Asymptotic Behavior for a Class of Modified -Potentials in a Half Space

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.

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 .

It is well known that (see, e.g., [1, Chapter 6]) the positive powers of the Laplace operator can be defined by

(1.1)

where , is a Schwarz function and

(1.2)

It follows that we can extend definition (1.1) to certain negative powers of , for and define an operator by

(1.3)

where and is a function in the Schwartz class.

If is defined as the inverse Fourier transform of (in the sense of distributions), one can show that

(1.4)

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

The function is known as the Riesz kernel. It follows immediately from the rules for manipulating Fourier transforms that any Schwartz function can be written as a Riesz potential,

(1.5)

where and .

This Riesz kernel in inspired us to introduce the modified Riesz kernel for . To do this, we first set

(1.6)

Let be the modified Riesz kernel for , that is,

(1.7)

where denotes reflection in the boundary plane just as .

We define the kernel function when and by

(1.8)

where if and if .

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

Next we use the following modified kernel function defined by

(1.9)

where is a nonnegative integer; is the ultraspherical (or Gegenbauer) polynomials (see [3]). The Gegenbauer polynomials come from the generating function

(1.10)

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.

First we consider the modified kernel function in case , which is defined by

(1.11)

In case , we define

(1.12)

where is a nonnegative integer, , and .

Then we define the modified kernel function by

(1.13)

Write

(1.14)

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

Following Fuglede (see [6]), we set

(1.15)

for a nonnegative Borel measurable function on and a nonnegative measure on a Borel set . We define a capacity by

(1.16)

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

(1.17)

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

(a) is nondecreasing on ,

(b) is nonincreasing on and ,

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

Let be a nonnegative measure on satisfying

(1.18)

Then there exists a Borel set with properties

(i)

(ii)

where .

Corollary 1.2.

Let be a nonnegative measure on satisfying

(1.19)

Then there exists a Borel set with properties

(i)

(ii)

where .

Theorem 1.3.

Let be defined as in Theorem 1.1 and a nonnegative measure on satisfying

(1.20)

Then there exists a Borel set with properties

(i)

(i)

where .

Remark 1.4.

In the case , .

Corollary 1.5.

Let be a nonnegative measure on satisfying

(1.21)

Then there exists a Borel set satisfying Corollary 1.2 (ii) such that

(1.22)

We define the modified -potentials on by

(1.23)

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.

Let be defined as in Theorem 1.1 and defined by (1.23). Then there exists a Borel set satisfying Corollary 1.2 (ii) such that

(1.24)

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.

Let be a Borel set satisfying Corollary 1.2 (ii), defined as in Theorem 1.1, and defined by (1.23). Then we can find a nonnegative measure defined on satisfying

(1.25)

such that

(1.26)

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.

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)

(ii)

(iii)

(iv).

Proof.

• (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:

(i)

(ii)

(iii)

(iv).

Lemma 2.4 (see [14]).

Let be a nonnegative integer and .

(i)If , then .

(ii)If and , then .

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

Lemma 2.5.

For any Borel set in , we have and

(2.1)

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

3. Proof of Theorem 1.1

For any , there exists such that

(3.1)

For fixed and , we write

(3.2)

where

(3.3)

We distinguish the following two cases.

Case 1 ().

Note that In view of (1.18), we can find a sequence of positive numbers such that and , where

(3.4)

Consider the sets

(3.5)

for If is a nonnegative measure on such that and for , then we have

(3.6)

So that

(3.7)

which yields

(3.8)

Setting , we see that Theorem 1.1 (ii) is satisfied and

(3.9)

Moreover by Lemma 2.1,

(3.10)

Note that . By (iii) and (iv) in Lemma 2.2, we take , in Lemma 2.2 (iv) and obtain

(3.11)

Similarly, we have by (iii) and (iv) in Lemma 2.2

(3.12)

By Lemma 2.1, we have

(3.13)

Similarly as , we obtain

(3.14)

Finally, by Lemma 2.1, we have

(3.15)

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

Case 2 ().

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.

Moreover we have by Lemma 2.3 (iii)

(3.16)

By Lemma 2.3 (iv), we have

(3.17)

Similarly as , we have

(3.18)

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

For any , there exists such that

(4.1)

For fixed and , we write

(4.2)

where

(4.3)

First note that

(4.4)

Write

(4.5)

where

(4.6)

We obtain by Lemma 2.4 (i)

(4.7)

For , by (4.7) we have

(4.8)

On the other hand, (4.7) yields that

(4.9)

Combining (4.8) and (4.9), we have

(4.10)

We have by Lemma 2.2 (iii)

(4.11)

By Lemma 2.4 (ii), we obtain

(4.12)

Note that . By the lower semicontinuity of , we can prove the following fact in the same way as in the proof of Theorem 1.1:

(4.13)

where , and .

Combining (4.4) and (4.10)–(4.13), we complete the proof of Theorem 1.1.