Some Estimates of Schrödinger-Type Operators with Certain Nonnegative Potentials

Abstract

We consider the Schrödinger-type operator 𝐻=(−Δ)²+𝑉², where the nonnegative potential 𝑉 belongs to the reverse Hölder class 𝐵_𝑞1 for 𝑞₁≥𝑛/2,  𝑛≥5. The 𝐿^𝑝 estimates of the operator ∇⁴𝐻⁻¹ related to 𝐻 are obtained when 𝑉∈𝐵_𝑞1 and 1<𝑝≤𝑞₁/2. We also obtain the weak-type estimates of the operator ∇⁴𝐻⁻¹ under the same condition of 𝑉.

1. Introduction

In recent years, there has been considerable activity in the study of Schrödinger operators (see [1–4]). In this paper, we consider the Schrödinger-type operator 𝐻=(−Δ)²+𝑉²onℝ^𝑛,𝑛≥5, (1.1) where the potential 𝑉 belongs to 𝐵_𝑞1 for 𝑞₁≥𝑛/2. We are interested in the 𝐿^𝑝 boundedness of the operator ∇⁴𝐻⁻¹, where the potential 𝑉 satisfies weaker condition than that in [5, Theorem 1, (2)]. The estimates of some other operators related to Schrödinger-type operators can be found in [2, 5].

Note that a nonnegative locally 𝐿^𝑞 integrable function 𝑉 on ℝ^𝑛 is said to belong to 𝐵_𝑞(1<𝑞<∞) if there exists 𝐶>0 such that the reverse Hölder inequality (1|𝐵|_𝐵𝑉(𝑥)^𝑞𝑑𝑥)^1/𝑞1≤𝐶(|𝐵|_𝐵𝑉(𝑥)𝑑𝑥) (1.2) holds for every ball 𝐵 in ℝ^𝑛.

It follows from [3] that the 𝐵_𝑞 class has a property of “self-improvement”, that is, if 𝑉∈𝐵_𝑞, then 𝑉∈𝐵_𝑞+𝜀 for some 𝜀>0.

We now give the main results for the operator ∇⁴𝐻⁻¹ in this paper.

Theorem 1.1. Suppose 𝑉∈𝐵_𝑞1,𝑞₁≥𝑛/2. Then for 1<𝑝≤𝑞₁/2 there exists a positive constant 𝐶_𝑝 such that ‖∇⁴𝐻⁻¹𝑓‖_𝐿^𝑝_(ℝ^𝑛₎≤𝐶_𝑝‖𝑓‖_𝐿^𝑝_(ℝ^𝑛₎. (1.3)

By the proof of Theorem 1.1, we obtain the following weak-type estimate.

Theorem 1.2. Suppose 𝑉∈𝐵_𝑞1, 𝑞₁≥𝑛/2. Then for 1<𝑝≤𝑞₁/2 there exists a positive constant 𝐶₁ such that |{𝑥∈ℝ^𝑛∶|∇⁴𝐻⁻¹𝐶𝑓(𝑥)|≥𝜆}|≤¹𝜆‖𝑓‖_𝐿¹_(ℝ^𝑛₎. (1.4)

Under a stronger condition on the potential 𝑉, Sugano [5] has obtained the following proposition.

Proposition 1.3. Suppose 𝑉∈𝐵_𝑛/2 and there exists a constant 𝐶 such that 𝑉(𝑥)≤𝐶𝑚(𝑥,𝑉)². Then for 1<𝑝<∞ there exists a positive constant 𝐶_𝑝 such that ‖∇⁴𝐻⁻¹𝑓‖_𝐿^𝑝_(ℝ^𝑛₎≤𝐶_𝑝‖𝑓‖_𝐿^𝑝_(ℝ^𝑛₎. (1.5)

As a direct consequence of our 𝐿^𝑝 estimates, we have the following corollary.

Corollary 1.4. Suppose 𝑉∈𝐵_𝑞1 for 𝑞₁≥𝑛/2. Assume that (−Δ)²𝑢+𝑉²𝑢=𝑓 in ℝ^𝑛. Then ‖∇⁴𝑢‖_𝐿^𝑝_(ℝ^𝑛₎≤𝐶_𝑝‖𝑓‖_𝐿^𝑝_(ℝ^𝑛₎𝑞for1<𝑝≤¹2. (1.6)

Throughout this paper, unless otherwise indicated, we will use 𝐶 to denote constants, which are not necessarily the same at each occurrence. By 𝐴∼𝐵, we mean that there exist constants 𝐶>0 and 𝑐>0 such that 𝑐≤𝐴/𝐵≤𝐶.

2. The Auxiliary Function 𝑚(𝑥,𝑉) and Estimates of Fundamental Solution

In this section, we firstly recall the definition of the auxiliary function 𝑚(𝑥,𝑉) and some lemmas about the auxiliary function 𝑚(𝑥,𝑉) which have been proven in [3].

Lemma 2.1. If 𝑉∈𝐵_𝑞, 𝑞>1, then the measure 𝑉(𝑥)𝑑𝑥 satisfies the doubling condition, that is, there exists 𝐶>0 such that _{𝐵(𝑥,2𝑟)}𝑉(𝑦)𝑑𝑦≤𝐶_{𝐵(𝑥,𝑟)}𝑉(𝑦)𝑑𝑦 (2.1) holds for all balls 𝐵(𝑥,𝑟) in ℝ^𝑛.

Lemma 2.2. For 0<𝑟<𝑅<∞ and 𝑉∈𝐵_𝑞1 for 𝑞₁≥𝑛/2, there exists 𝐶>0 such that 1𝑟^𝑛−2_{𝐵(𝑥,𝑟)}𝑟𝑉(𝑦)𝑑𝑦≤𝐶(𝑅)^{2−𝑛/𝑞1}1𝑅^𝑛−2_{𝐵(𝑥,𝑅)}𝑉(𝑦)𝑑𝑦. (2.2)

Assume that 𝑉∈𝐵_𝑞1, 𝑞₁≥𝑛/2. The auxiliary function 𝑚(𝑥,𝑉) is defined by 1𝑚(𝑥,𝑉)=̇sup_𝑟>01{𝑟∶𝑟^𝑛−2_{𝐵(𝑥,𝑟)}𝑉(𝑦)𝑑𝑦≤1},𝑥∈ℝ^𝑛. (2.3)

Lemma 2.3. If 𝑟=1/𝑚(𝑥,𝑉), then 1𝑟^𝑛−2_{𝐵(𝑥,𝑟)}𝑉(𝑦)𝑑𝑦=1. (2.4) Moreover, 1𝑟^𝑛−2_{𝐵(𝑥,𝑟)}1𝑉(𝑦)𝑑𝑦∼1,iff𝑟∼𝑚(𝑥,𝑉). (2.5)

Lemma 2.4. There exists 𝑙₀>0 such that for any 𝑥 and 𝑦 in ℝ^𝑛,1𝐶(1+𝑚(𝑥,𝑉)|𝑥−𝑦|)^−𝑙0≤𝑚(𝑥,𝑉)𝑚(𝑦,𝑉)≤𝐶(1+𝑚(𝑥,𝑉)|𝑥−𝑦|)^{𝑙0/(𝑙0+1)}. (2.6) In particular, 𝑚(𝑥,𝑉)∼𝑚(𝑦,𝑉), if |𝑥−𝑦|<𝐶/𝑚(𝑥,𝑉).

Lemma 2.5. There exists 𝑙₁>0 such that _{𝐵(𝑥,𝑅)}𝑉(𝑦)|𝑥−𝑦|^𝑛−2𝐶𝑑𝑦≤𝑅^𝑛−2_{𝐵(𝑥,𝑅)}𝑉(𝑦)𝑑𝑦≤𝐶(1+𝑅𝑚(𝑥,𝑉))^𝑙1. (2.7)

Lemma 2.6. There exists 𝐶>0,𝑐>0, and 𝑙₀>0 such that, for any 𝑥,𝑦∈ℝ^𝑛,𝑐{1+|𝑥−𝑦|𝑚(𝑦,𝑉)}^1/(𝑙0+1)≤1+|𝑥−𝑦|𝑚(𝑥,𝑉)≤𝐶{1+|𝑥−𝑦|𝑚(𝑦,𝑉)}^𝑙0+1. (2.8)

Refer to [3] for the proof of the above lemmas.

The next lemma has been obtained by Tao and Wang in [6].

Lemma 2.7. Let 𝑞>𝑠≥0,𝑞≥max{1,𝑠𝑛/𝛼},𝛼>0, and 𝑘 be sufficiently large, then there are positive constants 𝑘₀,𝐶, and 𝐶_𝑘 such that ∫_{|𝑥−𝑦|<𝑟}𝑉(𝑦)^𝑠|𝑥−𝑦|^𝑛−𝛼𝑑𝑦≤𝐶𝑟^𝛼−2𝑠{1+𝑟𝑚(𝑥,𝑉)}^𝑠𝑘0,∫_ℝ𝑛𝑉(𝑦)^𝑠{1+𝑚(𝑥,𝑉)|𝑥−𝑦|}^𝑘|𝑥−𝑦|^𝑛−𝛼𝑑𝑦≤𝐶_𝑘𝑚(𝑥,𝑉)^2𝑠−𝛼 (2.9) for any 𝑟>0,𝑥∈ℝ^𝑛, and 𝑉∈𝐵_𝑞.

In order to prove Theorem 1.1, we need to give the estimates of the fundamental solution of 𝐻. Zhong has established the estimates of the fundamental solution of 𝐻 in [2] when 𝑉(𝑥) is a nonnegative polynomial. Recently, Sugano [5] has obtained the polynomial decay estimates of the fundamental solution of 𝐻 under a weaker condition on 𝑉 in the following theorem.

Theorem 2.8. Assume 𝑉∈𝐵_𝑛/2 and let Γ_𝐻(𝑥,𝑦) be the fundamental solution of 𝐻. For any positive integer 𝑁, there exists a constant 𝐶_𝑁 such that 0≤Γ_𝐻𝐶(𝑥,𝑦)≤^𝑁(1+𝑚(𝑥,𝑉)|𝑥−𝑦|)^𝑁1|𝑥−𝑦|^𝑛−4. (2.10)

3. Proof of the Main Results

In this section, we will prove Theorems 1.1 and 1.2.

Theorem 3.1. Suppose 𝑉∈𝐵_𝑞1,𝑞₁≥𝑛/2. Then for 1<𝑝≤𝑞₁/2 there exists a positive constant 𝐶_𝑝 such that ‖𝑉²𝐻⁻¹𝑓‖_𝐿^𝑝_(ℝ^𝑛₎≤𝐶_𝑝‖𝑓‖_𝐿^𝑝_(ℝ^𝑛₎. (3.1)

Proof. Let 𝑓∈𝐿^𝑝(ℝ^𝑛) and 𝑢(𝑥)=_ℝ𝑛Γ_𝐻(𝑥,𝑦)𝑓(𝑦)𝑑𝑦. (3.2) We need to show that ‖𝑉²𝑢‖_𝐿^𝑝_(ℝ^𝑛₎≤𝐶‖𝑓‖_𝐿^𝑝_(ℝ^𝑛₎. (3.3) Write 𝑢(𝑥)=_{|𝑥−𝑦|<𝑟}Γ(𝑥,𝑦)𝑓(𝑦)𝑑𝑦+_{|𝑥−𝑦|≥𝑟}Γ(𝑥,𝑦)𝑓(𝑦)𝑑𝑦=𝑢₁(𝑥)+𝑢₂(𝑥), (3.4) where 𝑟=1/𝑚(𝑥,𝑉).
Because of the self-improvement of the 𝐵_𝑞1 class, 𝑉∈𝐵_𝑞0 for some 𝑞₀>𝑞₁, we have |𝑢₁(𝑥)|≤𝐶_{|𝑥−𝑦|<𝑟}|𝑓(𝑦)||𝑥−𝑦|^𝑛−4𝑑𝑦≤𝐶(_{|𝑥−𝑦|<𝑟}|𝑓(𝑦)|^𝑞0/2𝑑𝑦)^2/𝑞0(_{|𝑥−𝑦|<𝑟}|𝑥−𝑦|^{−(𝑛−4)𝑞′}𝑑𝑦)^1/𝑞′=𝐶𝑟^{4−2𝑛/𝑞0}(_{|𝑥−𝑦|<𝑟}|𝑓(𝑦)|^𝑞0/2𝑑𝑦)^2/𝑞0, (3.5) where 1/𝑞^′+2/𝑞₀=1.
Thus, _ℝ𝑛|𝑉²(𝑦)𝑢₁(𝑦)|^𝑞0/2𝑑𝑦≤𝐶_ℝ𝑛(_{|𝑥−𝑦|<𝑟}|𝑓(𝑦)|^𝑞0/2𝑑𝑦)𝑉(𝑥)^𝑞0𝑚(𝑥,𝑉)^{𝑛−2𝑞0}𝑑𝑥=𝐶_ℝ𝑛|𝑓(𝑦)|^𝑞0/2(_{|𝑥−𝑦|<1/𝑚(𝑥,𝑉)}𝑉(𝑥)^𝑞0𝑚(𝑥,𝑉)^{𝑛−2𝑞0}𝑑𝑥)𝑑𝑦. (3.6) Now, let 𝑅=1/𝑚(𝑦,𝑉). Then _{|𝑥−𝑦|<1/𝑚(𝑥,𝑉)}𝑉(𝑥)^𝑞0𝑚(𝑥,𝑉)^{𝑛−2𝑞0}𝑑𝑥≤𝐶𝑅^{2𝑞0−𝑛}_{|𝑥−𝑦|<𝐶𝑅}𝑉(𝑥)^𝑞0𝑑𝑥≤𝐶𝑅^2𝑞0(𝑅^−𝑛_{|𝑥−𝑦|<𝐶𝑅}𝑉(𝑥)𝑑𝑥)^𝑞01=𝐶(𝑅^𝑛−2_{|𝑥−𝑦|<𝐶𝑅}𝑉(𝑥)𝑑𝑥)^𝑞0≤𝐶, (3.7) where we used (1.2), Lemmas 2.3 and 2.4.
Hence, we have proved that for some 𝑞₀>𝑞₁≥𝑛/2, _ℝ𝑛|𝑉²(𝑥)𝑢₁(𝑥)|^𝑞0/2𝑑𝑥≤𝐶_ℝ𝑛|𝑓(𝑥)|^𝑞0/2𝑑𝑥. (3.8)By choosing 𝑠=2,𝛼=4, and 𝑟=1/𝑚(𝑥,𝑉) in Lemma 2.7, we immediately have _{|𝑥−𝑦|<1/𝑚(𝑥,𝑉)}𝑉²(𝑥)|𝑥−𝑦|^𝑛−4𝑑𝑥≤4^𝑘0. (3.9) Thus, _ℝ𝑛|𝑉²(𝑥)𝑢₁(𝑥)|𝑑𝑥≤𝐶_ℝ𝑛|𝑓(𝑦)|(_{|𝑥−𝑦|<1/𝑚(𝑥,𝑉)}𝑉²(𝑥)|𝑥−𝑦|^𝑛−4𝑑𝑥)𝑑𝑦≤𝐶_𝑘0ℝ𝑛|𝑓(𝑦)|𝑑𝑦. (3.10) Therefore, by using interpolation we have ‖𝑉²𝑢₁‖_𝐿^𝑝_1(ℝ^𝑛₎≤𝐶‖𝑓‖_𝐿^𝑝_1(ℝ^𝑛₎for1≤𝑝₁≤𝑞₀2. (3.11) Then we deal with 𝑢₂.
For 1<𝑝≤𝑞₀/2, by the Hölder inequality, |𝑢₂(𝑥)|≤𝐶_{|𝑥−𝑦|≥𝑟}|𝑓(𝑦)|𝑑𝑦(1+|𝑥−𝑦|𝑚(𝑥,𝑉))^𝑁|𝑥−𝑦|^𝑛−4≤𝐶(_{|𝑥−𝑦|≥𝑟}|𝑓(𝑦)|^𝑝𝑑𝑦(1+|𝑥−𝑦|𝑚(𝑥,𝑉))^𝑁|𝑥−𝑦|^𝑛−4)^1/𝑝×(_{|𝑥−𝑦|≥𝑟}𝑑𝑦(1+|𝑥−𝑦|𝑚(𝑥,𝑉))^𝑁|𝑥−𝑦|^𝑛−4)^1/𝑝′=𝐶𝑟^4/𝑝′(_{|𝑥−𝑦|≥𝑟}|𝑓(𝑦)|^𝑝𝑑𝑦(1+|𝑥−𝑦|𝑚(𝑥,𝑉))^𝑁|𝑥−𝑦|^𝑛−4)^1/𝑝, (3.12) where 𝑟=1/𝑚(𝑥,𝑉) and we apply the second inequality for 𝑠=0 and 𝛼=4 in Lemma 2.7 to the last step.
Thus, for 1≤𝑝≤𝑞₀/2, _ℝ𝑛|𝑉²(𝑥)𝑢₂(𝑥)|^𝑝𝑑𝑥≤𝐶_ℝ𝑛|𝑓(𝑦)|^𝑝(_{|𝑥−𝑦|≥1/𝑚(𝑥,𝑉)}|𝑉(𝑥)|^2𝑝𝑑𝑥𝑚(𝑥,𝑉)^4𝑝−4(1+|𝑥−𝑦|𝑚(𝑥,𝑉))^𝑁|𝑥−𝑦|^𝑛−4)𝑑𝑦. (3.13) Fix 𝑦∈ℝ^𝑛 and let 𝑅=1/𝑚(𝑦,𝑉). By Lemmas 2.4, 2.6, and 2.7, _{|𝑥−𝑦|≥1/𝑚(𝑥,𝑉)}|𝑉(𝑥)|^2𝑝𝑑𝑥𝑚(𝑥,𝑉)^4𝑝−4(1+|𝑥−𝑦|𝑚(𝑥,𝑉))^𝑁|𝑥−𝑦|^𝑛−4≤𝐶_{|𝑥−𝑦|≥1/𝑚(𝑥,𝑉)}|𝑉(𝑥)|^2𝑝𝑑𝑥𝑅^4−4𝑝(1+|𝑥−𝑦|𝑅⁻¹)^𝑁1|𝑥−𝑦|^𝑛−4(𝑁₁=𝑁−4(𝑝−1)𝑙₀𝑙₀)+1≤𝐶_𝑘1𝑅^4−4𝑝𝑚(𝑦,𝑉)^4𝑝−4≤𝐶 (3.14) if we choose 𝑁 large enough.
From this, we have _ℝ𝑛|𝑉²(𝑥)𝑢₂(𝑥)|^𝑝𝑑𝑥≤_ℝ𝑛|𝑓(𝑥)|^𝑝𝑞𝑑𝑥for1≤𝑝≤⁰2. (3.15) Thus the theorem is proved.

Now we give the proof of Theorem 1.1.

Proof of Theorem 1.1. Suppose 𝑉∈𝐵_𝑞1 for some 𝑞₁≥𝑛/2. By Theorem 3.1, we have‖𝑉²((−Δ)²+𝑉²)⁻¹𝑓‖_𝐿^𝑝_(ℝ^𝑛₎≤𝐶‖𝑓‖_𝐿^𝑝_(ℝ^𝑛₎𝑞for1≤𝑝≤¹2. (3.16) It follows that ‖(−Δ)²((−Δ)²+𝑉²)⁻¹𝑓‖_𝐿^𝑝_(ℝ^𝑛₎≤𝐶‖𝑓‖_𝐿^𝑝_(ℝ^𝑛₎𝑞for1≤𝑝≤¹2. (3.17) Because ∇⁴(−Δ)⁻² is a Calderón-Zygmund operator, for 1<𝑝≤𝑞₁/2, we have ‖∇⁴((−Δ)²+𝑉²)⁻¹𝑓‖_𝐿^𝑝_(ℝ^𝑛₎≤𝐶_𝑝‖(−Δ)²((−Δ)²+𝑉²)⁻¹𝑓‖_𝐿^𝑝_(ℝ^𝑛₎≤𝐶_𝑝‖𝑓‖_𝐿^𝑝_(ℝ^𝑛₎. (3.18)

Proof of Theorem 1.2. Note that ∇⁴(−Δ)⁻² satisfies |{𝑥∈ℝ^𝑛∶|∇⁴(−Δ)⁻²𝐶𝑓(𝑥)|≥𝜆}|≤¹𝜆‖𝑓‖_𝐿¹_(ℝ^𝑛₎. (3.19)
Thus, by the proof of Theorem 1.1, |{𝑥∈ℝ^𝑛∶|∇⁴((−Δ)²+𝑉²)⁻¹𝐶𝑓(𝑥)|≥𝜆}|≤¹𝜆‖(−Δ)²((−Δ)²+𝑉²)⁻¹𝑓‖_𝐿¹_(ℝ^𝑛₎≤𝐶₁𝜆‖𝑓‖_𝐿¹_(ℝ^𝑛₎. (3.20)

Acknowledgments

I would like to express my gratitude to the referee for the valuable comments. This work is supported by the Tian Yuan Project of the National Natural Science Foundation (NNSF) of China under Grant no. 10726064.