- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 825240, 9 pages
Nontangential Limits for Modified Poisson Integrals of Boundary Functions in a Cone
Department of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450002, China
Received 17 May 2012; Accepted 8 July 2012
Academic Editor: Dachun Yang
Copyright © 2012 Lei Qiao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Our aim in this paper is to deal with non-tangential limits for modified Poisson integrals of boundary functions in a cone, which generalized results obtained by Brundin and Mizuta-Shimomura.
1. Introduction and Main Results
Let and be the set of all real numbers and the set of all positive real numbers, respectively. We denote by the -dimensional Euclidean space. A point in is denoted by , where . The Euclidean distance of two points and in is denoted by . Also with the origin of is simply denoted by . The boundary, the closure, and the complement of a set in are denoted by , , and , respectively.
We introduce a system of spherical coordinates , , in which are related to cartesian coordinates by .
For positive functions and , we say that if for some constant . If and , we say that .
For and , let denote the open ball with center at and radius in . The unit sphere and the upper half unit sphere are denoted by and , respectively. For simplicity, a point on and the set for a set , are often identified with and , respectively. For two sets and , the set in is simply denoted by . In particular, the half space will be denoted by .
By , we denote the set in with the domain on . We call it a cone. Then is a special cone obtained by putting . We denote the sets and with an interval on by and . By we denote which is .
Let be a domain on with smooth boundary. Consider the Dirichlet problem: where is the spherical part of the Laplace operator We denote the least positive eigenvalue of this boundary value problem by and the normalized positive eigenfunction corresponding to by , where is the surface area on . We denote the solutions of the equation by (). If , then , and , where is the surface area of .
To simplify our consideration in the following, we will assume that if , then is a -domain on surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g., see (, pages 88-89) for the definition of -domain). Then by modifying Miranda’s method (, pages 7-8), we can prove the following inequality:
For any , we have (see ) which yields that where and .
Let be the Green function of . We define the Poisson kernel by where and denotes the differentiation at along the inward normal into .
In this paper, we consider functions , where . Then the Poisson integral is defined by where is the surface area element on .
Remark 1.1. Let . Then where , that is, is the mirror image of with respect to . Hence, for the two points and , we have
We fix an open, nonempty, and bounded set . In , we normalise the extension, with respect to , by where denotes the characteristic function of .
Let be a nontangential cone in with vertex .
Note that, if , then (a.e. point is a Lebesgue point).
In , the following conclusion was proved by Brundin (see (, pages 11–16)) and Mizuta and Shimomura (see (, Theorem 3)), respectively. In the unit disc, about related results, we refer the readers to the papers by Sjögren (see [6, 7]), Rönning (see ), and Brundin (see ).
Theorem A. For a.e. , (see Remark 1.1 for the definition of ) as along .
Our aim is to generalize Theorem A to the conical case.
Theorem 1.2. For any (in particular, for a.e. ) one has that as along .
2. Some Lemmas
Lemma 2.1. One has for any and any satisfying ; for any and any .
Lemma 2.2. One has
By (2.1), we have the following estimates
Next, we will estimate . Take a sufficiently small positive number such that where
Since , we only consider the case . Now put
Since , we have where is a positive integer satisfying .
By (1.6) we have for .
Combining (2.6)–(2.13), Lemma 2.2 is proved.
Lemma 2.3. One has
3. Proof of the Theorem 1.2
Now, let and be given. We may, without loss of generality, assume that . Furthermore, we assume that . For short, let . We write where , , and are sets on used in Lemma 2.2.
By using Hölder’s inequality, (2.1), we have the following estimates
Thus, it follows that
Using the fact that , we get
It is clear that is a convergent integral, since from the Hölder’s inequality.
Now, as , we also have . Since and since we have assumed that (and thus that ), it follows that as along . This concludes the proof.
This paper is supported by SRFDP (no. 20100003110004) and NSF of China (no. 11071020).
- D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, Germany, 1977.
- C. Miranda, Partial Differential Equations of Elliptic Type, Springer, Berlin, Germany, 1970.
- R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. I, Interscience Publishers., New York, NY, USA, 1953.
- M. Brundin, Boundary behavior of eigenfunctions for the hyperbolic Laplacian [Ph.D. thesis], Göteborg University and Chalmers University of Technology, Gothenburg, Sweden, 2002.
- Y. Mizuta and T. Shimomura, “Growth properties for modified Poisson integrals in a half space,” Pacific Journal of Mathematics, vol. 212, no. 2, pp. 333–346, 2003.
- P. Sjögren, “Une remarque sur la convergence des fonctions propres du laplacien à valeur propre critique,” in Théorie du Potentiel (Orsay, 1983), vol. 1096 of Lecture Notes in Mathematics, pp. 544–548, Springer, Berlin, Germany, 1984.
- P. Sjögren, “Approach regions for the square root of the Poisson kernel and bounded functions,” Bulletin of the Australian Mathematical Society, vol. 55, no. 3, pp. 521–527, 1997.
- J.-O. Rönning, “Convergence results for the square root of the Poisson kernel,” Mathematica Scandinavica, vol. 81, no. 2, pp. 219–235, 1997.
- M. Brundin, Approach regions for the square root of the Poisson kernel and weak Lp boundary functions [thesis], Göteborg University and Chalmers University of Technology, 1999.
- M. Essén and J. L. Lewis, “The generalized Ahlfors-Heins theorem in certain -dimensional cones,” Mathematica Scandinavica, vol. 33, pp. 111–129, 1973.
- V. S. Azarin, “Generalization of a theorem of Hayman on subharmonic functions in an m-dimensional cone,” American Mathematical Society Translations, vol. 2, no. 80, pp. 119–138, 1969.