A projection theorem and tangential boundary behavior of potentials

GowriSankaran, Kohur; Singman, David

doi:10.1090/S0002-9939-00-05524-6

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