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Abstract and Applied Analysis
Volume 2012, Article ID 765965, 13 pages
http://dx.doi.org/10.1155/2012/765965
Research Article

Modified Poisson Integral and Green Potential on a Half-Space

Department of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450002, China

Received 16 February 2012; Revised 5 May 2012; Accepted 16 May 2012

Academic Editor: Stefan Siegmund

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.

Abstract

We discuss the behavior at infinity of modified Poisson integral and Green potential on a half-space of the n-dimensional Euclidean space, which generalizes the growth properties of analytic functions, harmonic functions and superharmonic functions.

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 .

Set

Let be the green function of order for , that is, where denotes reflection in the boundary plane just as .

We define the Poisson kernel when and by where if and if . It has the expansion where is a Gegenbauer polynomial [1]. The series converges for . Each term in the series is a harmonic function of and vanishes on .

In case , we consider the modified kernel function defined by

In case , we define where is a nonnegative integer and is also the Gegenbauer polynomials. The expression arises from the generating function for Gegenbauer polynomials where , and . Each coefficient is called the Gegenbauer polynomial of degree associated with , the function is a polynomial of degree in .

Then we define the modified Poisson kernel and Green function respectively by where and . We remark that the new kernel will be of order as .

Write where (resp. ) is a nonnegative measure on (resp. ). Here note that (resp. ) is nothing but the general Poisson integral (resp. Green potential).

Let be a nonnegative Borel measurable function on , and set for a nonnegative measure on a Borel set . We define a capacity by where the supremum is taken over all nonnegative measures such that (the support of ) is contained in and for every .

For and , we consider the function defined by If , then is extended to be continuous on in the extended sense, where .

Recently, Siegel-Talvila [2] proved the following result.

Theorem A. Let be a measurable function on satisfying . Then the function satisfies Our first aim is to establish the following theorem.

Theorem 1.1. Let and . If is a nonnegative measure on satisfying then there exists a Borel set with properties: (1)(2), where .

Theorem 1.2. Let and . If is a nonnegative measure on satisfying then there exists a Borel set with properties:(1); (2)where .

Remark 1.3. If , then .
Next we generalize Theorem A to the modified -potentials on , which is defined by where and (resp. ) is a nonnegative measure on (resp. ) satisfying (1.15) (resp. (1.16) ()). Clearly, is a superharmonic function.

The following theorem follows readily from Theorems 1.1 and 1.2.

Theorem 1.4. Let be defined by (1.17). Then there exists a Borel set satisfying Theorem 1.1 such that

Remark 1.5. In the case , by using Lemma 2.5 below, we can easily show that in the notation of [3]. Thus, Theorem 1.1(2) with means that is -rarefied at infinity in the sense of [3]. In particular, this condition with , (resp. , ) means that is minimally thin at infinity (resp. rarefied at infinity) in the sense of [4].

Finally we are concerned with the best possibility of Theorem 1.4 as to the size of the exceptional set.

Theorem 1.6. Let , be a Borel set satisfying Theorem 1.1 and let be defined by (1.17). Then one can find a nonnegative measure defined on satisfying such that where and .

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 (see [5]). Let be a nonnegative integer and .(1)If , then .(2)If and , then .

Lemma 2.2. There exists a positive constant such that , where and in .

This can be proved by simple calculation.

Lemma 2.3. Gegenbauer polynomials have the following properties:(1); (2); (3); (4).

Proof. (1) and (2) can be derived from [1]. (3) follows by taking in (1.7); (4) follows by (1), (2) and the Mean Value Theorem for Derivatives.

Lemma 2.4. For , , one has the following properties:(1); (2); (3); (4).

The following lemma can be proved by using Fuglede (see [6], Théorèm 7.8).

Lemma 2.5. For any Borel set in , one has and where the infimum is taken over all nonnegative measures on (resp. ) such that for every .

3. Proof of Theorems

Proof of Theorem 1.1. For any , there exists such that For fixed and , we write where First note that By Lemma 2.1(1), we have
Write where
If , then we have from (3.5).
So
We have by Lemma 2.3(3)
By Lemma 2.1(2), we obtain
Note that . In view of (1.15), we can find a sequence of positive numbers such that and , where
Consider the sets for . If is a nonnegative measure on such that and for , then we have so that which yields Setting , we see that Theorem 1.1(2) is satisfied and Combining (3.4) and (3.9)–(3.17), Theorem 1.1(1) holds.
Then we complete the proof of Theorem 1.1.

Proof of Theorem 1.2. For any , there exists such that For fixed and , we write where
We distinguish the following two cases.
Case  1. .
Note that where .
By the lower semicontinuity of , then we can prove the following fact in the same way as in the proof of Theorem 1.1: where and .
Moreover, by Lemma 2.2
Note that . By (3) and (4) in Lemma 2.3, we take , in Lemma 2.3(4) and obtain
Similarly, we have by (3) and (4) in Lemma 2.3
By Lemma 2.2, we get
Similar to the estimate of , we obtain Combining (3.22)–(3.27), we see that Theorem 1.2(1) holds. Then we prove Case 1.
Case  2. .
Since the estimates of , , and are similar to those of Case 1, we omit them. (3.22), (3.23), and (3.26) still hold in Case 2.
Moreover, by Lemma 2.4(3), we find
By Lemma 2.4(4), we have Similar to the estimate of , we have Combining (3.22), (3.23), (3.26) and (3.28)–(3.30), we see that Theorem 1.2(1) holds. Then we prove Case 2.
Hence we complete the proof of Theorem 1.2.

Proof of Theorem 1.6. We prove the case , because the case can be proved similarly. Further, we only need prove By Lemma 2.5, for each we can find on such that and on . Denote by the restriction of to the set .
Set , where is a sequence of positive numbers such that but . Then If , then
We also have which implies that
Thus satisfies all the conditions in the Theorem 1.6.

Acknowledgments

The authors are extremely grateful to the referee for useful suggestions that improved the contents of the paper. This work is supported by the National Natural Science Foundation of China under Grant 11071020 and Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20100003110004.

References

  1. G. Szegő, Orthogonal Polynomials, vol. 23, American Mathematical Society Colloquium Publications, Providence, RI, USA, 4th edition, 1975.
  2. D. Siegel and E. Talvila, “Sharp growth estimates for modified Poisson integrals in a half space,” Potential Analysis, vol. 15, no. 4, pp. 333–360, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. H. Aikawa, “On the behavior at infinity of nonnegative superharmonic functions in a half space,” Hiroshima Mathematical Journal, vol. 11, no. 2, pp. 425–441, 1981. View at Google Scholar
  4. M. Essén and H. L. Jackson, “On the covering properties of certain exceptional sets in a half-space,” Hiroshima Mathematical Journal, vol. 10, no. 2, pp. 233–262, 1980. View at Google Scholar · View at Zentralblatt MATH
  5. W. K. Hayman and P. B. Kennedy, Subharmonic Functions, vol. 1, Academic Press, London, UK, 1976.
  6. B. Fuglede, “Le théorème du minimax et la théorie fine du potentiel,” Annales de l'Institut Fourier, vol. 15, pp. 65–88, 1965. View at Publisher · View at Google Scholar · View at Zentralblatt MATH