A projection theorem and tangential boundary behavior of potentials
HTML articles powered by AMS MathViewer
- by Kohur GowriSankaran and David Singman PDF
- Proc. Amer. Math. Soc. 129 (2001), 397-405 Request permission
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.References
- Bernadette Brelot-Collin and Marcel Brelot, Représentation intégrale des solutions positives de l’équation $L_{k}(u)=\sum _1^n \partial ^{2}u/\partial x_{1}^{2}+k/x_{n}\partial u/\partial x_{n}=0(k$ constante réelle) dans le demi-espace $E(x_{n}>0)$, de $\textbf {R}^{n}$, Acad. Roy. Belg. Bull. Cl. Sci. (5) 58 (1972), 317–326 (French). MR 318505, DOI 10.3406/barb.1972.60462
- Bernadette Brelot-Collin and Marcel Brelot, Allure à la frontière des solutions positives de l’équation de Weinstein $L_{k}(u)=\Delta u+(k/x_{n})$ $\partial u/\partial x_{n}=0$ dans le demi-espace $E$ $(x_{n}>0)$ de $\textbf {R}^{n}$ $(n\geq 2)$, Acad. Roy. Belg. Bull. Cl. Sci. (5) 59 (1973), 1100–1117 (French). MR 350035
- 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, DOI 10.1007/978-1-4612-5208-5
- 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
- Kohur GowriSankaran and David Singman, Minimal fine limits for a class of potentials, to appear, Potential Analysis.
- M. Gusman, Differentsirovanie integralov v $\textbf {R}^{n}$, 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
- W. K. Hayman, Subharmonic functions. Vol. 2, London Mathematical Society Monographs, vol. 20, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1989. MR 1049148
- W. K. Hayman and P. B. Kennedy, Subharmonic functions. Vol. I, London Mathematical Society Monographs, No. 9, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0460672
- R.-M. Hervé, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier (Grenoble) 12 (1962), 415–571 (French). MR 139756, DOI 10.5802/aif.125
- Erwin Kreyszig, Advanced Engineering Mathematics, seventh edition, Wiley, Appendix 3.
- Peter A. Loeb, Opening covering theorems of Besicovitch and Morse, to appear.
- 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, DOI 10.1016/0022-1236(84)90061-2
- A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8, DOI 10.1017/S0370164600012281
- I. I. Privalov, Sur un probleme limite des fonctions sous-harmoniques, Rec. Math. (Mat. Sbornik) N. S. 41 (1934), 3-10.
- C. A. Rogers, Hausdorff measures, Cambridge University Press, London-New York, 1970. MR 0281862
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
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
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 397-405
- MSC (2000): Primary 31B25
- DOI: https://doi.org/10.1090/S0002-9939-00-05524-6
- MathSciNet review: 1694863