Matsaev's type theorems for solutions of the stationary Schrödinger equation and its applications

Our aim in this paper is to give lower 
estimates for solutions of the stationary Schrodinger equation 
in a cone, which generalize and supplement the result obtained by 
Matsaev's type theorems for harmonic functions in a half space. Meanwhile, some 
applications of this conclusion are also given.

1. Introduction. 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 (n ≥ 2) the n-dimensional Euclidean space. A point in R n is denoted by P = (X, x n ), where X = (x 1 , x 2 , . . . , x n−1 ). The Euclidean distance between two points P and Q in R n is denoted by |P − Q|. Also |P − O| with the origin O of R n is simply denoted by |P |. The boundary and the closure of a set E in R n are denoted by ∂E and E, 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 where 0 ≤ r < +∞, − 1 2 π ≤ θ n−1 < 3 2 π, and if n ≥ 3, then 0 ≤ θ j ≤ π (1 ≤ j ≤ n − 2).
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 ; r ∈ Ξ, (1, Θ) ∈ Ω} in R n is simply denoted by Ξ×Ω. In particular, the half space R + ×S n−1 + = {(X, x n ) ∈ R n ; x n > 0} will be denoted by T n . We denote the sets I × Ω and I × ∂Ω with an interval on R 5710 LEI QIAO by C n (Ω; I) and S n (Ω; I). By S n (Ω; r) we denote C n (Ω) ∩ S r . By S n (Ω) we denote S n (Ω; (0, +∞)) which is ∂C n (Ω) − {O}. For P ∈ R n and r > 0, let B(P, r) denote the open ball with center at P and radius r in R n . S r = ∂B(O, r) We use the standard notations u + = max{u, 0} and u − = − min{u, 0}. Further, we denote by w n the surface area 2π n/2 {Γ(n/2)} −1 of S n−1 , by ∂/∂n Q denotes the differentiation at Q along the inward normal into C n (Ω), by dS r the (n − 1)dimensional volume elements induced by the Euclidean metric on S r . For positive functions h 1 and h 2 , we say that Let A a denote the class of nonnegative radial potentials a(P ), i.e. 0 ≤ a(P ) = a(r), P = (r, Θ) ∈ C n (Ω), such that a ∈ L b loc (C n (Ω)) with some b > n/2 if n ≥ 4 and with b = 2 if n = 2 or n = 3.
This article is devoted to the stationary Schrödinger equation where ∆ n is the Laplace operator and a ∈ A a . Note that solutions of equation (1) are the (classical) harmonic functions in a cone in the case a = 0. Under these assumptions the operator Sch a can be extended in the usual way from the space C ∞ 0 (C n (Ω)) to an essentially self-adjoint operator on L 2 (C n (Ω)) (see [8] ). We will denote it Sch a as well. This last one has a Green-Sch function G a Ω (P, Q). Here G a Ω (P, Q) is positive on C n (Ω) and its inner normal derivative ∂G a Ω (P, Q)/∂n Q ≥ 0, where ∂/∂n Q denotes the differentiation at Q along the inward normal into C n (Ω). We denote this derivative by PI a Ω (P, Q), which is called the Poisson-Sch kernel with respect to C n (Ω). We remark that G 0 Ω (P, Q) and PI 0 Ω (P, Q) are the Green function and Poisson kernel of the Laplacian in C n (Ω) respectively.
Let Ω be a domain on S n−1 with smooth boundary ∂Ω. Consider the Dirichlet problem (Λ n + λ)ϕ = 0 on Ω, where Λ n is the spherical part of the Laplace operata ∆ n . We denote the least positive eigenvlaue of this boundary value problem by λ and the normalized positive eigenfunction corresponding to λ by ϕ(Θ), Ω ϕ 2 (Θ)dS 1 = 1. In order to ensure the existence of λ and a smooth ϕ(Θ). We put a rather strong assumption on Ω: if n ≥ 3, 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. Then ϕ ∈ C 2 (Ω) and ∂ϕ/∂n > 0 on ∂Ω (here and below, ∂/∂n denotes differentiation along the interior normal).
Let V (r) and W (r) stand for solutions of the equation It is known (see, for example, [17]) that if the potential a ∈ A a , then equation (2) has a fundamental system of positive solutions {V, W } such that V is nondecreasing with (see [8,11,12,14]) and W is monotonically decreasing with +∞ = W (0+) > W (r) 0 as r → +∞.
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 solutions of the equation (1) are continuous (see [16]).
In the rest of paper, we assume that a ∈ B a and we shall suppress this assumption for simplicity. Further, We use H(a, Ω) to denote the class of solutions of equation (1), which are continuous in C n (Ω). We remark that H(0, Ω) denote the class of functions harmonic in C n (Ω) and continuous in C n (Ω). Denote then the solutions of equation (2) have the asymptotic (see [2,4]) Throughout this paper, let M denote various constants independent of the variables in questions, which may be different from line to line. Let ρ(t) be a function on a segment [1, +∞) and ρ(t) > ℵ + a . If a = 0 and Ω = S n−1 + , then ℵ + 0 = 1, The estimate we deal with has a long history which can be traced back to Matsaev's estimate of harmonic functions from below (see, for example, Matsaev [7, p. 209]).
where z = Re iθ ∈ T 2 and M is a constant independent of A 1 , R, θ and the function u(z).
Recently, Zhang, Kou and Deng (see [18]) consider Theorem 1.1 in the n−dimensional (n ≥ 2) case and obtain the following result. For related results in a cone, we refer the reader to the paper by Qiao and Pan (see [13]). and then where P ∈ T n and M is a constant independent of A 2 , R, θ 1 and the function u(P ).
Our aim is to prove the following result. For a related result with respect to Schrödinger operator, we refer the reader to the paper by Kheyfits (see [5]).
(II) For any P = (R, Θ) ∈ C n (Ω) and R ≤ 1, we have Then where P ∈ C n (Ω) and M is a constant independent of A 2 , R, ϕ(Θ) and the function u(P ).
). The conclusion of Theorem 1.3 remains valid if (7) in Theorem 1.3 is replaced by If, in addition, Ω = S n−1 + and ρ(R) ≡ ρ = const in Corollary 1, we immediately obtain Corollary 2. The conclusion of Theorem 1.2 remains valid if (4) and (5) in Theorem 1.2 are replaced by the following (I) and (II) respectively.
By taking a = 0 and ρ(R) ≡ ρ = const in Theorem 1.4, we obtain the following Corollary, which generalizes Theorem 1.2 to the conical case.

Remark 1.
On the one hand, by the proofs of Theorems 1.3 and 1.4, we know that (I) in Corollary 2 is weaker than (14) in Corollary 4, which is weaker than (4) in Theorem 1.2. On the other hand, (II) in Corollary 2 is weaker than (5) (7) and (8) and Sn(Ω;R) respectively.
Remark 2. From the proofs of Theorems 1.4 and 1.5, we can easily show that (7) and (8) in Theorem 1.3 are equivalent to (15) and (16) in Theorem 1.5.
The following result is due to Masaev (see [7, p. 212, Th. 3]). Theorem 1.6. Let u(z) (z = Re iα ) be a subharmonic function on T 2 , which satisfies the estimate where ρ > 1 and l ≥ 0. Then u(z) is of order ρ and finite type.
Masaev's inequalities like (17) are crucial in many problems, since they are intrinsically connected with the estimates of the Cauchy type integrals. Specifically, Theorem 1.6 has found important applications in operator theory, thus it may be of interest to extend these results onto more general classes of functions. Indeed, Govorov and Zhuravleva (see [3]) proved that Theorem 1.6 can be extended to analytic functions in a half plane, if in addition, they are continuous up to, and satisfy a similar upper bound at the boundary. For harmonic functions on T 2 , we refer readers to the following Theorem (see [7]), where the bounded condition is replaced by the weaker integral condition.
When Ω = S n−1 + and ρ(R) ≡ ρ = const, we have the following result, which generalize Theorem 1.8 to the higher dimensional half space. Then Remark 3. When n = 2, Corollary 6 reduces to Theorem 1.8. From (19) we know that u is at most of the growth order ρ and normal type in T n .
The following Carleman formula with respect to the Schrödinger operator plays an important role in our discussions, which is due to Levin and Kheyfits (see [8, p. 356]). Careleman formula for harmonic functions in a cone and its applications, we refer readers to the papers by Qiao (see [10]), Rashkovskii and Ronkin (see [15]). where 3. Proof of Theorem 1.3. By the Riesz Decomposition Theorem (see [15]), for any P = (r, Θ) ∈ C n (Ω; (0, R)) we have Now we distinguish three cases.
for a fixed R > 1. It immediately follows from (12) that and V (R) Notice that Hence from (35) from (8), (35) and (42). Hence, (43) implies (7) holds. The proof of it is similar to (41). So I omit it here. And then we obtain the result from Theorem 1.3.