- Research Article
- Open Access
© Lei Qiao and Guantie Deng. 2010
- Received: 13 March 2010
- Accepted: 21 June 2010
- Published: 8 July 2010
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.
- Green Function
- Nonnegative Integer
- General Order
- Lower Semicontinuity
- Negative Power
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
where is a certain constant (see, e.g., [1, page 414] for the exact value of ).
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]).
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 ), Siegel and Talvila (see ), and Mizuta and Shimomura (see ).
The following theorem follows readily from Theorems 1.1 and 1.3.
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 . In particular, This condition with , , and (resp., , , and ) means that is minimally thin at infinity (resp., rarefied at infinity) in the sense of .
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.
This can be proved by simple calculation.
Gegenbauer polynomials have the following properties:
and (ii) can be derived from . (iii) follows by taking in (1.10); (iv) follows by (i), (ii) and the Mean Value Theorem for Derivatives.
Lemma 2.4 (see ).
The following lemma can be proved by using Fuglede ([6, Théorèm ]).
We distinguish the following two cases.
Combining (3.9)–(3.15), we prove Case 1.
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.
Combining (4.4) and (4.10)–(4.13), we complete the proof of Theorem 1.1.
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.
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