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RETRACTED ARTICLE: Growth properties of Green-Sch potentials at infinity
Boundary Value Problems volume 2014, Article number: 245 (2014)
- The Retraction Note to this article has been published in Boundary Value Problems 2020 2020:19
This paper gives growth properties of Green-Sch potentials at infinity in a cone, which generalizes results obtained by Qiao-Deng. The proof is based on the fact that the estimations of Green-Sch potentials with measures are connected with a kind of densities of the measures modified by the measures.
MSC: 35J10, 35J25.
1 Introduction and main results
Let R and be the set of all real numbers and the set of all positive real numbers, respectively. We denote by () the n-dimensional Euclidean space. A point in is denoted by , . The Euclidean distance of two points P and Q in is denoted by . Also with the origin O of is simply denoted by . The boundary, the closure and the complement of a set S in are denoted by ∂ S, , and , respectively. For and , let denote the open ball with center at P and radius r in .
We introduce a system of spherical coordinates , , in which are related to cartesian coordinates by
and if , then
where , , and if , then ().
The unit sphere and the upper half unit sphere in are denoted by and , respectively. For simplicity, a point on and the set for a set Ω, , are often identified with Θ and Ω, respectively. For two sets and , the set in is simply denoted by . In particular, the half space will be denoted by .
By , we denote the set in with the domain Ω on (). We call it a cone. Then is a special cone obtained by putting . We denote the sets and with an interval on R by and . By we denote . By we denote , which is .
Let be an arbitrary domain in and denote the class of nonnegative radial potentials , i.e., , such that with some if and with if or .
If , then the stationary Schrödinger operator
where Δ is the Laplace operator and I is the identical operator, can be extended in the usual way from the space to an essentially self-adjoint operator on (see , Ch. 13]). We will denote it as well. This last one has a Green-Sch function . Here is positive on and its inner normal derivative , where denotes the differentiation at Q along the inward normal into . We denote this derivative by , which is called the Poisson-Sch kernel with respect to .
We shall say that a set has a covering if there exists a sequence of balls with centers in such that , where is the radius of and is the distance from the origin to the center of .
For positive functions and , we say that if for some constant . If and , we say that .
Let Ω be a domain on with smooth boundary. Consider the Dirichlet problem
where is the spherical part of the Laplace opera
We denote the least positive eigenvalue of this boundary value problem by λ and the normalized positive eigenfunction corresponding to λ by , . In order to ensure the existence of λ and a smooth . We put a rather strong assumption on Ω: if , then Ω is a -domain () on surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g. see , pp.88-89] for the definition of -domain).
For any , we have (see , pp.7-8])
where and .
Solutions of an ordinary differential equation
and W is monotonically decreasing with
We will also consider the class , consisting of the potentials such that there exists the finite limit , and moreover, . If , then the (sub)superfunctions are continuous (see ).
In the rest of paper, we assume that and we shall suppress this assumption for simplicity.
We denote the Green-Sch potential with a positive measure v on by
Let ν be any positive measure such that (resp. ) for . The positive measure (rep. ) on is defined by
Let , , and λ be any positive measure on having finite total mass. For each , the maximal function is defined by (see )
is denoted by .
If (), then for any positive number β. So we can find .
About the growth properties of Green potentials at infinity in a cone, Qiao-Deng (see , Theorem 1]) has proved the following result.
Let ν be a positive measure onsuch thatfor any. Then there exists a coveringof () satisfying
Now we state our first result.
Let ν be a positive measure on such that
Then there exists a coveringof () satisfying
By comparison the condition (1.4) is fairly briefer and easily applied. Moreover, is a set of 1-finite view in the sense of ,  (see , Definition 2.1] for the definition of 1-finite view). In the case , Theorem 1 (1.6) is just the result of Theorem A.
Let ν be a positive measure onsuch that (1.4) holds. Then for a sufficiently large L and a sufficiently small ϵ we have
2 Some lemmas
for anyand anysatisfying (resp. );
Further, for anyand any, we have
Let ν be a positive measure onsuch that there is a sequence of points, () satisfying ( ; ). Then, for a positive number l,
Take a positive number l satisfying , . Then from (2.2), we have
If we take a point , , then we have from (2.1)
If R () is sufficiently large, then
which gives (2.5). □
Let λ be any positive measure onhaving finite total mass. Thenhas a covering () satisfying
If , then there exists a positive number such that
Since can be covered by the union of a family of balls (). By the Vitali lemma (see ), there exists , which is at most countable, such that are disjoint and .
On the other hand, note that
Hence we obtain
Since . Then is finally covered by a sequence of balls ( ; ) satisfying
where () is the ball which covers . □
3 Proof of Theorem 1
For any point , where R () is a sufficiently large number and ϵ is a sufficiently small positive number.
Then by Lemma 2, we immediately get
To estimate , take a sufficiently small positive number c independent of P such that
and divide into two sets and , where
There exists a positive such that for any , and hence
from Lemma 2.
Now we estimate . Set
Since and hence from Remark 1, we can divide into
Since , we have
for any (). Then by (1.1)
Since , we obtain
By (3.4), we can take a positive integer satisfying
Since (), we have
Since , we have
Hence we obtain
Combining (3.1)-(3.3) and (3.5)-(3.7), we finally obtain the result that if R is sufficiently large and ϵ is a sufficiently small, then as , where . Finally, there exists an additional finite ball covering , which together with Lemma 3, gives the conclusion of Theorem 1.
Escassut A, Tutschke W, Yang CC: Some Topics on Value Distribution and Differentiability in Complex and P-Adic Analysis. Science Press, Beijing; 2008.
Gilbarg D, Trudinger NS: Elliptic Partial Differential Equations of Second Order. Springer, Berlin; 1977.
Miranda C: Partial Differential Equations of Elliptic Type. Springer, Berlin; 1970.
Verzhbinskii GM, Maz’ya VG: Asymptotic behavior of solutions of elliptic equations of the second order close to a boundary. I. Sib. Mat. Zh. 1971, 12(2):874-899.
Qiao L, Deng GT: A lower bound of harmonic functions in a cone and its application. Sci. Sin., Math. 2014, 44(6):671-684. (in Chinese) 10.1360/N012013-00108
Qiao L, Deng GT: Minimally thin sets at infinity with respect to the Schrödinger operator. Sci. Sin., Math. 2014, 44(12):1247-1256. (in Chinese)
Xue GX: A remark on the a-minimally thin sets associated with the Schrödinger operator. Bound. Value Probl. 2014., 2014: 10.1186/1687-2770-2014-133
Zhao T: Minimally thin sets associated with the stationary Schrodinger operator. J. Inequal. Appl. 2014., 2014: 10.1186/1029-242X-2014-67
Simon B: Schrödinger semigroups. Bull. Am. Math. Soc. 1982, 7(2):447-526. 10.1090/S0273-0979-1982-15041-8
Hartman P: Ordinary Differential Equations. Wiley, New York; 1964.
Qiao L, Ren YD: Integral representations for the solutions of infinite order of the stationary Schrödinger equation in a cone. Monatshefte Math. 2014, 173(4):593-603. 10.1007/s00605-013-0506-1
Qiao L, Deng GT: The Riesz decomposition theorem for superharmonic functions in a cone and its application. Sci. Sin., Math. 2012, 42(8):763-774. (in Chinese) 10.1360/012011-1018
Qiao L: Integral representations for harmonic functions of infinite order in a cone. Results Math. 2012, 61(4):63-74. 10.1007/s00025-010-0076-7
Qiao L, Deng GT: Integral representations of harmonic functions in a cone. Sci. Sin., Math. 2011, 41(6):535-546. (in Chinese)
Miyamoto I, Yoshida H: Two criteria of Wiener type for minimally thin sets and rarefied sets in a cone. J. Math. Soc. Jpn. 2002, 54: 487-512. 10.2969/jmsj/1191593906
Yoshida H: Harmonic majorant of a radial subharmonic function on a strip and their applications. Int. J. Pure Appl. Math. 2006, 30(2):259-286.
Stein EM: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton; 1970.
The authors are very thankful to the anonymous referees for their valuable comments and constructive suggestions, which helped to improve the quality of the paper. This work is supported by the Academy of Finland Grant No. 176512.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final manuscript.
The Editors-in-Chief have retracted this article  because it showed evidence of peer review manipulation. In addition, the identity of the corresponding author could not be verified: Stockholms Universitet have confirmed that Alexander Yamada has not been affiliated with their institution. The authors have not responded to any correspondence about this retraction.
 Zhao, T. & Yamada, A. Growth properties of Green-Sch potentials at infinity. Bound Value Probl (2014) 2014: 245. https://doi.org/10.1186/s13661-014-0245-9
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Zhao, T., Yamada, A. RETRACTED ARTICLE: Growth properties of Green-Sch potentials at infinity. Bound Value Probl 2014, 245 (2014). https://doi.org/10.1186/s13661-014-0245-9
- stationary Schrödinger operator
- Green-Sch potential
- growth property