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RETRACTED ARTICLE: A remark on the a-minimally thin sets associated with the Schrödinger operator

This article was retracted on 25 May 2021

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Abstract

The aim of this paper is to give a new criterion for a-minimally thin sets at infinity with respect to the Schrödinger operator in a cone, which supplement the results obtained by T. Zhao.

1 Introduction and results

Let R and R + be the set of all real numbers and the set of all positive real numbers, respectively. We denote by R n (n2) the n-dimensional Euclidean space. A point in R n is denoted by P=(X, x n ), X=( x 1 , x 2 ,, x n 1 ). The Euclidean distance between two points P and Q in R n is denoted by |PQ|. Also |PO| with the origin O of R n is simply denoted by |P|. The boundary and the closure of a set S in R n are denoted by ∂S and S ¯ , respectively. Further, intS, diamS, and dist( S 1 , S 2 ) stand for the interior of S, the diameter of S, and the distance between S 1 and S 2 , respectively.

We introduce a system of spherical coordinates (r,Θ), Θ=( θ 1 , θ 2 ,, θ n 1 ), in R n which are related to cartesian coordinates ( x 1 , x 2 ,, x n 1 , x n ) by x n =rcos θ 1 .

Let D be an arbitrary domain in R n and A a denote the class of non-negative radial potentials a(P), i.e. 0a(P)=a(r), P=(r,Θ)D, such that a L loc b (D) with some b>n/2 if n4 and with b=2 if n=2 or n=3 (see [[1], p.354] and [2]).

If a A a , then the stationary Schrödinger operator

Sc h a =Δ+a(P)I=0,

where Δ is the Laplace operator and I is the identical operator, can be extended in the usual way from the space C 0 (D) to an essentially self-adjoint operator on L 2 (D) (see [[1], Ch. 11]). We will denote it Sc h a as well. This last one has a Green a-function G D a (P,Q). Here G D a (P,Q) is positive on D and its inner normal derivative G D a (P,Q)/ n Q 0, where / n Q denotes the differentiation at Q along the inward normal into D.

We call a function u that is upper semi-continuous in D a subfunction with respect to the Schrödinger operator Sc h a if its values belong to the interval [,) and at each point PD with 0<r<r(P) we have the generalized mean-value inequality (see [[1], Ch. 11])

u(P) S ( P , r ) u(Q) G B ( P , r ) a ( P , Q ) n Q dσ(Q)

satisfied, where G B ( P , r ) a (P,Q) is the Green a-function of Sc h a in B(P,r) and dσ(Q) is a surface measure on the sphere S(P,r)=B(P,r). If −u is a subfunction, then we call u a superfunction (with respect to the Schrödinger operator Sc h a ).

The unit sphere and the upper half unit sphere in R n are denoted by S n 1 and S + n 1 , respectively. For simplicity, a point (1,Θ) on S n 1 and the set {Θ;(1,Θ)Ω} for a set Ω, Ω S n 1 , are often identified with Θ and Ω, respectively. For two sets Ξ R + and Ω S n 1 , the set {(r,Θ) R n ;rΞ,(1,Θ)Ω} in R n is simply denoted by Ξ×Ω. By C n (Ω), we denote the set R + ×Ω in R n with the domain Ω on S n 1 . We call it a cone. We denote the set I×Ω with an interval on R by C n (Ω;I).

From now on, we always assume D= C n (Ω). For the sake of brevity, we shall write G Ω a (P,Q) instead of G C n ( Ω ) a (P,Q). We shall also write g 1 g 2 for two positive functions g 1 and g 2 , if and only if there exists a positive constant c such that c 1 g 1 g 2 c g 1 .

Let Ω be a domain on S n 1 with smooth boundary. Consider the Dirichlet problem

( Λ n + λ ) φ = 0 on  Ω , φ = 0 on  Ω ,

where Λ n is the spherical part of the Laplace operata Δ n

Δ n = n 1 r r + 2 r 2 + Λ n r 2 .

We denote the least positive eigenvalue of this boundary value problem by λ and the normalized positive eigenfunction corresponding to λ by φ(Θ). In order to ensure the existence of λ and a smooth φ(Θ), we put a rather strong assumption on Ω: if n3, then Ω is a C 2 , α -domain (0<α<1) on S n 1 surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g. see [[3], pp.88-89] for the definition of C 2 , α -domain).

For any (1,Θ)Ω, we have (see [[4], pp.7-8])

δ(P)rφ(Θ),
(1)

where P=(r,Θ) C n (Ω) and δ(P)=dist(P, C n (Ω)).

Solutions of an ordinary differential equation (see [[5], p.217])

Q (r) n 1 r Q (r)+ ( λ r 2 + a ( r ) ) Q(r)=0,0<r<.
(2)

It is well known (see, for example, [6]) that if the potential a A a , then equation (2) has a fundamental system of positive solutions {V,W} such that V and W are increasing and decreasing, respectively.

We will also consider the class B a , consisting of the potentials a A a such that there exists the finite limit lim r r 2 a(r)=k[0,), and, moreover, r 1 | r 2 a(r)k|L(1,). If a B a , then the (sub)superfunctions are continuous (see [7]). In the rest of paper, we assume that a B a and we shall suppress this assumption for simplicity.

Denote

ι k ± = 2 n ± ( n 2 ) 2 + 4 ( k + λ ) 2 ,

then the solutions to equation (2) have the asymptotic (see [3])

V(r) r ι k + ,W(r) r ι k ,as r.
(3)

It is well known that the Martin boundary of C n (Ω) is the set C n (Ω){}, each of which is a minimal Martin boundary point. For P C n (Ω) and Q C n (Ω){}, the Martin kernel can be defined by M Ω a (P,Q). If the reference point P is chosen suitably, then we have

M Ω a (P,)=V(r)φ(Θ)and M Ω a (P,O)=cW(r)φ(Θ),
(4)

for any P=(r,Θ) C n (Ω).

In [[8], p.67], Zhao introduce the notations of a-thin (with respect to the Schrödinger operator Sc h a ) at a point, a-polar set (with respect to the Schrödinger operator Sc h a ) and a-minimal thin sets at infinity (with respect to the Schrödinger operator Sc h a ). A set H in R n is said to be a-thin at a point Q if there is a fine neighborhood E of Q which does not intersect H{Q}. Otherwise H is said to be not a-thin at Q on C n (Ω). A set H in R n is called a polar set if there is a superfunction u on some open set E such that H{PE;u(P)=}. A subset H of C n (Ω) is said to be a-minimal thin at Q C n (Ω){} on C n (Ω), if there exists a point P C n (Ω) such that

R ˆ M Ω a ( , Q ) H (P) M Ω a (P,Q),

where R ˆ M Ω a ( , Q ) H is the regularized reduced function of M Ω a (,Q) relative to H (with respect to the Schrödinger operator Sc h a ).

Let H be a bounded subset of C n (Ω). Then R ˆ M Ω a ( , ) H (P) is bounded on C n (Ω) and hence the greatest a-harmonic minorant of R ˆ M Ω a ( , ) H is zero. When by G Ω a μ(P) we denote the Green a-potential with a positive measure μ on C n (Ω), we see from the Riesz decomposition theorem that there exists a unique positive measure λ H a on C n (Ω) such that

R ˆ M Ω a ( , ) H (P)= G Ω a λ H a (P)

for any P C n (Ω) and λ H a is concentrated on I H , where

I H = { P C n ( Ω ) ; H  is not a-thin at  P } .

The Green a-energy γ Ω a (H) (with respect to the Schrödinger operator Sc h a ) of λ H a is defined by

γ Ω a (H)= C n ( Ω ) G Ω a λ H a d λ H a .

Also, we can define a measure σ Ω a on C n (Ω)

σ Ω a (H)= H ( M Ω a ( P , ) δ ( P ) ) 2 dP.

In [[8], Theorem 5.4.3], Long gave a criterion that characterizes a-minimally thin sets at infinity in a cone.

Theorem A A subset H of C n (Ω) is a-minimally thin at infinity on C n (Ω) if and only if

j = 0 γ Ω a ( H j )W ( 2 j ) V 1 ( 2 j ) <,

where H j =H C n (Ω;[ 2 j , 2 j + 1 )) and j=0,1,2, .

In recent work, Zhao (see [[2], Theorems 1 and 2]) proved the following results. For similar results in the half space with respect to the Schrödinger operator, we refer the reader to the papers by Ren and Su (see [9, 10]).

Theorem B The following statements are equivalent.

  1. (I)

    A subset H of C n (Ω) is a-minimally thin at infinity on C n (Ω).

  2. (II)

    There exists a positive superfunction v(P) on C n (Ω) such that

    inf P C n ( Ω ) v ( P ) M Ω a ( P , ) =0
    (5)

    and

    H { P C n ( Ω ) ; v ( P ) M Ω a ( P , ) } .
  3. (III)

    There exists a positive superfunction v(P) on C n (Ω) such that even if v(P)c M Ω a (P,) for any PH, there exists P 0 C n (Ω) satisfying v( P 0 )<c M Ω a ( P 0 ,).

Theorem C If a subset H of C n (Ω) is a-minimally thin at infinity on C n (Ω), then we have

H d P ( 1 + | P | ) n <.
(6)

Remark From equation (3), we immediately know that equation (6) is equivalent to

H V ( 1 + | P | ) W ( 1 + | P | ) ( 1 + | P | ) 2 dP<.
(7)

This paper aims to show that the sharpness of the characterization of an a-minimally thin set in Theorem C. In order to do this, we introduce the Whitney cubes in a cone.

A cube is the form

[ l 1 2 j , ( l 1 + 1 ) 2 j ] ×× [ l n 2 j , ( l n + 1 ) 2 j ] ,

where j, l 1 ,, l n are integers. The Whitney cubes of C n (Ω) are a family of cubes having the following properties:

  1. (I)

    k W k = C n (Ω).

  2. (II)

    int W j int W k = (jk).

  3. (III)

    diam W k dist( W k , R n C n (Ω))4diam W k .

Theorem 1 If H is a union of cubes from the Whitney cubes of C n (Ω), then equation (7) is also sufficient for H to be a-minimally thin at infinity with respect to C n (Ω).

From the Remark and Theorem 1, we have the following.

Corollary 1 Let v(P) be a positive superfunction on C n (Ω) such that equation (5) holds. Then we have

{ P C n ( Ω ) ; v ( P ) M Ω a ( P , ) } V ( 1 + | P | ) W ( 1 + | P | ) ( 1 + | P | ) 2 dP<.

Corollary 2 Let H be a Borel measurable subset of C n (Ω) satisfying

H V ( 1 + | P | ) W ( 1 + | P | ) ( 1 + | P | ) 2 dP=+.

If v(P) is a non-negative superfunction on C n (Ω) and c is a positive number such that v(P)c M Ω a (P,) for all PH, then v(P)c M Ω a (P,) for all P C n (Ω).

2 Lemmas

To prove our results, we need some lemmas.

Lemma 1 Let W k be a cube from the Whitney cubes of C n (Ω). Then there exists a constant c independent of k such that

γ Ω a ( W k )c σ Ω a ( W k ).

Proof If we apply a result of Long (see [[8], Theorem 6.1.3]) for compact set W ¯ k , we obtain a measure μ on C n (Ω), suppμ W ¯ k , μ( W ¯ k )=1 such that

{ C n ( Ω ) | P Q | 2 n d μ ( Q ) = { Cap ( W ¯ k ) } 1 if  n 3 , C 2 ( Ω ) log | P Q | d μ ( Q ) = log Cap ( W ¯ k ) if  n = 2
(8)

for any P W ¯ k . Also there exists a positive measure λ W ¯ k a on C n (Ω) such that

R ˆ M Ω a ( , ) W ¯ k (P)= G Ω a λ W ¯ k a (P)
(9)

for any P C n (Ω).

Let P k =( r k , Θ k ), ρ k , t k be the center of W k , the diameter of W j , the distance between W k and C n (Ω), respectively. Then we have ρ k t k 4 ρ k and ρ k r k . Then from equation (1) we have

r k M Ω a (P,)V( r k ) ρ k
(10)

for any P W ¯ k . We can also prove that

G Ω a (P,Q) { | P Q | 2 n if  n 3 , log ρ k | P Q | if  n = 2
(11)

for any P W ¯ k and any Q W ¯ k . Hence we obtain

r k λ W ¯ k a ( C n ( Ω ) ) { V ( r k ) ρ k Cap ( W ¯ k ) if  n 3 , V ( r k ) ρ k { log ρ k Cap ( W ¯ k ) } 1 if  n = 2
(12)

from equations (8), (9), (10), and (11). Since

γ Ω a ( W k )= G Ω a λ W ¯ k a d λ W ¯ k a W ¯ k M Ω a (P,)d λ W ¯ k a (P) r k ι k + 1 ρ k λ W ¯ k a ( C n ( Ω ) )

from equations (3), (9), and (10), we have from (12)

γ Ω a ( W k ) { r k 2 ι k + 2 ρ k 2 Cap ( W ¯ k ) if  n 3 , r k 2 ι k + 2 ρ k 2 { log ρ k Cap ( W ¯ k ) } 1 if  n = 2 .
(13)

Since

{ Cap ( W ¯ k ) ρ k n 2 if  n 3 , Cap ( W ¯ k ) ρ k if  n = 2 ,

we obtain from equation (13)

γ Ω a ( W k ) r k 2 ι k + 2 ρ k n .
(14)

On the other hand, we have from equation (1)

σ Ω a ( W k ) r k 2 ι k + 2 ρ k n ,

which, together with equation (14), gives the conclusion of Lemma 1. □

3 Proof of Theorem 1

Let { W k } be a family of cubes from the Whitney cubes of C n (Ω) such that H= k W k . Let { W k , j } be a subfamily of { W k } such that W k , j ( H j 1 H j H j + 1 ), where j=1,2,3, .

Since γ Ω a is a countably subadditive set function (see [[8], p.49]), we have

γ Ω a ( H j ) k γ Ω a ( W k , j )
(15)

for j=1,2, . Hence for j=1,2, we see from Lemma 1

k γ Ω a ( W k , j ) k σ Ω a ( W k , j ),
(16)

which, together with equation (1), gives

k σ Ω a ( W k , j ) ( H j 1 + H j + H j + 1 ) V 2 ( r ) r 2 d P ( H j 1 + H j + H j + 1 ) r 2 ( ι k + 1 ) d P r 2 ( j 1 ) ( ι k + 1 ) | H j 1 | + r 2 j ( ι k + 1 ) | H j | + r 2 ( j + 1 ) ( ι k + 1 ) | H j + 1 |
(17)

for j=1,2, . Thus equations (15), (16), and (17) give

γ Ω a ( H j ) r 2 ( j 1 ) ( ι k + 1 ) | H j 1 |+ r 2 j ( ι k + 1 ) | H j |+ r 2 ( j + 1 ) ( ι k + 1 ) | H j + 1 |

for j=1,2, . Finally we obtain from equation (1)

j = 0 γ Ω a ( H j ) W ( 2 j ) V 1 ( 2 j ) γ Ω a ( H 0 ) + j = 0 2 j ( 2 ι k + 2 ) 2 j ( ι k + + ι k ) | H j | γ Ω a ( H 0 ) + j = 0 2 2 j W ( 2 j ) V 1 ( 2 j ) | H j | γ Ω a ( H 0 ) + H V ( 1 + | P | ) W ( 1 + | P | ) ( 1 + | P | ) 2 d P < ,

which shows with Theorem A that H is a-minimally thin at infinity with respect to C n (Ω).

Change history

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Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grants Nos. 11301140 and U1304102.

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This article has been retracted. Please see the retraction notice for more detail:https://doi.org/10.1186/s13661-021-01530-9

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Xue, G. RETRACTED ARTICLE: A remark on the a-minimally thin sets associated with the Schrödinger operator. Bound Value Probl 2014, 133 (2014). https://doi.org/10.1186/1687-2770-2014-133

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