- Research Article
- Open Access

- Lei Qiao
^{1}and - Guantie Deng
^{2}Email author

**2010**:647627

https://doi.org/10.1155/2010/647627

© Lei Qiao and Guantie Deng. 2010

**Received:**13 March 2010**Accepted:**21 June 2010**Published:**8 July 2010

## Abstract

A class of -potentials represented as the sum of modified Green potential and modified Poisson integral are proved to have the growth estimates at infinity in the upper-half space of the -dimensional Euclidean space, where the function is a positive non-decreasing function on the interval satisfying certain conditions. This result generalizes the growth properties of analytic functions, harmonic functions, and superharmonic functions.

## Keywords

- Green Function
- Nonnegative Integer
- General Order
- Lower Semicontinuity
- Negative Power

## 1. Introduction and Main Results

Let denote the -dimensional Euclidean space with points , where and . The boundary and closure of an open of are denoted by and , respectively. The upper half-space is the set , whose boundary is . We identify with and with , writing typical points as where and putting

For and , let denote the open ball with center at and radius in .

where and is a function in the Schwartz class.

where is a certain constant (see, e.g., [1, page 414] for the exact value of ).

where denotes reflection in the boundary plane just as .

We remark that and are the classical Green function and classical Poisson kernel for respectively (see, e.g., [2, page 127]).

where , , and . The coefficients are called the ultraspherical (or Gegenbauer) polynomials of degree associated with , each function is a polynomial of degree in . Here note that is the modified Poisson kernel in , which has been used by several authors (see, e.g., [4–8]).

Motivated by this modified kernel function , it is natural to ask if the function can also be modified? In this paper, we give an affirmative answer to this question.

where is a nonnegative integer, , and .

where (resp., ) is a nonnegative measure on (resp., ). Here note that is nothing but the Green potential of general order (see [9–11]).

where the supremum is taken over all nonnegative measures such that (the support of ) is contained in and for every .

If , then is extended to be continuous on in the extended sense, where .

Now we will discuss the behavior at infinity of the modified Green potential and modified Poisson integral in the upper-half space, respectively. For related results, we refer the readers to the papers by Mizuta (see [9]), Siegel and Talvila (see [8]), and Mizuta and Shimomura (see [7]).

Theorem 1.1.

Let be a positive nondecreasing function on the interval such that

(c)there exists a positive constant such that for any .

Then there exists a Borel set with properties

Corollary 1.2.

Then there exists a Borel set with properties

Theorem 1.3.

Then there exists a Borel set with properties

Remark 1.4.

Corollary 1.5.

where and (resp., ) is a nonnegative measure on (resp., ) satisfying (1.18) ( ) (resp., (1.20)). Clearly, is a superharmonic function on .

The following theorem follows readily from Theorems 1.1 and 1.3.

Theorem 1.6.

Remark 1.7.

In the case , by using Lemma 2.5 below, we can easily show that Corollary 1.2 (ii) with means that is -rarefied at infinity in the sense of [12]. In particular, This condition with , , and (resp., , , and ) means that is minimally thin at infinity (resp., rarefied at infinity) in the sense of [13].

Theorem 1.6 is the best possibility as to the size of the exceptional set. In fact we have the following result. The proof of it is essentially due to Mizuta (see [9, Theorem ]), so we omit the proof here.

Theorem 1.8.

## 2. Some Lemmas

Throughout this paper, let denote various constants independent of the variables in questions, which may be different from line to line.

Lemma 2.1.

There exists a positive constant such that where , and in .

This can be proved by simple calculation.

Lemma 2.2.

Gegenbauer polynomials have the following properties:

- (i)
and (ii) can be derived from [3]. (iii) follows by taking in (1.10); (iv) follows by (i), (ii) and the Mean Value Theorem for Derivatives.

Lemma 2.3.

Let be a nonnegative integer and , , then one has the following properties:

Lemma 2.4 (see [14]).

Let be a nonnegative integer and .

The following lemma can be proved by using Fuglede ([6, Théorèm ]).

Lemma 2.5.

where the infimum is taken over all nonnegative measures on (resp., ) such that for every .

## 3. Proof of Theorem 1.1

We distinguish the following two cases.

Combining (3.9)–(3.15), we prove Case 1.

In this case, the growth estimates of , , and can be proved similarly as in Case 1. Inequations (3.9), (3.10), (3.13) and (3.15) still hold.

Combining (3.9), (3.10), (3.13), (3.15), and (3.16)–(3.18), we prove Case 2.

Hence we complete the proof of Theorem 1.1.

## 4. Proof of Theorem 1.3

## Declarations

### Acknowledgments

This work is supported by The National Natural Science Foundation of China under Grant 10671022 and Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20060027023.

## Authors’ Affiliations

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