Volume 2008, Article ID 214030,8pages doi:10.1155/2008/214030
Research Article
Some Estimates of Schr ¨odinger-Type Operators
with Certain Nonnegative Potentials
Yu Liu1 and Youzheng Ding2
1Department of Mathematics and Mechanics, University of Science and Technology Beijing, Beijing 100083, China
2College of Science, Shandong Jianzhu University, Jinan 250101, Shandong Province, China
Correspondence should be addressed to Yu Liu,[email protected]
Received 4 September 2008; Revised 1 November 2008; Accepted 3 November 2008
Recommended by Jie Xiao
We consider the Schr ¨odinger-type operatorH −Δ2V2, where the nonnegative potentialV
belongs to the reverse H ¨older classBq1forq1≥n/2,n≥5. TheLpestimates of the operator∇4H−1
related toHare obtained whenV ∈Bq1and 1< p≤q1/2. We also obtain the weak-type estimates
of the operator∇4H−1under the same condition ofV.
Copyrightq2008 Y. Liu and Y. Ding. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
In recent years, there has been considerable activity in the study of Schr ¨odinger operators see1–4. In this paper, we consider the Schr ¨odinger-type operator
H −Δ2V2 onRn, n≥5, 1.1
where the potentialV belongs toBq1forq1≥n/2. We are interested in theLpboundedness of
the operator∇4H−1, where the potentialVsatisfies weaker condition than that in5, Theorem 1,2. The estimates of some other operators related to Schr ¨odinger-type operators can be found in2,5.
Note that a nonnegative locallyLq integrable function V onRn is said to belong to
Bq 1< q <∞if there existsC >0 such that the reverse H ¨older inequality
1
|B|
B
Vxqdx
1/q ≤C
1
|B|
B
Vxdx
1.2
It follows from3that theBq class has a property of “self-improvement”, that is, if
V ∈Bq, thenV ∈Bqεfor someε >0.
We now give the main results for the operator∇4H−1in this paper.
Theorem 1.1. SupposeV ∈Bq1,q1≥n/2. Then for 1< p≤q1/2 there exists a positive constantCp such that
∇4H−1f
LpRn≤CpfLpRn. 1.3
By the proof ofTheorem 1.1, we obtain the following weak-type estimate.
Theorem 1.2. SupposeV ∈Bq1,q1≥n/2. Then for 1< p≤q1/2 there exists a positive constantC1 such that
x∈Rn:∇4H−1fx≥λ≤ C1
λ fL1Rn. 1.4
Under a stronger condition on the potentialV, Sugano5has obtained the following proposition.
Proposition 1.3. SupposeV ∈ Bn/2 and there exists a constant Csuch thatVx ≤ Cmx, V2.
Then for 1< p <∞there exists a positive constantCpsuch that
∇4H−1f
LpRn≤CpfLpRn. 1.5
As a direct consequence of ourLpestimates, we have the following corollary.
Corollary 1.4. SupposeV ∈Bq1forq1≥n/2. Assume that−Δ
2uV2ufinRn. Then
∇4u
LpRn≤CpfLpRn for 1< p≤
q1
2 . 1.6
Throughout this paper, unless otherwise indicated, we will useCto denote constants, which are not necessarily the same at each occurrence. ByA∼B, we mean that there exist constantsC >0 andc >0 such thatc≤A/B≤C.
2. The auxiliary functionmx, Vand estimates of fundamental solution
In this section, we firstly recall the definition of the auxiliary function mx, V and some lemmas about the auxiliary functionmx, Vwhich have been proven in3.
Lemma 2.1. IfV ∈ Bq,q > 1, then the measureVxdxsatisfies the doubling condition, that is, there existsC >0 such that
Bx,2rVydy≤C
Bx,rVydy 2.1
Lemma 2.2. For 0< r < R <∞andV ∈Bq1forq1≥n/2, there existsC >0 such that
1 rn−2
Bx,rVydy≤C
r R
2−n/q1 1
Rn−2
Bx,RVydy. 2.2
Assume thatV ∈Bq1,q1≥n/2. The auxiliary functionmx, Vis defined by
1
mx, V˙supr>0
r: 1 rn−2
Bx,rVydy≤1
, x∈Rn. 2.3
Lemma 2.3. Ifr 1/mx, V, then
1 rn−2
Bx,rVydy1. 2.4
Moreover,
1 rn−2
Bx,rVydy∼1, iffr∼
1
mx, V. 2.5
Lemma 2.4. There existsl0>0 such that for anyxandyinRn,
1 C
1mx, V|x−y|−l0≤ mx, V
my, V ≤C
1mx, V|x−y|l0/l01
. 2.6
In particular,mx, V∼my, V, if|x−y|< C/mx, V.
Lemma 2.5. There existsl1>0 such that
Bx,R
Vy
|x−y|n−2dy≤ C Rn−2
Bx,RVydy≤C
1Rmx, Vl1
. 2.7
Lemma 2.6. There existsC >0,c >0, andl0>0 such that, for anyx, y∈Rn,
c1|x−y|my, V1/l01 ≤1|
x−y|mx, V
≤C1|x−y|my, Vl01
.
2.8
The next lemma has been obtained by Tao and Wang in6.
Lemma 2.7. Letq > s ≥ 0,q ≥ max{1, sn/α},α > 0, andkbe sufficiently large, then there are positive constantsk0, C, andCksuch that
|x−y|<r
Vys
|x−y|n−αdy≤Cr
α−2s1rmx, Vsk0
,
Rn
Vys
1mx, V|x−y|k|x−y|n−αdy≤Ckmx, V
2s−α
2.9
for anyr >0, x∈Rn, andV ∈B q.
In order to prove Theorem 1.1, we need to give the estimates of the fundamental solution ofH. Zhong has established the estimates of the fundamental solution ofHin2
whenVxis a nonnegative polynomial. Recently, Sugano5has obtained the polynomial decay estimates of the fundamental solution of H under a weaker condition onV in the following theorem.
Theorem 2.8. AssumeV ∈Bn/2and letΓHx, ybe the fundamental solution ofH. For any positive integerN, there exists a constantCNsuch that
0≤ΓHx, y≤ CN
1mx, V|x−y|N 1
|x−y|n−4. 2.10
3. Proof of the main results
In this section, we will prove Theorems1.1and1.2.
Theorem 3.1. SupposeV ∈Bq1,q1≥n/2. Then for 1< p≤q1/2 there exists a positive constantCp such that
V2H−1f
LpRn≤CpfLpRn. 3.1
Proof. Letf∈LpRnand
ux
RnΓHx, yfydy.
3.2
We need to show that
V2u
Write
ux
|x−y|<r
Γx, yfydy
|x−y|≥r
Γx, yfydy
u1x u2x,
3.4
wherer1/mx, V.
Because of the self-improvement of theBq1class,V ∈Bq0for someq0> q1, we have
u1x≤C
|x−y|<r
|fy|
|x−y|n−4dy
≤C
|x−y|<r
fyq0/2
dy
2/q0
|x−y|<r
|x−y|−n−4qdy
1/q
Cr4−2n/q0
|x−y|<r
fyq0/2
dy
2/q0
,
3.5
where 1/q 2/q01. Thus,
Rn
V2yu1yq0/2
dy
≤C
Rn
|x−y|<r
fyq0/2
dy
Vxq0mx, Vn−2q0dx
C
Rn
fyq0/2
|x−y|<1/mx,VVx
q0mx, Vn−2q0dx
dy.
3.6
Now, letR1/my, V. Then
|x−y|<1/mx,VVx
q0mx, Vn−2q0dx≤CR2q0−n
|x−y|<CR
Vxq0dx
≤CR2q0
R−n
|x−y|<CR
Vxdx
q0
C
1 Rn−2
|x−y|<CR
Vxdx
q0
≤C,
3.7
Hence, we have proved that for someq0> q1≥n/2,
Rn
V2xu1xq0/2
dx≤C
Rn
fxq0/2
dx. 3.8
By choosings2, α4, andr 1/mx, VinLemma 2.7, we immediately have
|x−y|<1/mx,V
V2x
|x−y|n−4dx≤4k0. 3.9
Thus,
Rn
V2xu1xdx≤C
Rn
fy
|x−y|<1/mx,V
V2x
|x−y|n−4dx
dy
≤Ck0
Rn
fydy. 3.10
Therefore, by using interpolation we have
V2u
1Lp1Rn≤CfLp1Rn for 1≤p1≤ q0
2. 3.11
Then we deal withu2.
For 1< p≤q0/2, by the H ¨older inequality,
u2x≤C
|x−y|≥r
|fy|dy
1|x−y|mx, VN|x−y|n−4
≤C
|x−y|≥r
|fy|pdy
1|x−y|mx, VN|x−y|n−4
1/p
×
|x−y|≥r
dy
1|x−y|mx, VN|x−y|n−4
1/p
Cr4/p
|x−y|≥r
|fy|pdy
1|x−y|mx, VN|x−y|n−4
1/p
,
3.12
Thus, for 1≤p≤q0/2,
Rn
V2xu2xp
dx
≤C
Rn
fyp
|x−y|≥1/mx,V
Vx2p
dx
mx, V4p−41|x−y|mx, VN|x−y|n−4
dy. 3.13
Fixy∈Rnand letR1/my, V. By Lemmas2.4,2.6, and2.7,
|x−y|≥1/mx,V
Vx2p
dx
mx, V4p−41|x−y|mx, VN|x−y|n−4
≤C
|x−y|≥1/mx,V
Vx2p
dx R4−4p1|x−y|R−1N1|
x−y|n−4 N1
N−4p−1l0 l01
≤Ck 1
R4−4pmy, V
4p−4
≤C
3.14
if we chooseNlarge enough. From this, we have
Rn
V2xu2xp
dx≤
Rn
fxp
dx for 1≤p≤ q0
2. 3.15
Thus the theorem is proved.
Now we give the proof ofTheorem 1.1.
Proof of Theorem1.1. SupposeV ∈Bq1for someq1≥n/2. ByTheorem 3.1, we have
V2−Δ2V2−1f
LpRn≤CfLpRn for 1≤p≤
q1
2 . 3.16
It follows that
−Δ2−Δ2
V2−1fLpRn≤CfLpRn for 1≤p≤
q1
2. 3.17
Because∇4−Δ−2is a Calder ´on-Zygmund operator, for 1< p≤q
1/2, we have
∇4−Δ2V2−1f
LpRn ≤Cp−Δ2
−Δ2V2−1f
Proof of Theorem1.2. Note that∇4−Δ−2satisfies
x∈Rn:∇4−Δ−2fx≥λ≤ C1
λ fL1Rn. 3.19
Thus, by the proof ofTheorem 1.1,
x∈Rn:∇4−Δ2
V2−1fx≥λ≤ C1 λ −Δ
2−Δ2
V2−1fL1Rn
≤ C1
λ fL1Rn.
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 FoundationNNSFof China under Grant no. 10726064.
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