- Research Article
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Asymptotic Behavior for a Class of Modified
-Potentials in a Half Space
Journal of Inequalities and Applications volume 2010, Article number: 647627 (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.
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ1_HTML.gif)
where ,
is a Schwarz function and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ2_HTML.gif)
It follows that we can extend definition (1.1) to certain negative powers of ,
for
and define an operator
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ4_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ5_HTML.gif)
where and
.
This Riesz kernel in
inspired us to introduce the modified Riesz kernel for
. To do this, we first set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ6_HTML.gif)
Let be the modified Riesz kernel for
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ7_HTML.gif)
where denotes reflection in the boundary plane
just as
.
We define the kernel function when
and
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ8_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ9_HTML.gif)
where is a nonnegative integer;
is the ultraspherical (or Gegenbauer) polynomials (see [3]). The Gegenbauer polynomials come from the generating function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ10_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ11_HTML.gif)
In case , we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ12_HTML.gif)
where is a nonnegative integer,
, and
.
Then we define the modified kernel function by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ13_HTML.gif)
Write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ14_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ15_HTML.gif)
for a nonnegative Borel measurable function on
and a nonnegative measure
on a Borel set
. We define a capacity
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ16_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ17_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ18_HTML.gif)
Then there exists a Borel set with properties
(i)
(ii)
where .
Corollary 1.2.
Let be a nonnegative measure on
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ19_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ20_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ21_HTML.gif)
Then there exists a Borel set satisfying Corollary 1.2 (ii) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ22_HTML.gif)
We define the modified -potentials on
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ23_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ24_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ25_HTML.gif)
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ26_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ27_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ28_HTML.gif)
For fixed and
, we write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ29_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ30_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ31_HTML.gif)
Consider the sets
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ32_HTML.gif)
for If
is a nonnegative measure on
such that
and
for
, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ33_HTML.gif)
So that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ34_HTML.gif)
which yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ35_HTML.gif)
Setting , we see that Theorem 1.1 (ii) is satisfied and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ36_HTML.gif)
Moreover by Lemma 2.1,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ37_HTML.gif)
Note that . By (iii) and (iv) in Lemma 2.2, we take
,
in Lemma 2.2 (iv) and obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ38_HTML.gif)
Similarly, we have by (iii) and (iv) in Lemma 2.2
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ39_HTML.gif)
By Lemma 2.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ40_HTML.gif)
Similarly as , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ41_HTML.gif)
Finally, by Lemma 2.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ42_HTML.gif)
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)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ43_HTML.gif)
By Lemma 2.3 (iv), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ44_HTML.gif)
Similarly as , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ45_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ46_HTML.gif)
For fixed and
, we write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ47_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ48_HTML.gif)
First note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ49_HTML.gif)
Write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ50_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ51_HTML.gif)
We obtain by Lemma 2.4 (i)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ52_HTML.gif)
For , by (4.7) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ53_HTML.gif)
On the other hand, (4.7) yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ54_HTML.gif)
Combining (4.8) and (4.9), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ55_HTML.gif)
We have by Lemma 2.2 (iii)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ56_HTML.gif)
By Lemma 2.4 (ii), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ57_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F647627/MediaObjects/13660_2010_Article_2212_Equ58_HTML.gif)
where , and
.
Combining (4.4) and (4.10)–(4.13), we complete the proof of Theorem 1.1.
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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.
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Qiao, L., Deng, G. Asymptotic Behavior for a Class of Modified -Potentials in a Half Space.
J Inequal Appl 2010, 647627 (2010). https://doi.org/10.1155/2010/647627
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DOI: https://doi.org/10.1155/2010/647627