Research Article | Open Access
Criteria of Wiener Type for Minimally Thin Sets and Rarefied Sets Associated with the Stationary Schrödinger Operator in a Cone
We give some criteria for a-minimally thin sets and a-rarefied sets associated with the stationary Schrödinger operator at a fixed Martin boundary point or ∞ with respect to a cone. Moreover, we show that a positive superfunction on a cone behaves regularly outside an a-rarefied set. Finally we illustrate the relation between the a-minimally thin set and the a-rarefied set in a cone.
This paper is concerned with some properties for the generalized subharmonic functions associated with the stationary Schrödinger operator. More precisely the minimally thin sets and rarefied sets about these generalized subharmonic functions will be studied. The research on minimal thinness has been exploited a little and attracted many mathematicians. In 1949 Lelong-Ferrand  started the study of the thinness at boundary points for the subharmonic functions on the half-space. Then in 1957 Naïm  gave some criteria for minimally thin sets at a fixed boundary point with respect to half-space (see  for a survey of the results in [1, 2]). In 1980 Essén and Jackson  gave the criteria for minimally thin sets at with respect to half-space, and furthermore they introduced rarefied sets at with respect to half-space, which is more refined than minimally thin set. Later Miyamoto and Yoshida  extended these results of Essén and Jackson from half-space to a cone. In this paper, we will deal with the corresponding questions for the generalized subharmonic functions associated with the stationary Schrödinger operator.
To state our results, we will need some notations and preliminary results. As usual, denote by the -dimensional Euclidean space. For an open subset set , denote its boundary by and its closure by . Let , where , and let be the Euclidean norm of and the Euclidean distance of two points and in . The unit sphere and the upper half unit sphere are denoted by and , respectively. For and , let be the open ball of radius centered at in , then . Furthermore, denote by the -dimensional volume elements induced by the Euclidean metric on .
For , it can be reexpressed in spherical coordinates , via the following transforms: and if , where and .
Relative to the system of spherical coordinates, the Laplace operator may be written as where the explicit form of the Beltrami operator is given by Azarin (see ).
Let be an arbitrary domain in , and denotes the class of nonnegative radial potentials (i.e., for ) such that with some if and with if or .
For the identical operator , define the stationary Schrödinger operator with a potential by If , then can be extended in the usual way from the space to an essentially self-adjoint operator on (see [7, Chapter 13] for more details). Furthermore has a Green -function . Here is positive on , and its inner normal derivative is nonnegative, where denotes the differentiation at along the inward normal into . We write this derivative by , which is called the Poisson -kernel with respect to . Denote by the Green function of Laplacian. It is well known that for any potential . The “inverse” inequality in some sense is much more elaborate. When is a bounded domain in , Cranston (see , the case is implicitly contained in ) have proved that where is a positive constant and independent of points in . If , then obviously .
Suppose that a function is upper semicontinuous in . We call a subfunction for the Schrödinger operator if the generalized mean-value inequality is satisfied at each point with , where , is the Green -function of in , and the surface area element on (see ).
Denote by the class of subfunctions in . We call a superfunction associated with if , and denote by the class of superfunctions. If a function on is both subfunction and superfunction, then it is called an -harmonic function associated with the operator . The class of -harmonic functions is denoted by , and it is obviously . Here we follow the terminology from Levin and Kheyfits (see [11–13]).
For simplicity, the point on and the set for a set are often identified with and , respectively. For and , the set in is simply denoted by . In particular, the half space will be denoted by . We denote by the set in with the domain and call it a cone. For an interval and , write , , and . By we denote , which is . From now on, we always assume and write instead of .
Let be a domain on with smooth boundary. Suppose that is the least positive eigenvalue for on and the normalized positive eigenfunction corresponding to satisfies . Then (see [14, page 41]). In order to ensure the existence of and , we pose the assumption on : if , then is a -domain on surrounded by a finite number of mutually disjoint closed hypersurfaces (see e.g., [15, pages 88-89] for the definition of -domain).
Let be the class of the potential such that When , the subfunctions (superfunctions) associated with are continuous (see, e.g., ). In the rest of paper, we will always assume that .
An important role will be played by the solutions of the ordinary differential equation When the potential , these solutions are well known (see  for more references). Equation (1.10) has two specially linearly independent positive solutions and such that is increasing with and is decreasing with We remark that both and are harmonic on and vanish continuously on .
Remark 1.1. If and , then , and , where is the surface area of .
We recall that or for any and any satisfying or , where and are two positive constants (see Escassut et al. [11, Chapter 11], and for , see Azarin [6, Lemma 1], Essén, and Lewis [18, Lemma 2]).
The remainder of the paper is organized as follows: in Section 2 we will give our main theorems; in Section 3, some necessary lemmas are given; in Section 4, we will prove the main results.
2. Statement of the Main Results
In this section, we will state our main results. Before passing to our main results, we need some definitions.
Martin introduced the so-called Martin functions associated with the Laplace operator (see Brelot  or Martin ). Inspired by his spirit, we define the Martin function associated with the stationary Schrödinger operator as follows: which will be called the generalized Martin Kernel of (relative to ). If , the above quotient is interpreted as 0 (for , refer to Armitage and Gardiner ).
It is well known that the Martin boundary of is the set . When we denote the Martin kernel associated with the stationary Schrödinger operator by with respect to a reference point chosen suitably, we see for any , where is the origin of and a positive constant.
Let be a subset of and let be a nonnegative superfunction on . The reduced function of is defined by where . We define the regularized reduced function of relative to as follows: It is easy to see that is a superfunction on .
If and , then the Riesz decomposition and the generalized Martin representation allow us to express uniquely in the form , where and are the generalized Green potential and generalized Martin representation, respectively. We say that is -minimally thin at with respect to if . At last we remark that , where is the Martin boundary of .
Now we can state our main theorems.
Theorem 2.1. Let and a fixed point . The following are equivalent:(a)is -minimally thin at ;(b);(c).
If is a positive superfunction, then we will write for the measure appearing in the generalized Martin representation of the greatest a-harmonic minorant of .
Theorem 2.2. Let and a fixed point . Suppose that is a generalized Martin topology limit of . The following are equivalent:(a) is -minimally thin at ;(b)there exists a positive superfunction such that (c)there is an -potential on such that
A set in is said to be -thin at a point if there is a fine neighborhood of which does not intersect . Otherwise is said to be not -thin at . A set in is called -polar if there is a superfunction on some open set such that .
Let be a bounded subset of . Then is bounded on , and hence the greatest a-harmonic minorant of is zero. By the Riesz decomposition theorem there exists a unique positive measure associated with the stationary Schrödinger operator on such that for any , and is concentrated on , where For , see Brelot  and Doob . According to the Fatou's lemma, we easily know the condition (b) in Theorems 2.3 and 2.4.
Theorem 2.3. Let and a fixed point . Suppose that is a generalized Martin topology limit point of . The following are equivalent:(a) is -minimally thin at ;(b)there is an -potential such that (c)there is an -potential such that and
Theorem 2.4. Let , and a fixed point . Suppose that is a generalized Martin topology limit point of . Then is -minimally thin at if and only if there exists a positive superfunction such that
The generalized Green energy of is defined by Let be a subset of and , where . The previous theorems are concerned with the fixed boundary points. Next we will consider the case at infinity.
Theorem 2.5. A subset of is -minimally thin at with respect to if and only if
A subset of is -rarefied at with respect to , if there exists a positive superfunction in such that where
Theorem 2.6. A subset of is -rarefied at with respect to if and only if
Remark 2.7. When , Theorems 2.5 and 2.6 reduce to the results by Miyamoto and Yoshida . When and , these are exactly due to Aikawa and Essén .
Set for a positive superfunction on . We immediately know that . Actually let be a subfunction on satisfying for any and Then we see (for , see Yoshida ). If we apply this to , we may obtain .
Theorem 2.8. Let be a positive superfunction on . Then there exists an -rarefied set at with respect to such that uniformly converges to on as , where .
From the definition of -rarefied set, for any given -rarefied set at with respect to there exists a positive superfunction on such that on and . Hence does not converge to on as .
A cone is called a subcone of if , where is the closure of .
Theorem 2.10. Let be a subset of . If is an -rarefied set at with respect to , then is -minimally thin at with respect to . If is contained in a subcone of and is -minimally thin at with respect to , then is an -rarefied set at with respect to .
3. Some Lemmas
In our arguments we need the following results.
Lemma 3.1. Let and .(i)If and is -minimally thin at , then is -minimally thin at .(ii)If are -minimally thin at , then is -minimally thin at .(iii)If is -minimally thin at , then there is an open subset of such that and is -minimally thin at .
Proof. Since , we see (i) holds. To prove (ii) we note that is an -potential for each and so is an -potential. Finally, to prove (iii), let . Then is an -potential and on for some -polar set . Let be a nonzero -potential such that on , and let Then is open, and , so is an -potential and is -minimally thin at .
Lemma 3.2 (see ). Consider for any and any satisfying . In addition, for any and any .
Lemma 3.3 (see ). Let be a positive measure on such that there is a sequence of points , satisfying Then for a positive number ,
Lemma 3.4 (see ). Let be a positive measure on such that there is a sequence of points , satisfying Then for a positive number ,
Lemma 3.5. Let be a positive measure on for which is defined. Then for any positive number the set is -minimally thin at with respect to .
Lemma 3.6. Let be a positive superfunction on and put Then there are a unique positive measure on and a unique positive measure on such that where denotes the differentiation at along the inward normal into .
Proof. By the Riesz decomposition theorem, we have a unique measure on such that
where is the greatest a-harmonic minorant of on . Furthermore, by the generalized Martin representation theorem (Lemma 3.8) we have another positive measure on satisfying
We know from (3.11) that and .
Since where is a fixed reference point of the generalized Martin kernel, we also obtain by taking Hence by (3.13) and (3.16) we get the required.
Lemma 3.7. Let be a bounded subset of , and let be a positive superfunction on such that is represented as with two positive measures and on and , respectively, and satisfies for any . Then When , the equality holds in (3.19).
Proof. Since is concentrated on and for any , we see that
In addition, we have
for any , where and is the inward normal unit vector at , and
for any . Thus (3.19) follows from (3.20), (3.21), and (3.24). Because is bounded on , has the expression (3.18) by Lemma 3.6 when . Then the equalities in (3.20) hold because for any (Doob [21, page 169]). Hence we claim if
then the equality in (3.19) holds.
To see (3.25) we remark that To prove (3.26) we set Then is an -polar set, and hence for any . Consequently, for any , is not also -minimally thin at , and so for any positive measure on , where Take in (3.30). Since we obtain from (3.15) for any . Since for any , we have for any , which shows Let be the greatest a-harmonic minorant of , and let be the generalized Martin representing measure of . We claim if on , then . Since from (3.15), we also have , which gives (3.26) from (3.36).
To prove (3.37), we set . Then and hence from which (3.37) follows.
Lemma 3.8 (the generalized Martin representation). If is a positive a-harmonic function on , then there exists a measure on , uniquely determined by , such that and where is the same as the previous statement.
4. Proofs of the Main Theorems
Proof of Theorem 2.1. First we assume that (b) holds, and let . Since is minimal, the Riesz decomposition of is of the form , where is an -potential associated with the stationary Schrödinger operator on and . Since quasieverywhere on and quasieverywhere on ,
Hence , so by the hypothesis and (a) holds.
Next we assume (a) holds, and let be a decreasing sequence of compact neighborhoods of in the Martin topology such that . Then is a-harmonic on , and the decreasing sequence has a limit which is a-harmonic on . Since is majorized by , it follows that and (c) holds.
Finally we assume (c) holds, then there is a Martin topology neighborhood of such that . Since (b) implies (a), the set is -minimally thin at and so is an -potential. Then is an -potential and we yield (b).
Proof of Theorem 2.2. Obviously we see that (c) implies (b). If (b) holds, then there exist and a Martin topology neighborhood of such that on . If , then , and this yields contradictory conclusion that , where is the unit measure with support . Hence . Thus is -minimally thin at , and so (a) holds.
Finally we assume (a) holds. By Lemma 3.1 there is an open subset of such that and is -minimally thin at . By Theorem 2.1 there is a sequence of Martin topology open neighborhoods of such that . The function , being a sum of a-potentials, is an -potential since . Further, since on the open set , and so (c) holds.
Proof of Theorem 2.3. Clearly (c) implies (b). To prove that (b) implies (a), we suppose that (b) holds and choose A such that
Then on for some Martin topology neighborhood of . If denotes the swept measure of onto , where is the unit measure with support , then it follows that
on . Let be a sequence of compact subsets of such that , and let denote the -potential . Then
Letting , we see from our choice of that
then is -minimally thin at by Theorem 2.1, and so (a) holds.
Next we suppose that (a) holds. By Lemma 3.1 there is an open subset of such that and is -minimally thin at . By Theorem 2.1 there is a sequence of Martin topology open neighborhoods of such that Let , where is swept measure of onto . Then and (2.10) holds since on the open set , so (c) holds.
Proof of Theorem 2.4. Since (2.11) is independent of the choice of , we may multiply across by . Thus we may assume that and claim that
for any -potential . According to Fatou's lemma, we may yield
Since is not -minimally thin at , we know that
from Theorem 2.3. Hence the claim holds.
When is -minimally thin at , we see from (4.10) and the condition (b) of Theorem 2.3 that (2.11) holds for some -potential . Conversely, if (2.11) holds, then we can choose such that and define by . Then by (4.10) and it follows from Theorem 2.3 that is -minimally thin at .
Proof of Theorem 2.5. By applying the Riesz decomposition theorem to the superfunction on , we have a positive measure on satisfying for any and a nonnegative greatest a-harmonic minorant of such that We remark that is a minimal function at . If is -minimally thin at with respect to , then is an -potential, and hence on . Since for any , we see from (4.16) that for any . Take a sufficiently large from Lemma 3.3 such that Then from (1.16) or (1.17), for any