Nontangential Limits for Modified Poisson Integrals of Boundary Functions in a Cone
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 .
Proof. These immediately follow from (, Lemma 2), (, Lemma 4 and Remark), and (1.4).
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
Proof. In fact, we only need to prove
Write where , , and are sets on used in Lemma 2.2.
Further, we have by (2.2) where
Combining (2.17)–(2.19), (2.15) holds which gives the conclusion.
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
Similar to the estimate of in Lemma 2.2, we only consider the following inequality by (1.6) for , which is similar to the estimate of .
Notice that in the case . By (1.6) and (2.2), we have
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).
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