Poissontype inequalities for growth properties of positive superharmonic functions
Luan and Vieira Journal of Inequalities and Applications
Poissontype inequalities for growth properties of positive superharmonic functions
Kuan Luan 0
John Vieira 0
0 1 Introduction Cartesian coordinates of a point G of R
In this paper, we present new Poissontype inequalities for Poisson integrals with continuous data on the boundary. The obtained inequalities are used to obtain growth properties at infinity of positive superharmonic functions in a smooth cone. where ≤ r < +∞,  π ≤ θn < π and ≤ θj ≤ π for ≤ j ≤ n  (n ≥ ). We denote the unit sphere and the upper half unit sphere by Sn and Sn+, respectively. Let ⊂ Sn. The point (, ) and the set { ; (, ) ∈ } are identified with and , respectively. Let × denote the set {(r, ) ∈ Rn; r ∈ , (, ) ∈ }, where ⊂ R+. The set R+ × is denoted by n( ), which is called a cone. Especially, the set R+ × Sn+ is called the upperhalf space, which is denoted by Tn. Let I ⊂ R. Two sets I × and I × ∂ are denoted by n( ; I) and n( ; I), respectively. We denote n( ; R+) by n( ), which is ∂ n( )  {O}. Let B(G, l) denote the open ball, where G ∈ Rn is the center and l > is the radius.
Poissontype inequality; continuous data; growth property

xn = r cos θ, x = r( jn=– sin θj), n = ,
xn–m+ = r( jm=– sin θj) cos θm, n ≥ ,
E ⊂
In spherical coordinate the Laplace operator is
n = r– n + r–(n – )
h =
on ∂ .
where n is the Beltrami operator. Now we consider the boundary value problem
If the least positive eigenvalue of it is denoted by τ , then we can denote by h ( ) the
normalized positive eigenfunction corresponding to it.
We denote by ι (> ) and –κ (< ) two solutions of the problem t + (n – )t – τ = ,
Then ι + κ is denoted by for the sake of simplicity.
Remark In the case = Sn+–, it follows that
(I) ι = and κ = n – .
(II) h ( ) = wnn cos θ, where wn is the surface area of Sn–.
It is easy to see that the set ∂ n( ) ∪ {∞} is the Martin boundary of n( ). For any
G ∈ n( ) and any H ∈ ∂ n( ) ∪ {∞}, if the Martin kernel is denoted by MK(G, H),
where a reference point is chosen in advance, then we see that (see [])
MK(G, ∞) = rι h ( ) and
where G = (r, ) ∈ n( ) and c is a positive real number.
We shall say that two positive real valued functions f and g are comparable and write
f ≈ g if there exist two positive constants c ≤ c such that cg ≤ f ≤ cg.
Remark Let
∈ . Then h ( ) and dist( , ∂ ) are comparable.
Remark Let (G) = dist(G, ∂ n( )). Then h ( ) and (G) are comparable for any
(, ) ∈ (see []).
Remark Let ≤ α ≤ n. Then h ( ) ≤ c( , n){h ( )}–α , where c( , n) is a constant
depending on and n (e.g. see [], pp.).
Definition For any G ∈ n( ) and any H ∈ n( ). If the Green function in n( ) is
defined by GF (G, H), then:
(I) The Poisson kernel can be defined by
POI (G, H) =
∂nH
where G ∈ n( ) and ν is a positive measure in n( ).
(II) The Poisson integral with g can be defined by
POI [g](G) =
on Rn – n( ; (, +∞)).
h ( )t–κ dν on n( ; (, +∞)),
on Rn – n( ; (, +∞)).
Definition Let μ and ν be defined in Definitions and , respectively. Then the positive
measure ξ is defined by
Remark Let
= Sn+–. Then
GF Sn+– (G, H) =
log G – H∗ – log G – H if n = ,
G – H–n – G – H∗–n if n ≥ ,
where G = (X, xn), H∗ = (Y , –yn), that is, H∗ is the mirror image of H = (Y , yn) on ∂Tn.
Hence, for the two points G = (X, xn) ∈ Tn and H = (Y , yn) ∈ ∂Tn, we have
d =
Remark Let
= Sn+–. Then we define
d (y) =
Definition Let λ be any positive measure on Rn having finite total mass. Then the
maximal function M(G; λ, β) is defined by
EX( ; λ, β) = G = (r, ) ∈ Rn – {O}; M(G; λ, β)rβ >
where is a sufficiently small positive number.
Remark Let β > and λ({P}) > for any P = O. Then
(I) Then M(G; λ, β) = +∞.
(II) {G ∈ Rn – {O}; λ({P}) > } ⊂ EX( ; λ, β).
Recently, Qiao and Wang (see [], Corollary . with m = ) proved classical
Poissontype inequalities for Poisson integrals in a half space. Applications of them were also
developed by Pang and Ychussie (see []) and Xue and Wang (see []). In particular, Huang
(see []) further obtained SchrödingerPoissontype inequalities for PoissonSchrödinger
integrals and gave their related applications.
∂Tn
g(y) + y –n dy < ∞.
as x → ∞ in Tn.
2 Results
Our first aim in this paper is to prove the following result, which is a generalization of
Theorem A. For similar results with respect to Schrödinger operator, we refer the reader
to the literature (see [, ]).
Theorem Let POI μ(G) ≡ +∞ for any G = (r, ) ∈ n( ), where μ is a positive
measure on n( ). Then
< ∞.
dt < ∞.
Then the Poisson integral POI [g](G) is harmonic in n( ) and
Remark If = Sn+–, then it is easy to see that () is equivalent to () and () is a finite
sum, then the set EX( ; μ , ) is a bounded set and () reduces to () in the case α = n from
Remark .
Corollary If μ is a positive measure on ∂Tn satisfying POISn+– μ(x) ≡ +∞ for any x =
(X, xn) ∈ Tn, then
< ∞.
Let dμ = g dσH for any H = (t, ) ∈ n( ). Then we have the following result, which
generalizes Theorem A to the conical case.
Corollary If g is a measurable function on n( ) satisfying
< ∞.
The following result is very well known. We quote it from [].
Theorem B (see []) Let < w(G) be a superharmonic function in Tn. Then there exist a
positive measure μ on ∂Tn and a positive measure ν on Tn such that w(x) can be uniquely
decomposed as
w(x) = cxn + POISn+– μ(x) + GF Sn+– ν(x),
Theorem C (see [], Theorem ) Let < w(G) be a superharmonic function in n( ).
Then there exist a positive measure μ on n( ) and a positive measure ν in n( ) such
that w(G) can be uniquely decomposed as
c(w) =
and c(w) =
As an application of Theorem and Lemma in Section , we give the growth properties
of positive superharmonic functions at infinity in a cone.
w(G) – c(w)MK(G, ∞) – c(w)MK(G, O) = o rι
Theorem immediately gives the following corollary.
Corollary Let w(x) (≡ +∞) (x = (X, xn) ∈ Tn) be defined by (). Then w(x) – cxn = o(x)
for any x ∈ Tn – EX( ; , n – ) as x → ∞, where EX( ; , n – ) is a subset of n( ) and
has a covering satisfying ().
3 Lemmas
In order to prove our main results we need following lemmas. In this paper let M denote
various constants independent of the variables in questions, which may be different from
line to line.
Lemma (see [], Lemma ) Let any G = (r, ) ∈ n( ) and any H = (t, ) ∈ n( ), we
have the following estimates:
POI (G, H) ≤ Mr–κ tι –h ( )
POI (G, H) ≤ Mrι t–κ –h ( )
for < rt ≤ , and
for r < t ≤ r .
< ∞.
Lemma (see [], Lemma ) If β ≥ and λ is positive measure on Rn having finite total
mass, then exceptional set EX( ; λ, β) has a covering {rk, Rk} (k = , , . . .) satisfying
The estimation of the Green potential at infinity is the following, which is due to [].
4 Proof of Theorem 1
Let G = (r, ) be any point in the set n( ; (L, +∞)) – EX( ; μ , n – α), where r is a
sufficiently large number satisfying r ≥ l .
Put
POI μ(G) = POI (G) + POI (G) + POI (G),
POI (G) = n( ;(,r]) POI (G, H) dμ(H),
POI (G) = n( ;(r,r)) POI (G, H) dμ(H),
POI (G) = n( ;[r,∞)) POI (G, H) dμ(H).
We have the following estimates:
POI (G) ≤ Mrι h ( ) r –
from (), (), and [], Lemma .
By (), we write
POI (G) ≤ POI(G) + POI(G),
POI(G) = M
POI(G) = M
We first have
from [], Lemma .
Next, we shall estimate POI(G). We can find a number k satisfying k ≥ and
for any G = (r, ) ∈ (k), where
Then the set n( ) can be split into two sets (k) and n( ) – (k).
Let G = (r, ) ∈ n( ) – (k). Then
G – H ≥ kr,
where H ∈ n( ) and k is a positive number. So
n( ;( r,∞))
from [], Lemma .
If G ∈ (k), we put
Fl(G) = H ∈ n
; l– (G) ≤ G – H < l (G) .
i= Fl(G)
POI(G) = M
where l(G) is a positive integer satisfying l(G)– (G) ≤ r < l(G) (G).
By Remark we have rh ( ) ≤ M (G) (G = (r, ) ∈ n( )), and hence
dμ (H) ≤ Mrκ –α+ h ( ) –αμ Fl(G) l (G) α–n
for l = , , , . . . , l(G).
Since G = (r, ) ∈/ EX( ; μ , n – α), we have
μ Fl(G) l (G) α–n ≤ μ B G, l (G)
l (G) α–n ≤ M G; μ , n – α ≤ rα–n
for l = , , , . . . , l(G) – and
μ Fl(G)(G) l (G) α–n ≤ μ
B G, r
From (), (), (), (), (), and Remark , we obtain POI μ(G) = o(rι {h ( )}–α)
for any G = (r, ) ∈ n( ; (L, +∞)) – EX( ; μ , n – α) as r → ∞, where L is a sufficiently
large real number. With Lemma we have the conclusion of Theorem .
Let G = (r, ) be a fixed point in
n( ). Then there exists a number R satisfying
max{ r , } < R. There exists a positive constant M such that
P OI (G, H) ≤ M rι t–κ –h ( )
from Remark and (), where H = (t, ) ∈
Let M = M cn–rι h ( ). Then we have from () and ()
n( ) satisfying < rt ≤ .
P OI [g](G) is a harmonic function of G ∈
proof of Corollary is completed.
Competing interests
The authors declare that they have no competing interests.
n( ;(R,+∞))
dt < ∞.
For any G ∈
n( ), it is easy to see that P OI [g](G) is finite, which means that
n( ). Meanwhile, Theorem gives (). The
Authors’ contributions
JV completed the main study. KL pointed out some mistakes and verified the calculation. Both authors read and
approved the final manuscript.
Acknowledgements
The project is partially supported by the Applied Technology Research and the Development Foundation of Heilongjiang
Province (Grant No. GC13A308).The authors would like to thank the referees and the editor for their careful reading and
some useful comments on improving the presentation of this paper.
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