Solving Integral Representations Problems for the Stationary Schrödinger Equation

Abstract

When solutions of the stationary Schrödinger equation in a half-space belong to the weighted Lebesgue classes, we give integral representations of them, which imply known representation theorems of classical harmonic functions in a half-space.

1. Introduction and Results

Let 𝐑 and 𝐑₊ be the sets of all real numbers and of all positive real numbers, respectively. Let 𝐑^𝑛(𝑛≥2) denote the 𝑛-dimensional Euclidean space with points 𝑥=(𝑥^′,𝑥_𝑛), where 𝑥^′=(𝑥₁,𝑥₂,…,𝑥_𝑛−1)∈𝐑^𝑛−1 and 𝑥_𝑛∈𝐑. The unit sphere and the upper half unit sphere in 𝐑^𝑛 are denoted by 𝐒^𝑛−1 and 𝐒₊^𝑛−1, respectively. The boundary and closure of an open set 𝐷 of 𝐑^𝑛 are denoted by 𝜕𝐷 and 𝐷, respectively. The upper half space is the set 𝐻={(𝑥^′,𝑥_𝑛)∈𝐑^𝑛;𝑥_𝑛>0}, whose boundary is 𝜕𝐻.

We identify 𝐑^𝑛 with 𝐑^𝑛−1×𝐑 and 𝐑^𝑛−1 with 𝐑^𝑛−1×{0}, writing typical points 𝑥,𝑦∈𝐑^𝑛 as 𝑥=(𝑥^′,𝑥_𝑛),𝑦=(𝑦^′,𝑦_𝑛), where 𝑦^′=(𝑦₁,𝑦₂,…,𝑦_𝑛−1)∈𝐑^𝑛−1 and putting 𝑥⋅𝑦=^𝑛_𝑗=1𝑥_𝑗𝑦_𝑗,|𝑥|=√𝑥𝑥⋅𝑥,Θ=|𝑥|𝑦,Φ=𝑦. (1)

For 𝑥∈𝐑^𝑛 and 𝑟>0, let 𝐵(𝑥,𝑟) denote the open ball with center at 𝑥 and radius 𝑟(>0) in 𝐑^𝑛. 𝑆_𝑟=𝜕𝐵(𝑂,𝑟), where 𝑂 is the origin of 𝐑^𝑛. For a set 𝐸, 𝐸⊂𝐑₊∪{0}, we denote {𝑥∈𝐻;|𝑥|∈𝐸} and {𝑥∈𝜕𝐻;|𝑥|∈𝐸} by 𝐻𝐸 and 𝜕𝐻𝐸, respectively. By 𝐻𝑟 we denote 𝐻∩𝑆𝑟. We denote by 𝑑𝑆_𝑟 the (𝑛−1)-dimensional volume elements induced by the Euclidean metric on 𝑆_𝑟.

Let 𝒜_𝑏 denote the class of nonnegative radial potentials 𝑏(𝑥), that is, 0≤𝑏(𝑥)=𝑏(|𝑥|), 𝑥∈𝐻, such that 𝑏∈𝐿^𝑎_loc(𝐻) with some 𝑎>𝑛/2 if 𝑛≥4 and with 𝑎=2 if 𝑛=2 or 𝑛=3.

This paper is devoted to the stationary Schrödinger equation: 𝑆𝑆𝐸_𝑏ℎ(𝑥)=−Δℎ(𝑥)+𝑏(𝑥)ℎ(𝑥)=0, (2) where 𝑥∈𝐻, Δ is the Laplace operator and 𝑏∈𝒜_𝑏. Note that solutions of (2) are (classical) harmonic functions in the case 𝑏=0. Under these assumptions the operator 𝑆𝑆𝐸_𝑏 can be extended in the usual way from the space 𝐶^∞₀(𝐻) to an essentially self-adjoint operator on 𝐿²(𝐻) (see [1]). We will denote it by 𝑆𝑆𝐸_𝑏 as well. This last one has a Green function 𝐺_𝑏(𝑥,𝑦). Here 𝐺_𝑏(𝑥,𝑦) is positive on 𝐻 and its inner normal derivative 𝜕𝐺_𝑏(𝑥,𝑦)/𝜕𝑛(𝑦^′)≥0. We denote this derivative by 𝑃_𝑏(𝑥,𝑦^′), which is called the Poisson 𝑏-kernel with respect to 𝐻. If 𝐺(𝑥,𝑦) and 𝑃(𝑥,𝑦^′) are denoted by the Green function and Poisson kernel of the Laplace operator in 𝐻, respectively, then 𝑃𝑥,𝑦^′=−𝜕𝐺(𝑥,𝑦)𝜕𝑦_{𝑛𝑦𝑛=0}=2𝑥_𝑛𝑤_𝑛1𝑥−𝑦^′𝑛, (3) where 𝑥=(𝑥^′,𝑥_𝑛), 𝑦=(𝑦^′,𝑦_𝑛) and 𝑤_𝑛 is the area of the unit sphere in 𝐑^𝑛.

Let 𝑔 be a continuous function on 𝜕𝐻. We say that ℎ is a solution of the Dirichlet problem for the stationary Schrödinger operator 𝑆𝑆𝐸_𝑏 on 𝐻 with 𝑔, if ℎ∈𝐶²(𝐻)∩𝐶⁰𝐻,𝑆𝑆𝐸_𝑏ℎ=0in𝐻,ℎ=𝑔on𝜕𝐻. (4) Note that ℎ is a solution of the classical Dirichlet problem for the Laplace operator Δ on 𝐻 with 𝑔 in the case 𝑏=0.

Let Λ be a Laplace-Beltrami operator (spherical part of the Laplace) on the unit sphere. It is known (see, e.g., [2, page 41]) that the eigenvalue problem Λ𝜑(Θ)+𝜏𝜑(Θ)=0Θ∈𝐒₊^𝑛−1,𝜑(Θ)=0Θ∈𝜕𝐒₊^𝑛−1 (5) has the eigenvalues 𝜏_𝑗=𝑗(𝑗+𝑛−2), where 𝑗=0,1,2…. Corresponding eigenfunctions are denoted by 𝜑_𝑗𝑣(1≤𝑣≤𝑣_𝑗), where 𝑣_𝑗 is the multiplicity of 𝜏_𝑗. We norm the eigenfunctions in 𝐿²(𝐒₊^𝑛−1) and 𝜑₁=𝜑₁₁>0.

Let 𝑃_𝑗(𝑟) and 𝑄_𝑗(𝑟) stand, respectively, for the increasing and nonincreasing, as 𝑟→+∞, solutions of the equation: −𝑇^′′(𝑟)−𝑛−1𝑟𝑇^′(𝑟)+𝜏_𝑗𝑟²+𝑏(𝑟)𝑇(𝑟)=0,0<𝑟<∞, (6) normalized under the condition 𝑃_𝑗(1)=𝑄_𝑗(1)=1.

We shall also consider the class ℬ_𝑏, consisting of the potentials 𝑏∈𝒜_𝑏 such that there exists a finite limit lim_𝑟→∞𝑟²𝑏(𝑟)=𝑠∈[0,∞); moreover, 𝑟⁻¹|𝑟²𝑏(𝑟)−𝑠|∈𝐿(1,∞). If 𝑏∈ℬ_𝑏, then solutions of (2) are continuous (see [3]).

In the rest of paper, we assume that 𝑏∈ℬ_𝑏 and we shall suppress this assumption for simplicity. Further, we use the standard notations 𝑢⁺=max{𝑢,0}, 𝑢⁻=−min{𝑢,0}, and [𝑑] is the integer part of 𝑑 and 𝑑=[𝑑]+{𝑑}, where 𝑑 is a positive real number.

Denote 𝜆^±_𝑗,𝑠=2−𝑛±(𝑛−2)²+4𝑠+𝜏_𝑗2𝑗=0,1,2,3,…. (7)

Remark 1. 𝜆⁺_𝑗,0=𝑗 in the case 𝑏=0, where 𝑗=0,1,2,3,….

It is known (see [4]) that in the case under consideration the solutions to (6) have the asymptotics 𝑃_𝑗(𝑟)∼𝑑₁𝑟^{𝜆+𝑗,𝑠},𝑄_𝑗(𝑟)∼𝑑₂𝑟^{𝜆−𝑗,𝑠},as𝑟⟶∞, (8) where 𝑑₁ and 𝑑₂ are some positive constants.

If 𝑏∈𝒜_𝑏, it is known that the following expansion for the Green function 𝐺_𝑏(𝑥,𝑦) (see [5, Chapter 11], [6]): 𝐺_𝑏𝑥,𝑦=^∞_𝑗=01𝜒^′(1)𝑃_𝑗min|𝑥|,𝑦×𝑄_𝑗max|𝑥|,𝑦^𝑣𝑗_𝑣=1𝜑_𝑗𝑣(Θ)𝜑_𝑗𝑣(Φ), (9) where |𝑥|≠|𝑦| and 𝜒^′(1)=𝑤(𝑄₁(𝑟),𝑃₁(𝑟))|_𝑟=1 is its Wronskian. The series converges uniformly if either |𝑥|≤𝑘|𝑦| or |𝑦|≤𝑘|𝑥|, where 0<𝑘<1.

For a nonnegative integer 𝑚 and two points 𝑥,𝑦∈𝐻, we define a modified Green function: 𝐺(𝑏,𝑚)𝑥,𝑦=𝐺_𝑏𝑥,𝑦−𝑉(𝑏,𝑚)𝑥,𝑦, (10) where 𝑉()=𝑦𝑏,𝑚𝑥,𝑦0if<1,_𝑚𝑗∑₌₀1𝜒^′(1)𝑃_𝑗(|𝑥|)𝑄_𝑗𝑦×∑^𝑣𝑗_𝑣=1𝜑_𝑗𝑣(Θ)𝜑_𝑗𝑣(Φ)if1≤𝑦<∞. (11)

If the generalized Poisson kernel 𝑃(𝑏,𝑚)(𝑥,𝑦^′) with respect to 𝐻 is defined by 𝑃(𝑏,𝑚)𝑥,𝑦^′=𝜕𝐺(𝑏,𝑚)𝑥,𝑦𝜕𝑛𝑦^′, (12) then we have 𝑃(𝑏,0)(𝑥,𝑦^′)=𝑃_𝑏(𝑥,𝑦^′) and 𝑃(0,𝑚)(𝑥,𝑦^′) coincides with ones in Finkelstein and Scheinberg [7], Siegel and Talvila [8], Deng [9], Qiao and Deng [10], and Qiao [11] (see [5, Chapter 11]).

Put 𝑈(𝑏,𝑚;𝑢)(𝑥)=_𝜕𝐻𝑃(𝑏,𝑚)𝑥,𝑦^′𝑢𝑦^′𝑑𝑦^′, (13) where 𝑢(𝑦^′) is a continuous function on 𝜕𝐻.

For real numbers 𝛽≥1, we denote 𝒞(𝛽,𝑏) the class of all measurable functions 𝑓(𝑦) satisfying the following inequality: _𝐻𝑓𝑦𝜑₁1+𝑃_[𝛽]𝑦𝑦^𝑛+{𝛽}𝑑𝑦<∞, (14) and the class 𝒟(𝛽,𝑏) consists of all measurable functions 𝑔(𝑦^′)(𝑦^′∈𝜕𝐻) satisfying _𝜕𝐻𝑔𝑦^′𝑃₁𝑦^′𝑄₁𝑦^′1+𝜒^′𝑦^′𝑃_[𝛽]𝑦^′|𝑦^′|^{𝑛+{𝛽}−1}𝜕𝜑₁𝜕𝑛𝑑𝑦^′<∞. (15)

We will also consider the class of all continuous functions ℎ(𝑦), which is the solution of (2), with ℎ⁺(𝑦)∈𝒞(𝛽,𝑏) and ℎ⁺(𝑦^′)∈𝒟(𝛽,𝑏), is denoted by ℰ(𝛽,𝑏).

Remark 2. If 𝑏=0 and 𝛽=𝛼+1, then (14) and (15) are equivalent to _𝐻𝑦_𝑛𝑓𝑦1+𝑦^𝑛+𝛼+2𝑑𝑦<∞,_𝜕𝐻𝑔𝑦^′1+𝑦^′𝑛+𝛼𝑑𝑦^′<∞, (16) respectively, from Remark 1 and (14), which yield that ℰ(𝛼+1,0) is equivalent to (𝐶𝐻)_𝛼 in the notation of [9].
Let us recall the classical case 𝑏=0. If ℎ(𝑥)≥0 is harmonic in 𝐻, continuous on 𝐻, and ℎ∈ℰ(1,0), then there exists a constant 𝑑₃≥0 such that (see [12]) ℎ(𝑥)=𝑑₃𝑥_𝑛+_𝜕𝐻𝑃𝑥,𝑦^′𝑢𝑦^′𝑑𝑦^′, (17) where 𝑥=(𝑥^′,𝑥_𝑛)∈𝐻.

Deng (see [9]) has constructed a similar representation to (17) for ℎ∈ℰ(𝛼+1,0)(𝛼≥0), which is the integral with a modified Poisson kernel derived by subtracting some special harmonic polynomials from 𝑃(𝑥,𝑦^′). By virtue of this modified Poisson kernel, Qiao (see [11, 13]) and Qiao and Deng (see [14–18]) have constructed different integral representations for harmonic functions of finite order and infinite order.

Especially, Su (see [6, 19]) recently writes solutions to the half-space Dirichlet problem with respect to the stationary Schrödinger operator 𝑆𝑆𝐸_𝑏. Now we state our main results as follows.

Theorem 3. If ℎ∈ℰ(𝛽,𝑏), then ℎ∈𝒟(𝛽,𝑏).

Theorem 4. If ℎ∈ℰ(𝛽,𝑏), 𝑚 is an integer such that 𝑃_𝑚(|𝑦|)<𝑃_[𝛽](|𝑦|)|𝑦|^{𝛽}≤𝑃_𝑚+1(|𝑦|)(|𝑦|≥1), and then the following properties hold.(I) If 𝛽=1, then there exists a constant 𝑑₄ such that ℎ(𝑥)=𝑑₄𝑃₁(|𝑥|)𝜑₁(Θ)+𝑈(𝑏,0;ℎ)(𝑥) (18) for 𝑥∈𝐻.(II) If 𝛽>1, then we have ℎ(𝑥)=𝑈(𝑏,𝑚;ℎ)(𝑥)+𝑢(𝑥), where 𝑢(𝑥)=^𝑚_𝑗=0^𝑣𝑗_𝑣=1𝑑_𝑗𝑣𝜑_𝑗𝑣(Θ)𝑃_𝑗(|𝑥|) (19) vanishing continuously on 𝜕𝐻, 𝑥=(𝑥^′,𝑥_𝑛)∈𝐻 and 𝑑_𝑗𝑣 are constants.

2. Lemmas

The following Lemma plays an important role in our discussions, which is due to Levin and Kheyfits (see [5, page 356]).

Lemma 5. If 𝑅>1 and ℎ(𝑦) is a solution of (2) on a domain containing 𝐻(1,𝑅), then _𝐻𝑅𝜒^′(𝑅)𝑃₁(𝑅)ℎ𝜑₁(Φ)𝑑𝑆_𝑅+_{𝜕𝐻(1,𝑅)}ℎ𝑦^′𝜕𝜑₁𝑊𝑦𝜕𝑛^′𝑑𝑦^′+𝑑₅+𝑑₆𝑄₁(𝑅)𝑃₁(𝑅)=0, (20) where 𝑊𝑦^′=𝑄₁𝑦^′−𝑄₁(𝑅)𝑃₁(𝑅)𝑃₁𝑦^′,𝑑₅=_𝐻1ℎ𝜑₁(Φ)𝑄^′₁(𝑟)−𝑄₁(1)𝜑₁(Φ)𝜕ℎ𝜕𝑛𝑑𝑆₁,𝑑₆=_𝐻1𝑃₁(1)𝜑₁(Φ)𝜕ℎ𝜕𝑛−ℎ𝜑₁(Φ)𝑃^′₁(1)𝑑𝑆₁. (21)

Lemma 6 (see [6, Corollary 1.6]). If 𝑢 is a continuous function on 𝜕𝐻 satisfying _𝜕𝐻𝑢𝑦^′1+𝑦^{′𝜆+𝑚+1,𝑠+𝑛−1}𝑑𝑦^′<∞, (22) then 𝑈(𝑏,𝑚;𝑢)(𝑥) is a solution of the Dirichlet problem for 𝑆𝑆𝐸_𝑏 on 𝐻 with ℎ and lim_{|𝑥|→∞,𝑥∈𝐻}|𝑥|^{−𝜆+𝑚+1,𝑠}𝑈(𝑏,𝑚;𝑢)(𝑥)=0. (23)

Lemma 7 (see [6, Lemma 2.1] or [20, Theorem 1]). If 𝑢(𝑥) is a solution of (2) on 𝐻 satisfying lim_{|𝑥|→∞,𝑥∈𝐻}|𝑥|^{−𝜆+𝑚+1,𝑠}𝑢⁺(𝑥)=0, (24) then (19) holds.

3. Proof of Theorem 3

We apply the formula (20) to ℎ=ℎ⁺−ℎ⁻ in 𝐻(1,𝑅): 𝑚₊(𝑅)+_{𝜕𝐻(1,𝑅)}ℎ⁺𝑦^′𝑊𝑦^′𝜕𝜑₁𝜕𝑛𝑑𝑦^′+𝑑₅+𝑄₁(𝑅)𝑃₁(𝑅)𝑑₆=𝑚₋(𝑅)+_{𝜕𝐻(1,𝑅)}ℎ⁻𝑦^′𝑊𝑦^′𝜕𝜑₁𝜕𝑛𝑑𝑦^′, (25) where𝑚_±(𝑅)=_𝐻𝑅𝜒^′(𝑅)𝑃₁(𝑅)ℎ^±𝜑₁𝑑𝑆_𝑅. (26)

Since 𝑢∈ℰ(𝛽,𝑏), we obtain by (8) ^∞₁𝑚₊(𝑅)𝑃₁(𝑅)𝜒^′(𝑅)𝑃_[𝛽](𝑅)𝑅^𝑛+{𝛽}=𝑑𝑅_𝐻(1,∞)ℎ⁺𝜑₁𝑃_[𝛽]𝑦𝑦^𝑛+{𝛽}𝑑𝑦≤2_𝐻ℎ⁺𝜑₁1+𝑃_[𝛽]𝑦𝑦^𝑛+{𝛽}𝑑𝑦<∞. (27)

From (15), we conclude that ^∞₁𝑃₁(𝑅)𝜒^′(𝑅)𝑃_[𝛽](𝑅)𝑅^𝑛+{𝛽}×_{𝜕𝐻(1,𝑅)}ℎ⁺𝑦^′𝑊𝑦^′𝜕𝜑₁𝜕𝑛𝑑𝑦^′=𝑑𝑅_{𝜕𝐻(1,∞)}ℎ⁺𝑦^′𝑃₁𝑦^′×^∞_𝑦′𝑃₁(𝑅)𝜒^′(𝑅)𝑃_[𝛽](𝑅)𝑅^𝑛+{𝛽}×𝑄₁𝑦^′𝑃₁𝑦^′−𝑊₁(𝑅)𝑃₁(𝑅)𝑑𝑅𝜕𝜑₁𝜕𝑛𝑑𝑦^′≤𝑀_{𝜕𝐻(1,∞)}ℎ⁺𝑦^′𝑃₁𝑦^′𝑄₁𝑦^′𝜒^′𝑦^′𝑃_[𝛽]𝑦^′𝑦^{′𝑛+{𝛽}−1}𝜕𝜑₁𝜕𝑛𝑑𝑦^′≤𝑀_𝜕𝐻ℎ⁺𝑦^′𝑃₁𝑦^′𝑄₁𝑦^′1+𝜒^′𝑦^′𝑃_[𝛽]𝑦^′𝑦^{′𝑛+{𝛽}−1}𝜕𝜑₁𝜕𝑛𝑑𝑦^′<∞. (28)

Combining (25), (27), and (28), we obtain ^∞₁𝑃₁(𝑅)𝜒^′(𝑅)𝑃_[𝛽](𝑅)𝑅^{𝑛+({𝛽}/2)}×_{𝜕𝐻(1,𝑅)}ℎ⁻𝑦^′𝑊𝑦^′𝜕𝜑₁𝜕𝑛𝑑𝑦^′≤𝑑𝑅_∞1𝑚₊(𝑅)𝑃₁(𝑅)𝜒^′(𝑅)𝑃_[𝛽](𝑅)𝑅^{𝑛+({𝛽}/2)}+𝑑𝑅_∞1𝑃₁(𝑅)𝜒^′(𝑅)𝑃_[𝛽](𝑅)𝑅^{𝑛+({𝛽}/2)}×_{𝜕𝐻(1,𝑅)}ℎ⁺𝑦^′𝑊𝑦^′𝜕𝜑₁𝜕𝑛𝑑𝑦^′+𝑑𝑅_∞11𝜒^′(𝑅)𝑃_[𝛽](𝑅)𝑅^{𝑛+({𝛽}/2)}×𝑃₁(𝑅)𝑑₅+𝑄₁(𝑅)𝑑₆𝑑𝑅<∞. (29)

Set ℋ𝛽=lim_{𝑦′→∞}𝜒^′𝑦^′𝑃_[𝛽]𝑦^′𝑦^{′𝑛+{𝛽}−1}𝑄₁𝑦^′×^∞_𝑦′𝑃₁(𝑅)𝜒^′(𝑅)𝑃_[𝛽](𝑅)𝑅^{𝑛+({𝛽}/2)}×𝑄₁𝑦^′𝑃₁𝑦^′−𝑊₁(𝑅)𝑃₁(𝑅)𝑑𝑅=lim_{𝑦′→∞}𝑦^{′𝜆+[𝛽],𝑠+𝜆+1,𝑠+𝑛+{𝛽}−2}×^∞_𝑦′1𝑅^{𝜆+[𝛽],𝑠−𝜆+1,𝑠+({𝛽}/2)+1}1𝑦^{′𝜒1,𝑠}−1𝑅^𝜒1,𝑠𝑑𝑅, (30) where 𝜒_1,𝑠=𝜆⁺_1,𝑠−𝜆⁻_1,𝑠.

By the L’hospital’s rule, we have ℋ𝛽=𝜒_1,𝑠𝜆⁺_[𝛽],𝑠−𝜆⁺_1,𝑠𝜆⁺_[𝛽],𝑠+𝜆⁺_1,𝑠𝛽𝛽+𝑛−2if=0,+∞if≠0, (31) which yields that there exists a positive constant 𝑀 such that, for any |𝑦^′|≥1, ^∞_𝑦′𝑃₁(𝑅)𝜒^′(𝑅)𝑃_[𝛽](𝑅)𝑅^{𝑛+({𝛽}/2)}𝑊𝑦^′≥𝑑𝑅𝑀𝑃₁𝑦^′𝑄₁𝑦^′𝜒^′𝑦^′𝑃_[𝛽]𝑦^′𝑦^{′𝑛+{𝛽}−1}. (32)

Then 𝑀_{𝜕𝐻(1,∞)}ℎ⁻𝑦^′𝑃₁𝑦^′𝑄₁𝑦^′𝜒^′𝑦^′𝑃_[𝛽]𝑦^′𝑦^{′𝑛+{𝛽}−1}𝜕𝜑₁𝜕𝑛𝑑𝑦^′≤_{𝜕𝐻(1,∞)}ℎ^−∞_𝑦′𝑃₁(𝑅)𝜒^′(𝑅)𝑃_[𝛽](𝑅)𝑅^{𝑛+({𝛽}/2)}×𝑊𝑦^′𝑑𝑅𝜕𝜑₁𝜕𝑛𝑑𝑦^′<∞, (33) which shows that ℎ∈𝒟(𝛽,𝑏) from |ℎ|=ℎ⁺+ℎ⁻. Then Theorem 3 is proved.

4. Proof of Theorem 4

To prove (II), notice that 𝑃_𝑚(|𝑦|)<𝑃_[𝛽](|𝑦|)|𝑦|^{𝛽}≤𝑃_𝑚+1(|𝑦|)(|𝑦|≥1) and condition (15) is stronger than condition (22) from Theorem 3.

Consider the function ℎ(𝑥)−𝑈(𝑎,𝑚;ℎ)(𝑥). Then it follows from Lemma 6 and Theorem 3 that this is a solution of (2) in 𝐻 and vanishes continuously on 𝜕𝐻.

Then0≤(ℎ(𝑥)−𝑈(𝑏,𝑚;ℎ)(𝑥))⁺≤ℎ⁺(𝑥)+(𝑈(𝑏,𝑚;ℎ))⁻(𝑥) (34) for any 𝑃∈𝐻. Further, (8) gives that lim_{|𝑥|→∞,𝑥∈𝐻}|𝑥|^{−𝜆+𝑚+1,𝑠}ℎ⁺(𝑥)=0, (35) which together with (34) and Lemmas 6 and 7 give the result of (II).

If 𝑢∈ℰ(1,𝑏), then ℎ∈ℰ(𝛽,𝑏) for each 𝛽>1 and there exists a constant 𝑑₇ such that ℎ(𝑥)=𝑑₇𝑃₁(|𝑥|)𝜑₁(Θ)+𝑈(𝑏,1;ℎ)(𝑥) (36) for all 𝑥∈𝐻. So if we take 𝑑₄=𝑑₇−_{𝜕𝐻[1,∞)}𝑃(𝑏,1)0,𝑦^′ℎ𝑦^′𝑑𝑦^′, (37) we see that (18) holds for all 𝑥∈𝐻.

Thus we complete the proof of Theorem 4.

Acknowledgments

The author wishes to expresseses his appreciation to Professor Jingben Yin for very useful conversations related to this problem. He is grateful to the referee for her or his careful reading and helpful suggestions which led to the improvement of the original paper. This work is supported by the National Natural Science Foundation of China under Grant 11226093.