#### Abstract

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 , β, 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 ([1], pages 88-89) for the definition of -domain). Then by modifying Mirandaβs method ([2], pages 7-8), we can prove the following inequality:

For any , we have (see [3]) 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 .

We define

Note that, if , then (a.e. point is a Lebesgue point).

In , the following conclusion was proved by Brundin (see ([4], pages 11β16)) and Mizuta and Shimomura (see ([5], Theorem 3)), respectively. In the unit disc, about related results, we refer the readers to the papers by SjΓΆgren (see [6, 7]), RΓΆnning (see [8]), and Brundin (see [9]).

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 ([10], Lemma 2), ([11], Lemma 4 and Remark), and (1.4).

Lemma 2.2. One has

Proof. Write where
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 .
So
Combining (2.6)β(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.
Obviously,
Further, we have by (2.2) where
Combining (2.17)β(2.19), (2.15) holds which gives the conclusion.

#### 3. Proof of the Theorem 1.2

As , from Lemmas 2.2 and 2.3.

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 HΓΆlderβ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 .

So

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 HΓΆlderβ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.

#### Acknowledgments

This paper is supported by SRFDP (no. 20100003110004) and NSF of China (no. 11071020).