] Mme. R.-M. Hervé, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier t. 12 (1962), 415-571. MR 25:3186
  • [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, J. C. Taylor, Projection theorems for hitting probabilities and a theorem of Littlewood, J. Funct. Anal. 59(1984), no. 3 470-489. MR 86c:31002
  • [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, 1970. MR 43:7576
  • [St] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, 1970. MR 44:7280
  • [W] A. Weinstein, Generalized axially symmetric potential theory, Bull. Am. Math. Soc. 69 (1953), 20-38. MR 14:749c

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 31B25

Retrieve articles in all journals with MSC (2000): 31B25

Additional Information

Kohur GowriSankaran
Affiliation: Department of Mathematics, McGill University, Montreal, Quebec, Canada H3A 2K6

David Singman
Affiliation: Department of Mathematics, George Mason University, Fairfax, Virginia 22030

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

American Mathematical Society