Estimates of the Fundamental Solution for Higher Order Schrödinger Type Operators and Their Applications

Abstract

We consider higher order Schrödinger type operators with nonnegative potentials. We assume that the potential belongs to the reverse Hölder class which includes nonnegative polynomials. We establish estimates of the fundamental solution and show 𝐿^𝑝 boundedness of some Schrödinger type operators.

1. Introduction

Let 𝑉(𝑥) be a nonnegative potential and consider the Schrödinger type operators 𝐻_𝑘=(−Δ)^𝑘+𝑉^𝑘 on 𝐑^𝑛, where 𝑘 is a positive integer and 𝑛≥2𝑘+1. When 𝑉 is a nonnegative polynomial, Zhong proved estimates of the fundamental solution for 𝐻₁ and 𝐻₂ and showed some estimates for 𝐻₁ and 𝐻₂ (see [1]). More precisely, he showed the 𝐿^𝑝 boundedness of the operators 𝑉𝐻₁⁻¹, 𝑉^1/2∇𝐻₁⁻¹, and 𝑉^2−𝑗/2∇^𝑗𝐻₂⁻¹, where 𝑗=0,1,2,3,4. He also proved that the operators ∇²𝐻₁⁻¹ and ∇⁴𝐻₂⁻¹ are Calderón-Zygmund operators. Recently, in [2], the authors showed the 𝐿^𝑝 boundedness of the operators 𝑉^𝑘𝐻_𝑘⁻¹, ∇^2𝑘𝐻_𝑘⁻¹, and ∇^𝑘𝐻_𝑘^−1/2 for nonnegative polynomial potential  𝑉.

For the potential 𝑉 which belongs to the reverse Hölder class, which includes nonnegative polynomials, Shen generalized Zhong's results on 𝐻₁ (see [3]). Actually, he established estimates of the fundamental solution for 𝐻₁ and showed the 𝐿^𝑝 estimates for the operators 𝑉𝐻₁⁻¹, 𝑉^1/2∇𝐻₁⁻¹, ∇²𝐻₁⁻¹, and so on. For the operator 𝐻₁ with reverse Hölder class potentials, further results have been investigated by many researchers. See [4–7], for example. For the operator 𝐻₂ with reverse Hölder class potentials, in [8], the author established estimates of the fundamental solution for 𝐻₂ and showed the 𝐿^𝑝 boundedness of the operators 𝑉^2−𝑗/2∇^𝑗𝐻₂⁻¹, where 𝑗=0,1,2,3,4. In [8], the author also showed that the operator ∇⁴𝐻₂⁻¹ is a Calderón-Zygmund operator. Recently, in [9], the authors showed 𝐿^𝑝 boundedness of the operator ∇²𝐻₂^−1/2 and that the operator ∇²𝐻₂^−1/2 is a Calderón-Zygmund operator.

In this paper, we study 𝐻₂^𝑚=(−Δ)^2𝑚+𝑉^2𝑚 with reverse Hölder class potential 𝑉 on 𝐑^𝑛, where 𝑚 is a nonnegative integer and 𝑛≥2^𝑚+1+1. We establish estimates of the fundamental solution for 𝐻₂^𝑚 and show the 𝐿^𝑝 boundedness of the operators 𝑉^{2𝑚−𝑗/2}∇^𝑗𝐻₂^−1𝑚, where 𝑗 is an integer satisfying 0≤𝑗≤2^𝑚+1.

As mentioned above, in [2], the authors proved some results on 𝐻_𝑘=(−Δ)^𝑘+𝑉^𝑘, where 𝑉 is a nonnegative polynomial and 𝑘 is an integer, 𝑘≥3. They proved their results by making use of [2, Lemma 3.3 and Corollary 3.1] which have been proved only for nonnegative polynomial 𝑉. In this paper, our strategy is different from the one in [2], since the question whether the above two results can be proved for reverse Hölder class potentials 𝑉 is yet to be settled. The purpose of this paper is to show some results on 𝐻_𝑘=(−Δ)^𝑘+𝑉^𝑘 with potential 𝑉 which belongs to the reverse Hölder class, which includes nonnegative polynomials. However, our results are only for 𝑘=2^𝑚, where 𝑚 is an integer, 𝑚≥2.

We recall the definitions of the reverse Hölder class (e.g., [3]). We denote by 𝐵_𝑟(𝑥) the ball centered at 𝑥 with radius 𝑟.

Definition 1 (reverse Hölder class). Let 𝑉≥0.
(1) For 1<𝑝<∞ one says that 𝑉∈(𝑅𝐻)_𝑝 if 𝑉∈𝐿^𝑝_loc(𝐑^𝑛) and there exists a positive constant 𝐶 such that 1𝐵_𝑟(𝑥)_{𝐵𝑟(𝑥)}𝑉𝑦^𝑝𝑑𝑦^1/𝑝≤𝐶𝐵_𝑟(𝑥)_{𝐵𝑟(𝑥)}𝑉𝑦𝑑𝑦 (1) holds for every 𝑥∈𝐑^𝑛 and 0<𝑟<∞.
(2) One says that 𝑉∈(𝑅𝐻)_∞ if 𝑉∈𝐿^∞_loc(𝐑^𝑛) and there exists a positive constant 𝐶 such that ‖𝑉‖_𝐿^∞_{(𝐵𝑟(𝑥))}≤𝐶𝐵_𝑟(𝑥)_{𝐵𝑟(𝑥)}𝑉𝑦𝑑𝑦 (2) holds for every 𝑥∈𝐑^𝑛 and 0<𝑟<∞.

Remark 2. (1) If 𝑃(𝑥) is a polynomial and 𝛼>0, then 𝑉(𝑥)=|𝑃(𝑥)|^𝛼 belongs to (𝑅𝐻)_∞ (see [10, page 146]).
(2) For 1<𝑝<∞, it is easy to see (𝑅𝐻)_∞⊂(𝑅𝐻)_𝑝.

Definition 3 (see [3, Definition 1.3]). (1) Let 𝑉∈(𝑅𝐻)_𝑛/2 and 𝑉≢0. Then it is well known that there exists a positive 𝜖 such that 𝑉∈(𝑅𝐻)_𝑛/2+𝜖 (see [11, Lemma 2]). Then the function 𝜌(𝑥,𝑉) is well defined by 1𝜌()𝑟𝑥,𝑉=sup𝑟>0∶²𝐵_𝑟(𝑥)_{𝐵𝑟(𝑥)}𝑉𝑦𝑑𝑦≤1 (3) and satisfies 0<𝜌(𝑥,𝑉)<∞ for every 𝑥∈𝐑^𝑛.

One recalls that 𝑟²𝐵_𝑟(𝑥)_{𝐵𝑟(𝑥)}𝑉𝑦1𝑑𝑦=1for𝑟=𝜌(𝑥,𝑉) (4) (see [3, page 519]).

Remark 4. (1) If 𝑉∈(𝑅𝐻)_𝑝, 𝑝≥𝑛/2, then there exists a positive constant 𝐶 such that 1𝐵_𝑟(𝑥)_{𝐵𝑟(𝑥)}𝑉𝑦^𝑝𝑑𝑦^1/𝑝≤𝐶𝜌(𝑥,𝑉)² (5) (cf. [3, Lemma 1.8] and [5, Lemma 2.2 (a)]).
(2) If 𝑉∈(𝑅𝐻)_∞ then there exists a positive constant 𝐶 such that 𝑉(𝑥)≤𝐶𝜌(𝑥,𝑉)² (see [3, Remark 2.9]).
Let 𝛼=(𝛼₁,…,𝛼_𝑛) denote the multi-index with 𝛼_𝑖∈𝐍, 𝑖∈𝐍, and 1≤𝑖≤𝑛. Define 𝑥^𝛼=𝑥^𝛼1₁⋯𝑥^𝛼𝑛_𝑛 and ∇^𝑗=𝜕^𝛼=𝜕^|𝛼|/(𝜕𝑥^𝛼1₁⋯𝜕𝑥^𝛼𝑛_𝑛) for 𝑗=|𝛼|=𝛼₁+⋯+𝛼_𝑛. For any positive integer 𝑗 and a function 𝑢∈𝐶^𝑗(𝐑^𝑛), denote that ∇^𝑗𝑢(𝑥)=(𝜕^𝛼𝑢(𝑥)∶|𝛼|=𝑗) and |∇^𝑗𝑢(𝑥)|²=∑_|𝛼|=𝑗|𝜕^𝛼𝑢(𝑥)|². We denote by Γ_𝐻𝑘(𝑥,𝑦) the fundamental solution for 𝐻_𝑘. The operator 𝐻_𝑘⁻¹ is the integral operator with Γ_𝐻𝑘(𝑥,𝑦) as its kernel.

Now we state our theorems.

Theorem 5. Let 𝑗, 𝑚, and 𝑛 be integers, 𝑚≥0, 𝑛≥2^𝑚+1+1, and 0≤𝑗≤2^𝑚+1−1. Suppose that 𝑉∈(𝑅𝐻)_𝑞0 for some 𝑞₀≥2^𝑚𝑛/(2^𝑚+1−𝑗). Then there exist positive constants 𝐶_𝑗 such that 𝑉^{2𝑚−𝑗/2}∇^𝑗𝐻₂^−1𝑚𝑓_{𝐿𝑝(𝐑𝑛)}≤𝐶_𝑗𝑓_{𝐿𝑝(𝐑𝑛)}, (6) where 1≤𝑝≤2𝑞₀/(2^𝑚+1−𝑗).

In Theorem 5, the case 𝑚=0 was shown in [3, Theorems 2.7 and 4.13].

Remark 6. In Theorem 5, for the case 𝑚=𝑗=0, inequality (6) was shown in [3, Remark 2.9] under the assumption 𝑉∈(𝑅𝐻)_∞. If we take the limit 𝑞₀→∞, then class (𝑅𝐻)_𝑞0 becomes (𝑅𝐻)_∞ and 𝑉∈(𝑅𝐻)_∞ implies “𝑉∈(𝑅𝐻)_𝑛/2 and 𝑉(𝑥)≤𝐶𝜌(𝑥,𝑉)².” See [8, Remark 3] and also [8, Theorem 1].

Remark 7. We assume “𝑉∈(𝑅𝐻)_𝑛/2 and 𝑉(𝑥)≤𝐶𝜌(𝑥,𝑉)².” Then inequality (6) is true for 1≤𝑝≤∞. See Section 7 for the details. We also remark that the inequality (6) is true for 𝑗=2^𝑚+1 and 1<𝑝<∞ under the assumption “𝑉∈(𝑅𝐻)_𝑛/2 and 𝑉(𝑥)≤𝐶𝜌(𝑥,𝑉)².” See [3, Remark 2.9] and also [8, Theorem 1 (2)].

Theorem 8. Let 𝑗, 𝑚, and 𝑛 be integers, 𝑚≥0, 𝑛≥2^𝑚+1+1, and 1≤𝑗≤2^𝑚+1−1. Suppose that 𝑉∈(𝑅𝐻)_𝑞0 for some 𝑞₀, 𝑛/2≤𝑞₀<2^𝑚𝑛/(2^𝑚+1−𝑗). Then there exist positive constants 𝐶_𝑗 such that (6) holds, where 1≤𝑝≤𝑝₀ and 1/𝑝₀=(2^𝑚+2−𝑗)/(2𝑞₀)−(2^𝑚+1−𝑗)/𝑛.

To prove Theorems 5 and 8 we need the estimates of the fundamental solution (Theorems 9 and 10). The following Theorem 9 generalizes the results in [2, Theorem 3.1] to the operator 𝐻₂^𝑚 with potential 𝑉 which belongs to the reverse Hölder class.

Theorem 9. Let 𝑚 and 𝑛 be integers, 𝑚≥0, and 𝑛≥2^𝑚+1+1. Suppose that 𝑉∈(𝑅𝐻)_𝑛/2. Then for any positive integer 𝑁 there exists a positive constant 𝐶_𝑁 such that (0≤)Γ_𝐻2𝑚≤𝐶𝑥,𝑦^𝑁1+𝜌(𝑥,𝑉)𝑥−𝑦^𝑁⋅1𝑥−𝑦^{𝑛−2𝑚+1}. (7)

In Theorem 9, the cases 𝑚=0 and 𝑚=1 were shown in [3, Theorem 2.7] and [8, Theorem 2], respectively. We prove Theorem 9 by induction; that is, we assume that Theorem 9 is true for 𝑚=𝑙 and show the case 𝑚=𝑙+1. We also prove the following theorem which states derivative estimates of the fundamental solution.

Theorem 10. Let 𝑗, 𝑚, and 𝑛 be integers, 𝑚≥0, 𝑛≥2^𝑚+1+1, and 1≤𝑗≤2^𝑚+1−1. Suppose that 𝑉∈(𝑅𝐻)₂^𝑚_𝑛/(2^𝑚+1_−𝑗). Then for any positive integer 𝑁 there exists a positive constant 𝐶_𝑁 such that ∇^𝑗_𝑥Γ_𝐻2𝑚≤𝐶𝑥,𝑦^𝑁1+𝜌(𝑥,𝑉)𝑥−𝑦^𝑁⋅1𝑥−𝑦^{𝑛−2𝑚+1+𝑗}. (8)

In Theorem 10, the cases 𝑚=0 and 𝑚=1 were shown in [3, page 537] and [8, Theorem 6], respectively.

The plan of this paper is as follows. In Section 2, we describe some lemmas needed later. In Section 3, we assume that Theorem 9 is true for 𝑚=𝑙 and show some estimates for 𝐻₂^𝑙 which are needed to prove the case 𝑚=𝑙+1 in Theorem 9. In Section 4, we prove the case 𝑚=𝑙+1 in Theorem 9. Section 5 is devoted to proof of Theorem 10. In Section 6, we prove Theorems 5 and 8. Finally, in Section 7, we state some remarks. Throughout this paper the letter 𝐶 stands for a constant not necessarily the same at each occurrence.

2. Preliminaries

In this section, we describe some lemmas needed later. First, we remark that 𝑉∈(𝑅𝐻)_𝑞, 𝑞>1, implies that 𝑉(𝑥)𝑑𝑥 is a doubling measure; that is, there exists a positive constant 𝐶₀ such that _{𝐵2𝑟(𝑥)}𝑉𝑦𝑑𝑦≤𝐶_{0𝐵𝑟(𝑥)}𝑉𝑦𝑑𝑦 (9) (see [3, page 518] and also [5, Remark 1.2 (1)]).

Lemma 11 (see [3, Lemma 1.2]). Assume that 𝑉∈(𝑅𝐻)_𝑞 for some 𝑞>𝑛/2. Then there exists a positive constant 𝐶 such that, for 0<𝑟<𝑅<∞, 𝑟²𝐵_𝑟(𝑥)_{𝐵𝑟(𝑥)}𝑉𝑦𝑅𝑑𝑦≤𝐶𝑟^{𝑛/𝑞−2}𝑅²𝐵_𝑅(𝑥)_{𝐵𝑅(𝑥)}𝑉𝑦𝑑𝑦. (10)

Lemma 12 (see [3, Lemma 1.4 (b), (c)]). (1) Suppose that 𝑉∈(𝑅𝐻)_𝑛/2. Then there exist positive constants 𝐶 and 𝑘₀ such that, for 𝑥,𝑦∈𝐑^𝑛, 𝜌𝑦,𝑉≤𝐶1+𝜌(𝑥,𝑉)𝑥−𝑦^𝑘0𝜌(𝑥,𝑉). (11)(2) Suppose 𝑉∈(𝑅𝐻)_𝑛/2. Then there exist positive constants 𝐶^′ and 𝑘₀ such that, for 𝑥, 𝑦∈𝐑^𝑛, 𝜌≥𝐶𝑦,𝑉^′𝜌(𝑥,𝑉)1+𝜌(𝑥,𝑉)𝑥−𝑦^{𝑘0/(𝑘0+1)}. (12)

Lemma 13 (Caccioppoli type inequality). Let 𝑎, 𝑗, and 𝑘 be integers, 𝑘≥2, 1≤𝑎≤𝑘/2, and 1≤𝑗≤2𝑎. Assume that (−Δ)^𝑘𝑢+𝑉^𝑘𝑢=0 in 𝐵_𝑅(𝑥₀) for some 𝑥₀∈𝐑^𝑛. Then there exists a positive constant 𝐶 such that _{𝐵𝑅/2(𝑥0)}Δ^𝑘−𝑎𝑢(𝑥)²+𝑑𝑥_{𝐵𝑅/2(𝑥0)}𝑉(𝑥)^𝑘|𝑢(𝑥)|Δ^𝑘−2𝑎𝑢(𝑥)𝑑𝑥≤𝐶^2𝑎_𝑗=11𝑅^2𝑗_{𝐵𝑅(𝑥0)}∇^2𝑎−𝑗Δ^𝑘−2𝑎𝑢(𝑥)²𝑑𝑥. (13)

For readers’ convenience, we give the proof of Lemma 13 at the end of this section. If 𝑘 is an even number, then letting 𝑎=𝑘/2 in Lemma 13, we have the following.

Corollary 14 (see [2, Lemma 3.5]). Let 𝑗 be an integer, 𝑘 an even number, 𝑘≥2, and 0≤𝑗≤𝑘−1. Assume that (−Δ)^𝑘𝑢+𝑉^𝑘𝑢=0 in 𝐵_𝑅(𝑥₀) for some 𝑥₀∈𝐑^𝑛. Then there exists a positive constant 𝐶 such that _{𝐵𝑅/2(𝑥0)}Δ^𝑘/2𝑢(𝑥)²𝑑𝑥+_{𝐵𝑅/2(𝑥0)}𝑉(𝑥)^𝑘|𝑢(𝑥)|²𝑑𝑥≤𝐶^𝑘−1_𝑗=01𝑅^{2(𝑘−𝑗)𝐵}_𝑅^(𝑥₀⁾∇^𝑗𝑢(𝑥)²𝑑𝑥. (14)

Lemma 15 (Caccioppoli type inequality). Let 𝑎, 𝑗, and 𝑘 be integers, 𝑘≥1, 1≤𝑎≤(𝑘+1)/2, and 1≤𝑗≤2𝑎−1. Assume that (−Δ)^𝑘𝑢+𝑉^𝑘𝑢=0 in 𝐵_𝑅(𝑥₀) for some 𝑥₀∈𝐑^𝑛. Then there exists a positive constant 𝐶 such that _{𝐵𝑅/2(𝑥0)}∇Δ^𝑘−𝑎𝑢(𝑥)²+𝑑𝑥_{𝐵𝑅/2(𝑥0)}𝑉(𝑥)^𝑘|𝑢(𝑥)|Δ^{𝑘−2𝑎+1}𝑢(𝑥)𝑑𝑥≤𝐶^2𝑎−1_𝑗=11𝑅^2𝑗_{𝐵𝑅(𝑥0)}∇^{2𝑎−𝑗−1}Δ^{𝑘−2𝑎+1}𝑢(𝑥)²𝑑𝑥. (15)

If 𝑘 is an odd number, then letting 𝑎=(𝑘+1)/2 in Lemma 15, we have the following.

Corollary 16 (see [2, Lemma 3.5]). Let 𝑗 be an integer, 𝑘 an odd number, 𝑘≥1, and 0≤𝑗≤𝑘−1. Assume that (−Δ)^𝑘𝑢+𝑉^𝑘𝑢=0 in 𝐵_𝑅(𝑥₀) for some 𝑥₀∈𝐑^𝑛. Then there exists a positive constant 𝐶 such that _{𝐵𝑅/2(𝑥0)}∇Δ^(𝑘−1)/2𝑢(𝑥)²𝑑𝑥+_{𝐵𝑅/2(𝑥0)}𝑉(𝑥)^𝑘|𝑢(𝑥)|²𝑑𝑥≤𝐶^𝑘−1_𝑗=01𝑅^{2(𝑘−𝑗)𝐵}_𝑅^(𝑥₀⁾∇^𝑗𝑢(𝑥)²𝑑𝑥. (16)

Lemma 17. Let 𝑗, 𝑘, and 𝑛 be integers, 𝑘≥1, 𝑛≥2𝑘+1, and 0≤𝑗≤𝑘−1. Assume that (−Δ)^𝑘𝑢+𝑉^𝑘𝑢=0, 𝑢≥0, in 𝐵_𝑅(𝑥₀) for some 𝑥₀∈𝐑^𝑛. Then there exists a positive constant 𝐶 such that sup_{𝑦∈𝐵𝑅/2𝑥0}𝑢𝑦≤𝐶^𝑘−1_𝑗=0𝑅^𝑗1𝐵_𝑅𝑥_{0𝐵𝑅(𝑥0)}∇^𝑗𝑢(𝑥)²𝑑𝑥^1/2. (17)

In Lemma 17, the cases 𝑘=1 and 𝑘=2 have been proved (see [3, page 523] and [1, Corollary 5.6]).

Proof of Lemma 17. We proceed following the proof of [1, Corollary 5.6]. We prove Lemma 17 for 𝑢 which satisfies (−Δ)^𝑘𝑢+𝑉^𝑘𝑢=0, 𝑢≥0, in 𝐵_𝑅(𝑥₀). Let 𝑎 and 𝑏 be nonnegative integers and choose 𝜂∈𝐶^∞₀(𝐵_𝑅/4(𝑥₀)) such that 𝜂≡1 on 𝐵_𝑅/8(𝑥₀) and |∇^𝑏(Δ^𝑎𝜂)|≤𝐶/𝑅^2𝑎+𝑏. Let Γ_{𝐻𝑘−1,0}(𝑥,𝑦)=𝐶|𝑥−𝑦|^{2(𝑘−1)−𝑛} be the fundamental solution for (−Δ)^𝑘−1. Note that Δ^𝑘−1𝑢𝑦𝜂𝑦=_𝑎,𝑏2()∇𝑘−12𝑎+𝑏^𝑏Δ^{𝑘−1−𝑎−𝑏}𝑢𝑦⋅∇^𝑏Δ^𝑎𝜂𝑦, (18) where ∇^𝑏Δ^{𝑘−1−𝑎−𝑏}𝑢𝑦⋅∇^𝑏Δ^𝑎𝜂𝑦=^𝑛_{𝑖1,𝑖2,…,𝑖𝑘−1=1}𝜕^𝑏𝜕𝑦_𝑖1𝜕𝑦_𝑖2⋯𝜕𝑦_𝑖𝑏×𝜕²𝜕𝑦²_𝑖1𝜕²𝜕𝑦²_𝑖2⋯𝜕²𝜕𝑦²_{𝑖𝑘−1−𝑎−𝑏}𝑢⋯⋅𝜕^𝑏𝜕𝑦_𝑖1𝜕𝑦_𝑖2⋯𝜕𝑦_𝑖𝑏𝜕²𝜕𝑦²_𝑖1𝜕²𝜕𝑦²_𝑖2⋯𝜕²𝜕𝑦²_𝑖𝑎𝜂⋯ (19) and the summation is taken over all integers 𝑎 and 𝑏 satisfying 𝑎≥0, 𝑏≥0, and 0≤𝑎+𝑏≤𝑘−1. Then, integrating by parts, for 𝑥∈𝐵_𝑅/4(𝑥₀) we have 𝑢(𝑥)𝜂(𝑥)=_𝐑𝑛Γ_{𝐻𝑘−1,0}𝑥,𝑦(−Δ)^𝑘−1𝑦=𝑢𝜂𝑑𝑦_𝐑𝑛Γ_{𝐻𝑘−1,0}×𝑥,𝑦_𝑎,𝑏(−1)^𝑘−12()∇𝑘−12𝑎+𝑏^𝑏Δ^{𝑘−1−𝑎−𝑏}𝑢𝑦⋅∇^𝑏Δ^𝑎𝜂𝑦=𝑑𝑦_𝐑𝑛Γ_{𝐻𝑘−1,0}𝑥,𝑦(−Δ)^𝑘−1𝑢𝑦𝜂𝑦𝑑𝑦+𝐶_{𝐑𝑛()≠()𝑎,𝑏0,0}∇^𝑏Δ^{𝑘−1−𝑎−𝑏}Γ_{𝐻𝑘−1,0}𝑥,𝑦⋅∇^𝑏Δ^𝑎𝜂𝑦𝑢𝑦𝑑𝑦. (20) Let 𝑢𝑥₀≤_{𝐵𝑅/4(𝑥0)}(−Δ)^𝑘−1𝑢𝑦𝑥₀−𝑦^{𝑛−2(𝑘−1)}+𝐶𝑑𝑦𝑅^2𝑎+𝑏_{()≠()𝑎,𝑏0,0}^𝐵_𝑅/4^(𝑥₀^)⧵𝐵_𝑅/8^(𝑥₀⁾𝑢𝑦𝑥₀−𝑦^{𝑛−2𝑎−𝑏}𝑑𝑦=𝐼₁+𝐼₂. (21) Then we have 𝐼₂≤𝐶𝑅^𝑛_{𝐵𝑅/4(𝑥0)}𝑢𝑦1𝑑𝑦≤𝐶𝐵_𝑅𝑥_{0𝐵𝑅(𝑥0)}𝑢𝑦²𝑑𝑦^1/2. (22) It remains to estimate 𝐼₁. Since 𝑢≥0, it follows that Δ((−Δ)^𝑘−1𝑢)=−(−Δ)^𝑘𝑢=𝑉^𝑘𝑢≥0. Hence (−Δ)^𝑘−1𝑢 is subharmonic. Then for all 𝑥∈𝐵_𝑅/4(𝑥₀), (−Δ)^𝑘−1𝑢(𝑥)≤1𝐵_𝑅/4(𝑥)^𝐵_𝑅/4^(𝑥)(−Δ)^𝑘−1𝑢𝑦1𝑑𝑦≤𝐶𝐵_𝑅/2𝑥₀^𝐵_𝑅/2^(𝑥₀⁾Δ^𝑘−1𝑢𝑦²𝑑𝑦^1/2. (23) Then we have 𝐼₁≤𝐶sup_{𝑦∈𝐵𝑅/4𝑥0}(−Δ)^𝑘−1𝑢𝑦_{𝐵𝑅/4(𝑥0)}𝑑𝑦𝑥₀−𝑦^{𝑛−2(𝑘−1)}≤𝐶𝑅^2(𝑘−1)sup_{𝑦∈𝐵𝑅/4𝑥0}Δ^𝑘−1𝑢𝑦≤𝐶𝑅^2𝑘−21𝐵_𝑅/2𝑥₀^𝐵_𝑅/2^(𝑥₀⁾Δ^𝑘−1𝑢𝑦²𝑑𝑦^1/2. (24) Then using Lemmas 13 and 15 repeatedly, we have 𝐼₁≤𝐶𝑅^{2𝑘−2−𝑛/2}1𝑅²_{𝐵𝑅(𝑥0)}∇Δ^𝑘−2𝑢(𝑥)²+1𝑑𝑥𝑅⁴_{𝐵𝑅(𝑥0)}Δ^𝑘−2𝑢(𝑥)²𝑑𝑥^1/2≤⋯≤𝐶^𝑘−1_𝑗=0𝑅^𝑗1𝑅^𝑛_{𝐵𝑅(𝑥0)}∇^𝑗𝑢(𝑥)²𝑑𝑥^1/2. (25) It follows that 𝑢𝑥₀≤𝐶^𝑘−1_𝑗=0𝑅^𝑗1𝐵_𝑅𝑥_{0𝐵𝑅(𝑥0)}∇^𝑗𝑢(𝑥)²𝑑𝑥^1/2. (26) From (26) we have, for all 𝑦∈𝐵_𝑅/2(𝑥₀), 𝑢𝑦≤𝐶^𝑘−1_𝑗=0𝑅^𝑗1𝐵_𝑅/4𝑦^𝐵_𝑅/4^(𝑦)∇^𝑗𝑢(𝑥)²𝑑𝑥^1/2. (27) Then we arrive at the desired estimate.

Lemma 18. Let 𝑗, 𝑘, and 𝑛 be integers, 𝑘≥1, 𝑛≥2𝑘+1, and 1≤𝑗≤2𝑘−1. Suppose that 𝑉∈(𝑅𝐻)_𝑞0 for some 𝑞₀, 𝑛/2≤𝑞₀<𝑘𝑛/(2𝑘−𝑗). Assume also that (−Δ)^𝑘𝑢+𝑉^𝑘𝑢=0 in 𝐵_𝑅(𝑥₀) for some 𝑥₀∈𝐑^𝑛. Then there exists a positive constant 𝐶 such that _{𝐵𝑅/2(𝑥0)}∇^𝑗𝑢(𝑥)^𝑡𝑑𝑥^1/𝑡≤𝐶𝑅^{𝑘𝑛/𝑞0−2𝑘}1+𝑅𝜌𝑥₀,𝑉^2𝑘sup_{𝑦∈𝐵𝑅𝑥0}𝑢𝑦, (28) where 1/𝑡=𝑘/𝑞₀−(2𝑘−𝑗)/𝑛.

In Lemma 18, the cases 𝑘=1 and 𝑘=2 were shown in [3, Lemma 4.6] and [8, Lemma 7], respectively.

Proof of Lemma 18. We show Lemma 18 by a method similar to the one used in the proof of [3, Lemma 4.6]. Let 𝜂∈𝐶^∞₀(𝐵_𝑅(𝑥₀)) such that 𝜂≡1 on 𝐵_3𝑅/4(𝑥₀) and |∇^𝑏(Δ^𝑎𝜂)|≤𝐶/𝑅^2𝑎+𝑏, where 𝑎 and 𝑏 are nonnegative integers. We denote by Γ_𝐻𝑘,0(𝑥,𝑦) the fundamental solution for (−Δ)^𝑘. Note that 𝑢(𝑥)𝜂(𝑥)=_𝐑𝑛Γ_𝐻𝑘,0𝑥,𝑦−𝑉𝑦^𝑘𝑢𝑦𝜂𝑦+(−1)^𝑘×_𝑎,𝑏∇2𝑘2𝑎+𝑏^𝑏Δ^{𝑘−𝑎−𝑏}𝑢𝑦⋅∇^𝑏Δ^𝑎𝜂𝑦𝑑𝑦, (29) where the summation is taken over all integers 𝑎 and 𝑏 satisfying 𝑎≥0, 𝑏≥0, and 1≤𝑎+𝑏≤𝑘. Then integrating by parts, for 𝑥∈𝐵_𝑅/2(𝑥₀), we have ∇^𝑗𝑢(𝑥)≤𝐶_{𝐵𝑅(𝑥0)}𝑉𝑦^𝑘𝑢𝑦𝜂𝑦𝑥−𝑦^{𝑛−2𝑘+𝑗}+𝐶𝑑𝑦𝑅^{𝑛−2𝑘+𝑗+𝑏+2(𝑘−𝑎−𝑏)+2𝑎+𝑏𝐵}_𝑅^(𝑥₀⁾𝑢𝑦𝑑𝑦≤𝐶sup_{𝑦∈𝐵𝑅𝑥0}𝑢𝑦⋅_{𝐵𝑅(𝑥0)}𝑉𝑦^𝑘𝑥−𝑦^{𝑛−2𝑘+𝑗}+𝐶𝑑𝑦𝑅^𝑛+𝑗_{𝐵𝑅(𝑥0)}𝑢𝑦𝑑𝑦. (30) It then follows from the well-known theorem on fractional integrals that _{𝐵𝑅/2(𝑥0)}∇^𝑗𝑢(𝑥)^𝑡𝑑𝑥^1/𝑡≤𝐶sup_{𝑦∈𝐵𝑅𝑥0}𝑢𝑦_{𝐵𝑅(𝑥0)}𝑉(𝑥)^𝑞0𝑑𝑥^𝑘/𝑞0+𝐶𝑅^{𝑘𝑛/𝑞0−2𝑘}sup_{𝑦∈𝐵𝑅𝑥0}𝑢𝑦≤𝐶𝑅^{𝑘𝑛/𝑞0−2𝑘}1+𝑅𝜌𝑥₀,𝑉^2𝑘sup_{𝑦∈𝐵𝑅𝑥0}𝑢𝑦, (31) where 1/𝑡=𝑘/𝑞₀−(2𝑘−𝑗)/𝑛 and we have used Remark 4 (1).

At the end of this section, we give the following.

Proof of Lemma 13. Let 𝑙 and 𝑚 be nonnegative integers and choose 𝜂∈𝐶^∞₀(𝐵_𝑅(𝑥₀)) such that 𝜂≡1 on 𝐵_𝑅/2(𝑥₀) and |∇^𝑚(Δ^𝑙𝜂)|≤𝐶/𝑅^2𝑙+𝑚. Multiplying (−Δ)^𝑘𝑢+𝑉^𝑘𝑢=0 by ((−Δ)^𝑘−2𝑎𝑢)𝜂^𝑘+1 and integrating over 𝐑^𝑛 by integrating by parts, we have _{𝐵𝑅/2(𝑥0)}Δ^𝑘−𝑎𝑢(𝑥)²+𝑑𝑥_{𝐵𝑅/2(𝑥0)}𝑉(𝑥)^𝑘|𝑢(𝑥)|Δ^𝑘−2𝑎𝑢(𝑥)𝑑𝑥≤𝐶_{𝐵𝑅(𝑥0)𝑙,𝑚}Δ^𝑘−𝑎𝑢(𝑥)∇^𝑚Δ^{𝑘−𝑎−𝑙−𝑚}𝑢(𝑥)⋅∇^𝑚Δ^𝑙𝜂^𝑘+1(𝑥)𝑑𝑥, (32) where the summation is taken over all integers 𝑙 and 𝑚 satisfying 𝑙≥0, 𝑚≥0, and 1≤𝑙+𝑚≤𝑎. Let 𝜖 be a positive real number which will be determined later. Then the right-hand side of (32) is bounded by 𝐶_{𝐵𝑅(𝑥0)𝑙,𝑚}√𝜖Δ^𝑘−𝑎𝑢(𝑥⋅1)√𝜖⋅1𝑅^2𝑙+𝑚∇^𝑚Δ^{𝑘−𝑎−𝑙−𝑚}𝑢(𝑥)𝑑𝑥≤𝐶_{𝐵𝑅(𝑥0)𝑙,𝑚}𝜖Δ^𝑘−𝑎𝑢(𝑥)²+1𝜖⋅1𝑅^{2(2𝑙+𝑚)}×∇^𝑚Δ^{𝑘−𝑎−𝑙−𝑚}𝑢(𝑥)²𝑑𝑥. (33) Then choosing 𝜖 such that 𝐶(𝑎−1)(𝑎+2)𝜖=1, we arrive at the desired estimate.

3. Estimates for 𝐻₂^𝑙

In this section, we assume that Theorem 9 is true for 𝑚=𝑙 and show estimates for 𝐻₂^𝑙 which is needed to prove the case 𝑚=𝑙+1 in Theorem 9.

Lemma 19. Let 𝑗, 𝑙, and 𝑛 be integers, 𝑙≥0, and 𝑛≥2^𝑙+1+1. Suppose that 𝑉∈(𝑅𝐻)_𝑛/2. Assume also that Theorem 9 is true for 𝑚=𝑙. Then there exists a positive constant 𝐶 such that 𝜌(⋅,𝑉)^2𝑙+1𝐻₂⁻¹_𝑙𝑓_{𝐿𝑝(𝐑𝑛)}≤𝐶𝑓_{𝐿𝑝(𝐑𝑛)}, (34) where 1≤𝑝≤∞.

Proof. We can prove Lemma 19 by the same way as in the proof of [3, Corollary 2.8]. It follows from the inductive assumption that _𝐑𝑛Γ_𝐻2𝑙𝑥,𝑦𝑑𝑦≤𝐶_𝐑𝑛𝑑𝑦1+𝜌(𝑥,𝑉)𝑥−𝑦^2𝑙+1+1𝑥−𝑦^{𝑛−2𝑙+1}=𝐶𝜌(𝑥,𝑉)^2𝑙+1. (35) If 𝑓∈𝐿^𝑝(𝐑^𝑛), then 𝑢(𝑥)=𝐻₂⁻¹_𝑙𝑓(𝑥)=_𝐑𝑛Γ_𝐻2𝑙𝑥,𝑦𝑓𝑦𝑑𝑦. (36) Let 𝑝^′=𝑝/(𝑝−1). Then using Hölder's inequality and (35) we have |𝑢(𝑥)|≤_𝐑𝑛Γ_𝐻2𝑙𝑥,𝑦𝑑𝑦^1/𝑝′×_𝐑𝑛Γ_𝐻2𝑙𝑥,𝑦𝑓𝑦^𝑝𝑑𝑦^1/𝑝≤𝐶𝜌(𝑥,𝑉)^{2𝑙+1/𝑝′𝐑𝑛}Γ_𝐻2𝑙𝑥,𝑦𝑓𝑦^𝑝𝑑𝑦^1/𝑝. (37) It follows that _𝐑𝑛𝜌(𝑥,𝑉)^2𝑙+1𝑢(𝑥)^𝑝𝑑𝑥≤𝐶_𝐑𝑛𝑓𝑦^𝑝_𝐑𝑛𝜌(𝑥,𝑉)^2𝑙+1Γ_𝐻2𝑙𝑥,𝑦𝑑𝑥𝑑𝑦. (38) Finally, note that, by the inductive assumption and Lemma 12 (1), _𝐑𝑛𝜌(𝑥,𝑉)^2𝑙+1Γ_𝐻2𝑙𝑥,𝑦𝑑𝑥≤𝐶𝐶_𝑁𝐑𝑛𝜌𝑦,𝑉^2𝑙+11+𝜌𝑦,𝑉𝑥−𝑦^{𝑁−2𝑙+1𝑘0}𝑥−𝑦^{𝑛−2𝑙+1}𝑑𝑥≤𝐶 (39) if we choose 𝑁=2^𝑙+1𝑘₀+2^𝑙+1+1. Then Lemma 19 follows.

We also need the following lemma.

Lemma 20. Let 𝑗, 𝑙, and 𝑛 be integers, 𝑙≥0, 𝑛≥2^𝑙+1+1, and 1≤𝑗≤2^𝑙+1−1. Suppose that 𝑉∈(𝑅𝐻)_𝑞0 for some 𝑞₀, 𝑛/2≤𝑞₀<2^𝑙𝑛/(2^𝑙+1−𝑗). Assume also that Theorem 9 is true for 𝑚=𝑙. Then for 1≤𝑝≤𝑝₀ there exists a positive constant 𝐶 such that 𝜌(⋅,𝑉)^{2𝑙+1−𝑗}∇^𝑗𝐻₂⁻¹_𝑙𝑓_{𝐿𝑝(𝐑𝑛)}≤𝐶𝑓_{𝐿𝑝(𝐑𝑛)}, (40) where 1/𝑝₀=2^𝑙/𝑞₀−(2^𝑙+1−𝑗)/𝑛.

Proof. We show Lemma 20 by a method similar to the one used in the proof of [3, Theorem 4.13]. Suppose that 𝑉∈(𝑅𝐻)_𝑞0 for some 𝑞₀≥𝑛/2. Then 𝑉∈(𝑅𝐻)_𝑞1 for some 𝑞₁, 2^𝑙𝑛/(2^𝑙+1−𝑗)>𝑞₁>𝑞₀. Let 𝑇₂^𝑙𝑓(𝑥)=𝜌(𝑥,𝑉)^{2𝑙+1−𝑗}_𝐑𝑛∇^𝑗_𝑥Γ_𝐻2𝑙𝑥,𝑦𝑓𝑦𝑑𝑦. (41) The adjoint of 𝑇₂^𝑙 is given by 𝑇^∗_2𝑙𝑓(𝑥)=_𝐑𝑛∇^𝑗_𝑦Γ_𝐻2𝑙𝑦,𝑥𝜌𝑦,𝑉^{2𝑙+1−𝑗}𝑓𝑦𝑑𝑦. (42) By duality, it suffices to show that 𝑇^∗_2𝑙𝑓_{𝐿𝑝(𝐑𝑛)}≤𝐶𝑓_{𝐿𝑝(𝐑𝑛)}for𝑝^′₀≤𝑝≤∞, (43) where 1/𝑝₀+1/𝑝^′₀=1. Let 𝑟=1/𝜌(𝑥,𝑉). We choose 𝑡 and 𝑝₁ such that 1/𝑡=2^𝑙/𝑞₁−(2^𝑙+1−𝑗)/𝑛, 1/𝑝₁=1−2^𝑙/𝑞₁+(2^𝑙+1−𝑗)/𝑛. Thus 1/𝑡+1/𝑝₁=1. By Hölder’s inequality we have 𝑇^∗_2𝑙𝑓(𝑥≤_{∞𝑖=−∞})_{2𝑖−1𝑟<|𝑦−𝑥|≤2𝑖𝑟}∇^𝑗_𝑦Γ_𝐻2𝑙𝑦,𝑥×𝜌𝑦,𝑉^{2𝑙+1−𝑗}𝑓𝑦≤𝑑𝑦_{∞𝑖=−∞2𝑖−1𝑟<|𝑦−𝑥|≤2𝑖𝑟}∇^𝑗_𝑦Γ_𝐻2𝑙𝑦,𝑥×𝜌𝑦,𝑉^{2𝑙+1−𝑗𝑡}𝑑𝑦^1/𝑡⋅_{|𝑦−𝑥|≤2𝑖𝑟}𝑓𝑦^𝑝1𝑑𝑦^1/𝑝1. (44) Let 𝑎 be a positive integer and 𝑦_𝑎∈𝐑^𝑛. It follows from Lemma 18, Lemma 12 (1), and the inductive assumption that _{2𝑖−1𝑟<|𝑦−𝑥|≤2𝑖𝑟}∇^𝑗_𝑦Γ_𝐻2𝑙𝑦,𝑥𝜌𝑦,𝑉^{2𝑙+1−𝑗𝑡}𝑑𝑦^1/𝑡≤^𝐴_{𝑎=1𝑎𝑟|𝑦−𝑥|=3⋅2𝑖−2|𝑧−𝑦𝑎|≤2𝑖−1𝑟}∇^𝑗_𝑧Γ_𝐻2𝑙(𝑧,𝑥)𝜌(𝑧,𝑉)^{2𝑙+1−𝑗𝑡}𝑑𝑧^1/𝑡≤𝐶2^𝑖−1𝑟^{2𝑙𝑛/𝑞1−2𝑙+1}1+2^𝑖𝑟𝜌(𝑥,𝑉)^{2𝑙+1+(2𝑙+1−𝑗)𝑘0}×𝜌(𝑥,𝑉)^{2𝑙+1−𝑗}⋅^𝐴_𝑎=1sup_{𝑧∈𝐵𝑟5⋅2𝑖−3𝑦𝑎}Γ_𝐻2𝑙(2𝑧,𝑥)≤𝐶^𝑖𝑟^{2𝑙𝑛/𝑞1−2𝑙+1}1+2^{𝑖2𝑙+1+(2𝑙+1−𝑗)𝑘0}⋅1𝑟^{2𝑙+1−𝑗}⋅𝐶_𝑁1+𝜌(𝑥,𝑉)|𝑧−𝑥|^𝑁⋅^𝐴_𝑎=1sup_{𝑧∈𝐵𝑟5⋅2𝑖−3𝑦𝑎}1|𝑧−𝑥|^{𝑛−2𝑙+1}≤𝐶𝐶_𝑁2^𝑖𝑟^{2𝑙𝑛/𝑞1−2𝑙+1}1+2^{𝑖2𝑙+1+(2𝑙+1−𝑗)𝑘0}𝑟^{2𝑙+1−𝑗}1+2^𝑖−3𝑁2^𝑖−3𝑟^{𝑛−2𝑙+1}≤𝐶𝐶_𝑁2^𝑖𝑟^{2𝑙𝑛/𝑞1−𝑛}𝑟^{2𝑙+1−𝑗}1+2^{𝑖𝑁−2𝑙+1−(2𝑙+1−𝑗)𝑘0}, (45) where 𝐴 is a finite integer not depending on 𝑖 and 𝑟. Thus 𝑇^∗_2𝑙𝑓(𝑥)≤𝐶𝐶_𝑁^∞_𝑖=−∞2^{𝑖2𝑙+1−𝑗}1+2^{𝑖𝑁−2𝑙+1−(2𝑙+1−𝑗)𝑘0}×12^𝑖𝑟^𝑛𝐵_2𝑖𝑟^(𝑥)𝑓𝑦^𝑝1𝑑𝑦^1/𝑝1≤𝐶𝑀𝑓^𝑝1(𝑥)^1/𝑝1, (46) where we choose 𝑁≥2^𝑙+2+(2^𝑙+1−𝑗)𝑘₀−𝑗+1 and 𝑀 is the Hardy-Littlewood maximal operator. Hence it follows that 𝑇^∗_2𝑙𝑓_{𝐿𝑝(𝐑𝑛)}≤𝐶𝑓_{𝐿𝑝(𝐑𝑛)}for𝑝₁<𝑝≤∞. (47) Then (43) follows since 𝑝^′₀>𝑝₁.

For the case 𝑞₀=𝑛/2 in Lemma 20, if 𝑛≥2^𝑙+2+1, letting 𝑝=2, we have the following.

Corollary 21. Let 𝑗, 𝑙, and 𝑛 be integers, 𝑙≥0, 𝑛≥2^𝑙+2+1, and 1≤𝑗≤2^𝑙+1−1. Suppose that 𝑉∈(𝑅𝐻)_𝑛/2. Assume also that Theorem 9 is true for 𝑚=𝑙. Then there exists a positive constant 𝐶 such that 𝜌(⋅,𝑉)^{2𝑙+1−𝑗}∇^𝑗𝐻₂⁻¹_𝑙𝑓_{𝐿2(𝐑𝑛)}≤𝐶𝑓_{𝐿2(𝐑𝑛)}. (48)

4. Proof of Theorem 9 (Case𝑚=𝑙+1)

In this section, we assume that Theorem 9 is true for 𝑚=𝑙 and prove the case 𝑚=𝑙+1 in Theorem 9. It follows easily from the following lemma.

Lemma 22. Let 𝑗, 𝑙, and 𝑛 be integers, 𝑙≥0, and 𝑛≥2^𝑙+2+1. Suppose that 𝑉∈(𝑅𝐻)_𝑛/2. Assume also that (−Δ)^2𝑙+1𝑢+𝑉^2𝑙+1𝑢=0 in 𝐵_𝑅(𝑥₀) for some 𝑥₀∈𝐑^𝑛 and Theorem 9 is true for 𝑚=𝑙. Then for any positive integer 𝑁 there exists a positive constant 𝐶_𝑁 such that sup_{𝑦∈𝐵𝑅/2𝑥0}𝑢𝑦≤𝐶_𝑁1+𝑅𝜌𝑥₀,𝑉^𝑁sup_{𝑦∈𝐵𝑅𝑥0}𝑢𝑦. (49)

Assuming Lemma 22 for the moment, we give

Proof of Theorem 9  (𝐶𝑎𝑠𝑒𝑚=𝑙+1). Fix 𝑥₀, 𝑦₀∈𝐑^𝑛 and put 𝑅=|𝑥₀−𝑦₀|. Then 𝑢(𝑥)=Γ_𝐻2𝑙+1(𝑥,𝑦₀) is a solution of (−Δ)^2𝑙+1𝑢+𝑉^2𝑙+1𝑢=0 on 𝐵_𝑅/2(𝑥₀). Using the estimate 0≤Γ_𝐻2𝑙+1(𝑥,𝑦)≤𝐶|𝑥−𝑦|^{2𝑙+2−𝑛} and (49), we arrive at the desired estimate.

To prove Lemma 22, we need some lemmas.

Lemma 23. Let 𝑗, 𝑙, and 𝑛 be integers, 𝑙≥0, 𝑛≥2^𝑙+2+1, and 0≤𝑗≤2^𝑙+1−1. Suppose that 𝑉∈(𝑅𝐻)_𝑛/2. Assume also that Theorem 9 is true for 𝑚=𝑙. Then there exists a positive constant 𝐶 such that ^2𝑙+1−1_{𝑗=0𝐑𝑛}𝜌(𝑥,𝑉)^{2(2𝑙+1−𝑗)}∇^𝑗𝑢(𝑥)²𝑑𝑥≤𝐶_𝐑𝑛Δ^2𝑙𝑢(𝑥)²𝑑𝑥+_𝐑𝑛𝑉(𝑥)^2𝑙+1|𝑢(𝑥)|²𝑑𝑥, (50) where 𝑢∈𝐶^∞₀(𝐑^𝑛).

Proof. By the case 𝑝=2 in Lemma 19 and Corollary 21, we have ^2𝑙+1−1_{𝑗=0𝐑𝑛}𝜌(𝑥,𝑉)^{2(2𝑙+1−𝑗)}∇^𝑗𝑢(𝑥)²𝑑𝑥≤𝐶_𝐑𝑛(−Δ)^2𝑙+𝑉^2𝑙𝑢(𝑥)²𝑑𝑥≤𝐶_𝐑𝑛Δ^2𝑙𝑢(𝑥)²𝑑𝑥+_𝐑𝑛𝑉(𝑥)^2𝑙+1|𝑢(𝑥)|²𝑑𝑥. (51)

If 𝑛≥2^𝑙+2+1 and 1≤𝑗≤2^𝑙+1−1, then letting 𝑘=2^𝑙+1 and 𝑞₀=𝑛/2 in Lemma 18 we have the following.

Corollary 24. Let 𝑗, 𝑙, and 𝑛 be integers, 𝑙≥0, 𝑛≥2^𝑙+2+1, and 1≤𝑗≤2^𝑙+1−1. Suppose that 𝑉∈(𝑅𝐻)_𝑛/2. Assume also that (−Δ)^2𝑙+1𝑢+𝑉^2𝑙+1𝑢=0 in 𝐵_𝑅(𝑥₀) for some 𝑥₀∈𝐑^𝑛. Then there exists a positive constant 𝐶 such that 1𝐵_𝑅/2𝑥₀^𝐵_𝑅/2^(𝑥₀⁾∇^𝑗𝑢(𝑥)²𝑑𝑥^1/2≤𝐶1+𝑅𝜌𝑥₀,𝑉^2𝑙+2𝑅^𝑗sup_{𝑦∈𝐵𝑅𝑥0}𝑢𝑦. (52)

Now we are ready to give the following.

Proof of Lemma 22. Let 𝜂∈𝐶^∞₀(𝐵_𝑅/2(𝑥₀)) such that 𝜂≡1 on 𝐵_𝑅/4(𝑥₀), |∇^𝑗𝜂|≤𝐶/𝑅^𝑗, where 𝑗 is an integer satisfying 1≤𝑗≤2^𝑙+1−1. Applying Lemma 23 to 𝑢𝜂 and using Corollary 14 we have ^2𝑙+1−1_{𝑗=0𝐵𝑅/4(𝑥0)}𝜌(𝑥,𝑉)^{2(2𝑙+1−𝑗)}∇^𝑗𝑢(𝑥)²𝑑𝑥≤𝐶^2𝑙+1−1_𝑗=01𝑅^{2(2𝑙+1−𝑗)𝐵}_𝑅^(𝑥₀⁾∇^𝑗𝑢(𝑥)²𝑑𝑥. (53) From Lemma 12 (2) it follows for each integer 𝑗 satisfying 1≤𝑗≤2^𝑙+1−1 that _{𝐵𝑅/4(𝑥0)}∇^𝑗𝑢(𝑥)²≤𝐶𝑥𝑑𝑥1+𝑅𝜌₀,𝑉^{2(2𝑙+1−𝑗)𝑘0/(𝑘0+1)}𝑅𝜌𝑥₀,𝑉^{2(2𝑙+1−𝑗)}⋅𝑅^2𝑙+2𝑅^2𝑗⋅^2𝑙+1−1_𝑖=0𝑅^2𝑖𝑅^2𝑙+2𝐵_𝑅^(𝑥₀⁾∇^𝑖𝑢(𝑥)²𝑑𝑥. (54) Then we have ^2𝑙+1−1_𝑗=0𝑅^𝑗1𝐵_𝑅/4𝑥₀^𝐵_𝑅/4^(𝑥₀⁾∇^𝑗𝑢(𝑥)²𝑑𝑥^1/2≤𝐶1+𝑅𝜌𝑥₀,𝑉^{1/(𝑘02+1)𝑙+1−1}_𝑖=0𝑅^𝑖⋅1𝐵_𝑅𝑥_{0𝐵𝑅(𝑥0)}∇^𝑖𝑢(𝑥)²𝑑𝑥^1/2. (55) Repeating the above argument, for any positive integer 𝐾 we have ^2𝑙+1−1_𝑗=0𝑅^𝑗1𝐵_𝑅/4^𝐾𝑥₀^𝐵_𝐾𝑅/4^(𝑥₀⁾∇^𝑗𝑢(𝑥)²𝑑𝑥^1/2≤𝐶_𝐾1+𝑅𝜌𝑥₀,𝑉^{𝐾/(𝑘02+1)𝑙+1−1}_𝑗=0𝑅^𝑗⋅1𝐵_𝑅𝑥_{0𝐵𝑅(𝑥0)}∇^𝑗𝑢(𝑥)²𝑑𝑥^1/2. (56) Then using Lemma 17, Corollary 24, and estimate (56), we arrive at the desired estimate.

5. Proof of Theorem 10

In this section, we prove Theorem 10 which states derivative estimates of the fundamental solution for 𝐻₂^𝑚. We arrive at Theorem 10 combining Lemmas 25 with 22.

Lemma 25. Let 𝑗, 𝑚, and 𝑛 be integers, 𝑚≥0, 𝑛≥2^𝑚+1+1, and 1≤𝑗≤2^𝑚+1−1. Suppose that 𝑉∈(𝑅𝐻)₂^𝑚_𝑛/(2^𝑚+1_−𝑗). Assume also that (−Δ)^2𝑚𝑢+𝑉^2𝑚𝑢=0 in 𝐵_𝑅(𝑥₀) for some 𝑥₀∈𝐑^𝑛. Then there exist positive constants 𝐶_𝑗 and 𝐶^′_𝑗 such that sup_{𝑦∈𝐵𝑅/2𝑥0}∇^𝑗𝑢𝑦≤𝐶_𝑗1+𝑅𝜌𝑥₀,𝑉^𝐶′𝑗𝑅^𝑗sup_{𝑦∈𝐵𝑅𝑥0}𝑢𝑦. (57)

Proof. Let 𝜂∈𝐶^∞₀(𝐵_𝑅(𝑥₀)) such that 𝜂≡1 on 𝐵_3𝑅/4(𝑥₀) and |∇^𝑏(Δ^𝑎𝜂)|≤𝐶/𝑅^2𝑎+𝑏, where 𝑎 and 𝑏 are nonnegative integers. We use (29); integrating by parts, we have ∇^𝑗𝑢𝑥₀≤𝐶_{𝐵𝑅(𝑥0)}𝑉𝑦^2𝑚𝑢𝑦𝑥₀−𝑦^{𝑛−2𝑚+1+𝑗}+𝐶𝑑𝑦𝑅^𝑛+𝑗_{𝐵𝑅(𝑥0)}𝑢𝑦𝑑𝑦. (58) Since 𝑉∈(𝑅𝐻)₂^𝑚_𝑛/(2^𝑚+1_−𝑗), it follows that 𝑉∈(𝑅𝐻)_𝑞 for some 𝑞>2^𝑚𝑛/(2^𝑚+1−𝑗). We choose 𝑟 such that 2^𝑚/𝑞+1/𝑟=1 and 𝑟>1. By Hölder’s inequality we have ∇^𝑗𝑢𝑥₀≤𝐶𝑅^𝑛1𝑅^𝑛_{𝐵𝑅(𝑥0)}𝑉𝑦^𝑞𝑑𝑦^2𝑚/𝑞⋅1𝑅^𝑛_{𝐵𝑅(𝑥0)}𝑑𝑦𝑥₀−𝑦^{(𝑛−2𝑚+1+𝑗)𝑟1/𝑟}sup_{𝑦∈𝐵𝑅𝑥0}𝑢𝑦+𝐶𝑅^𝑛+𝑗_{𝐵𝑅(𝑥0)}𝑢𝑦≤𝐶𝑥𝑑𝑦1+𝑅𝜌₀,𝑉^2𝑚+1𝑅^𝑗sup_{𝑦∈𝐵𝑅𝑥0}𝑢𝑦, (59) where we have used Remark 4 (1). From (59) we have, for all 𝑦∈𝐵_𝑅/2(𝑥₀), ∇^𝑗𝑢𝑦≤𝐶1+𝑅𝜌𝑦,𝑉^2𝑚+1𝑅^𝑗sup_{𝑥∈𝐵𝑅/4𝑦}|𝑢(𝑥)|. (60) Using Lemma 12 (1), we have sup_{𝑦∈𝐵𝑅/2𝑥0}∇^𝑗𝑢𝑦≤𝐶1+𝑅𝜌𝑥₀,𝑉^{2𝑚+1(𝑘0+1)}𝑅^𝑗sup_{𝑦∈𝐵𝑅𝑥0}𝑢𝑦. (61) Then the proof is complete.

6. Proofs of Theorems 5 and 8

In this section, we prove Theorems 5 and 8.

Proof of Theorem 5. We assume that 𝑉∈(𝑅𝐻)_𝑞0 for some 𝑞₀≥2^𝑚𝑛/(2^𝑚+1−𝑗). Then 𝑉∈(𝑅𝐻)_𝑞1 for some 𝑞₁>𝑞₀. Let 𝑇₂^𝑚𝑓(𝑥)=𝑉(𝑥)^{(2𝑚+1−𝑗)/2}_𝐑𝑛∇^𝑗_𝑥Γ_𝐻2𝑚𝑥,𝑦𝑓𝑦𝑑𝑦. (62) The adjoint of 𝑇₂^𝑚 is given by 𝑇^∗₂^𝑚𝑓(𝑥)=_𝐑𝑛∇^𝑗_𝑦Γ_𝐻2𝑚𝑦,𝑥𝑉𝑦^{(2𝑚+1−𝑗)/2}𝑓𝑦𝑑𝑦. (63) Let 𝑝₀=2𝑞₀/(2^𝑚+1−𝑗). By duality, it suffices to show that 𝑇^∗₂^𝑚𝑓_{𝐿𝑝(𝐑𝑛)}≤𝐶𝑓_{𝐿𝑝(𝐑𝑛)}for𝑝^′₀≤𝑝≤∞, (64) where 1/𝑝₀+1/𝑝^′₀=1. Let 𝑟=1/𝜌(𝑥,𝑉). By Theorems 9 and 10 and Hölder's inequality we have 𝑇^∗₂^𝑚𝑓(𝑥≤_{∞𝑖=−∞})_{2𝑖−1𝑟<|𝑦−𝑥|≤2𝑖𝑟}∇^𝑗_𝑦Γ_𝐻2𝑚𝑦𝑦,𝑥×𝑉^{(2𝑚+1−𝑗)/2}𝑓𝑦𝑑𝑦≤𝐶𝐶_𝑁^∞_𝑖=−∞11+2^𝑖𝑁⋅12^𝑖𝑟^{𝑛−2𝑚+1+𝑗}×_{𝐵2𝑖𝑟(𝑥)}𝑉𝑦^{(2𝑚+1−𝑗)/2}𝑓𝑦𝑑𝑦≤𝐶𝐶_𝑁^∞_𝑖=−∞2^𝑖𝑟^{2𝑚+1−𝑗}1+2^𝑖𝑁×12^𝑖𝑟^𝑛𝐵_2𝑖𝑟^(𝑥)𝑉𝑦^𝑞1𝑑𝑦^{(2𝑚+1−𝑗)/(2𝑞1)}⋅12^𝑖𝑟^𝑛𝐵_2𝑖𝑟^(𝑥)𝑓𝑦^𝑝1𝑑𝑦^1/𝑝1, (65) where 1/𝑝₁=1−(2^𝑚+1−𝑗)/(2𝑞₁). Since 𝑉∈(𝑅𝐻)_𝑞1 we have 𝑇^∗₂^𝑚𝑓(𝑥)≤𝐶𝐶_𝑁𝑀𝑓^𝑝1(𝑥)^1/𝑝1⋅^∞_𝑖=−∞11+2^𝑖𝑁12^𝑖𝑟^𝑛−2𝐵₂^𝑖_𝑟^(𝑥)𝑉𝑦𝑑𝑦^{(2𝑚+1−𝑗)/2}. (66) For the case 𝑗≤0, by Lemma 11 and (4), we have 12^𝑖𝑟^𝑛−2𝐵₂^𝑖_𝑟^(𝑥)𝑉𝑦𝑟𝑑𝑦≤𝐶2^𝑖𝑟^{𝑛/𝑞1−2}⋅1𝑟^𝑛−2_{𝐵𝑟(𝑥)}𝑉𝑦2𝑑𝑦=𝐶^{𝑖2−𝑛/𝑞1}. (67) For the case 𝑗≥1, by the doubling condition (9) and (4), we have 12^𝑖𝑟^𝑛−2𝐵₂^𝑖_𝑟^(𝑥)𝑉𝑦𝑑𝑦≤2^2𝑖⋅2^−𝑖𝑛𝐶^𝑖₀⋅1𝑟^𝑛−2_{𝐵𝑟(𝑥)}𝑉𝑦2𝑑𝑦≤𝐶^𝑖𝑘1, (68) where 𝑘₁=2−𝑛+log₂𝐶₀. Hence, choosing 𝑁≥𝑘₁(2^𝑚+1−𝑗)/2+1, we obtain 𝑇^∗₂^𝑚𝑓(𝑥)≤𝐶𝑀𝑓^𝑝1(𝑥)^1/𝑝1. (69) It follows that 𝑇^∗₂^𝑚𝑓_{𝐿𝑝(𝐑𝑛)}≤𝐶𝑓_{𝐿𝑝(𝐑𝑛)}for𝑝₁<𝑝≤∞. (70) Then (64) follows since 𝑝₁={2𝑞₁/(2^𝑚+1−𝑗)}^′<{2𝑞₀/(2^𝑚+1−𝑗)}^′=𝑝^′₀.

Proof of Theorem 8. We assume that 𝑉∈(𝑅𝐻)_𝑞0 for some 𝑞₀, 𝑛/2≤𝑞₀<2^𝑚𝑛/(2^𝑚+1−𝑗). Then 𝑉∈(𝑅𝐻)_𝑞1 for some 𝑞₁, 2^𝑚𝑛/(2^𝑚+1−𝑗)>𝑞₁>𝑞₀. We use (62) and (63) again and it suffices to show (64) for 1/𝑝₀=(2^𝑚+2−𝑗)/(2𝑞₀)−(2^𝑚+1−𝑗)/𝑛 and 1/𝑝₀+1/𝑝^′₀=1. We choose 𝑡 and 𝑝₁ such that 1/𝑡=2^𝑚/𝑞₁−(2^𝑚+1−𝑗)/𝑛 and 1/𝑝₁=1−(2^𝑚+2−𝑗)/(2𝑞₁)+(2^𝑚+1−𝑗)/𝑛. Thus, 1𝑡+2^𝑚+1−𝑗2𝑞₁+1𝑝₁=1. (71) Let 𝑟=1/𝜌(𝑥,𝑉). By Hölder’s inequality we have 𝑇^∗₂^𝑚𝑓(𝑥≤_{∞𝑖=−∞})_{2𝑖−1𝑟<|𝑦−𝑥|≤2𝑖𝑟}∇^𝑗_𝑦Γ_𝐻2𝑚𝑦,𝑥^𝑡𝑑𝑦^1/𝑡⋅_{|𝑦−𝑥|≤2𝑖𝑟}𝑉𝑦^𝑞1𝑑𝑦^{(2𝑚+1−𝑗)/(2𝑞1)}×_{|𝑦−𝑥|≤2𝑖𝑟}𝑓𝑦^𝑝1𝑑𝑦^1/𝑝1. (72) It follows from Lemma 18 and Theorem 9 that _{2𝑖−1𝑟<|𝑦−𝑥|≤2𝑖𝑟}∇^𝑗_𝑦Γ_𝐻2𝑚𝑦,𝑥^𝑡𝑑𝑦^1/𝑡≤𝐶𝐶_𝑁2^𝑖𝑟^{2𝑚𝑛/𝑞1−𝑛}1+2^𝑖𝑁. (73) Thus, as in the proof of Theorem 5, we have 𝑇^∗₂^𝑚𝑓(𝑥)≤𝐶𝐶_𝑁𝑀𝑓^𝑝1(𝑥)^1/𝑝1⋅^∞_𝑖=−∞11+2^𝑖𝑁12^𝑖𝑟^𝑛−2𝐵₂^𝑖_𝑟^(𝑥)𝑉𝑦𝑑𝑦^{(2𝑚+1−𝑗)/2}≤𝐶𝑀𝑓^𝑝1(𝑥)^1/𝑝1. (74) Then (64) follows since 𝑝^′₀>𝑝₁.

7. Remarks

We show estimate (57) under another assumption.

Lemma 26. Let 𝑘, 𝑙, and 𝑛 be integers, 𝑚≥0, 𝑛≥2^𝑚+1+1, and 1≤𝑗≤2^𝑚+1−1. Suppose that 𝑉∈(𝑅𝐻)_𝑛/2 and there exists a positive constant 𝐶 such that 𝑉(𝑥)≤𝐶𝜌(𝑥,𝑉)². Assume also that (−Δ)^2𝑚𝑢+𝑉^2𝑚𝑢=0 in 𝐵_𝑅(𝑥₀) for some 𝑥₀∈𝐑^𝑛. Then there exist constants 𝐶_𝑗 and 𝐶^′_𝑗 such that sup_{𝑦∈𝐵𝑅/2𝑥0}∇^𝑗𝑢𝑦≤𝐶_𝑗1+𝑅𝜌𝑥₀,𝑉^𝐶′𝑗𝑅^𝑗sup_{𝑦∈𝐵𝑅𝑥0}𝑢𝑦. (75)

Proof. We use (58). From Lemma 12 (1) we have ∇^𝑗𝑢𝑥₀≤𝐶1+𝑅𝜌𝑥₀,𝑉^2𝑚+1𝑘0×𝜌𝑥₀,𝑉^2𝑚+1𝑅^{2𝑚+1−𝑗}sup_{𝑦∈𝐵𝑅𝑥0}𝑢𝑦+𝐶𝑅^𝑗1𝑅^𝑛_{𝐵𝑅(𝑥0)}|𝑢(𝑥)|²𝑑𝑥^1/2≤𝐶1+𝑅𝜌𝑥₀,𝑉^{2𝑚+1(𝑘0+1)}𝑅^𝑗sup_{𝑦∈𝐵𝑅𝑥0}𝑢𝑦. (76) Then by the same argument as in the proof of Lemma 25, we arrive at the desired estimate.

The following theorem immediately follows from Lemmas 22 and 26.

Theorem 27. Let 𝑚 and 𝑛 be integers, 𝑚≥0, 𝑛≥2^𝑚+1+1, and 1≤𝑗≤2^𝑚+1+1. Suppose 𝑉∈(𝑅𝐻)_𝑛/2 and there exists a positive constant 𝐶 such that 𝑉(𝑥)≤𝐶𝜌(𝑥,𝑉)². Then for any positive integer 𝑁 there exists a constant 𝐶_𝑁 such that ∇^𝑗_𝑥Γ_𝐻2𝑚≤𝐶𝑥,𝑦^𝑁1+𝜌(𝑥,𝑉)𝑥−𝑦^𝑁⋅1𝑥−𝑦^{𝑛−2𝑚+1+𝑗}. (77)

Using Theorem 27 we can prove the following theorem by the same way as the proof of Lemma 19.

Theorem 28. Let 𝑗, 𝑚, and 𝑛 be integers, 𝑚≥0, 𝑛≥2^𝑚+1+1, and 0≤𝑗≤2^𝑚+1−1. Suppose that 𝑉∈(𝑅𝐻)_𝑛/2 and there exists a positive constant 𝐶 such that 𝑉(𝑥)≤𝐶𝜌(𝑥,𝑉)². Then there exist positive constants 𝐶_𝑗 such that (6) holds, where 1≤𝑝≤∞.

We obtain the following pointwise estimate stated with the Hardy-Littlewood maximal operator.

Theorem 29. Let 𝑗, 𝑚, and 𝑛 be integers, 𝑚≥0, 𝑛≥2^𝑚+1+1, and 0≤𝑗≤2^𝑚+1−1. Suppose that 𝑉∈(𝑅𝐻)_𝑛/2 and there exists a positive constant 𝐶 such that 𝑉(𝑥)≤𝐶𝜌(𝑥,𝑉)². Then there exists a constant 𝐶^′ such that 𝑉^{2𝑚−𝑗/2}∇^𝑗𝐻₂^−1𝑚𝑓(𝑥)≤𝐶^′𝑀𝑓(𝑥). (78)

In Theorem 29, the cases 𝑚=0 and 𝑚=1 were shown in [5, Theorem 1.3] and [8, Lemma 10], respectively.

Proof of Theorem 29. Let 𝑟=1/𝜌(𝑥,𝑉). Since 𝑉(𝑥)≤𝐶𝜌(𝑥,𝑉)², it follows from Theorems 9 and 27 that 𝑉(𝑥)^{2𝑚−𝑗/2}∇^𝑗𝐻₂^−1𝑚𝑓(𝑥)≤𝐶𝐶_𝑁𝐑𝑛𝜌(𝑥,𝑉)^{2𝑚+1−𝑗}𝑓𝑦1+𝜌(𝑥,𝑉)𝑥−𝑦^𝑁𝑥−𝑦^{𝑛−2𝑚+1+𝑗}𝑑𝑦≤𝐶𝐶_𝑁^∞_{𝑖=−∞2𝑖−1𝑟<|𝑥−𝑦|≤2𝑖𝑟}×𝑓𝑦𝑑𝑦𝑟^{2𝑚+1−𝑗}1+𝑟⁻¹𝑥−𝑦^𝑁𝑥−𝑦^{𝑛−2𝑚+1+𝑗}≤𝐶𝐶_𝑁^∞_𝑖=−∞2^{𝑖−12𝑚+1−𝑗}1+2^𝑖−1𝑁⋅12^𝑖−1𝑟^{𝑛|𝑥−𝑦|≤2𝑖𝑟}𝑓𝑦𝑑𝑦≤𝐶𝐶_𝑁^∞_𝑖=−∞2^{𝑖2𝑚+1−𝑗}1+2^𝑖𝑁𝑀𝑓(𝑥). (79) Then choosing 𝑁≥2^𝑚+1−𝑗+1 we arrive at the desired estimate.

Remark 30 (see [5, Remark 1.6]). Once we obtain the estimate (78), we can obtain boundedness of the operator 𝑉^{2𝑚−𝑗/2}∇^𝑗𝐻₂^−1𝑚 on any space on which 𝑀 is bounded, for example, Morrey spaces (see [12]).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is grateful to the referee(s) for the helpful comment. The author is also grateful to Professor Kazuhiro Kurata, who read the first version of this paper and gave useful comment. This work was supported by Grant-in-Aid for Scientific Research (C) (no. 25400152), Japan Society for the Promotion of Science.