# Some Estimates of Schrödinger-Type Operators with Certain Nonnegative Potentials

Authors: Yu Liu; Youzheng Ding
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

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## Abstract

We consider the Schrödinger-type operatorH=(−Δ)2+V2, where the nonnegative potential V belongs to the reverse Hölder class Bq1 for q1≥n/2,  n≥5. The Lp estimates of the operator ∇4H−1 related to H are obtained when V∈Bq1 and 1<p≤q1/2. We also obtain the weak-type estimates of the operator ∇4H−1 under the same condition of V.


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## Body

## 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 (1.1)H=(−Δ)2+V2onℝn,n≥5, where the potential V belongs to Bq1 for q1≥n/2.
We are interested in the Lp 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 locallyLq integrable function V on ℝn is said to belong to Bq(1<q<∞) if there exists C>0 such that the reverse Hölder
inequality (1.2)(1|B|∫BV(x)qdx)1/q≤C(1|B|∫BV(x)dx) holds for every ball B in ℝn.It follows from [3] that the Bq class has a property of
“self-improvement”, that is, if V∈Bq,
then V∈Bq+ε for some ε>0.We now give the main results for the operator∇4H−1 in this paper.Theorem 1.1.
SupposeV∈Bq1,q1≥n/2.
Then for 1<p≤q1/2 there exists a positive constant Cp such that (1.3)∥∇4H−1f∥Lp(ℝn)≤Cp∥f∥Lp(ℝn).By the proof of Theorem1.1, 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 constant C1 such that (1.4)|{x∈ℝn:|∇4H−1f(x)|≥λ}|≤C1λ∥f∥L1(ℝn).Under a stronger condition on the potentialV,
Sugano [5] has
obtained the following proposition.Proposition 1.3.
SupposeV∈Bn/2 and there exists a constant C such that V(x)≤Cm(x,V)2.
Then for 1<p<∞ there exists a positive constant Cp such that (1.5)∥∇4H−1f∥Lp(ℝn)≤Cp∥f∥Lp(ℝn).As a direct consequence of ourLp estimates,  we have the following
corollary.Corollary 1.4.
SupposeV∈Bq1 for q1≥n/2.
Assume that (−Δ)2u+V2u=f in ℝn.
Then (1.6)∥∇4u∥Lp(ℝn)≤Cp∥f∥Lp(ℝn)for1<p≤q12.Throughout this paper, unless otherwise indicated, we
will useC 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 Functionm(x,V) and Estimates of Fundamental Solution

In this section, we firstly recall the definition of
the auxiliary functionm(x,V) and some lemmas about the auxiliary function m(x,V) which have been proven in [3].Lemma 2.1.
IfV∈Bq, q>1,
then the measure V(x)dx satisfies the doubling condition, that is,
there exists C>0 such that (2.1)∫B(x,2r)V(y)dy≤C∫B(x,r)V(y)dy holds for all balls B(x,r) in ℝn.Lemma 2.2.
For0<r<R<∞ and V∈Bq1 for q1≥n/2,
there exists C>0 such that (2.2)1rn−2∫B(x,r)V(y)dy≤C(rR)2−n/q11Rn−2∫B(x,R)V(y)dy.Assume thatV∈Bq1, q1≥n/2.
The auxiliary function m(x,V) is defined by (2.3)1m(x,V)=˙supr>0{r:1rn−2∫B(x,r)V(y)dy≤1},x∈ℝn.Lemma 2.3.
Ifr=1/m(x,V),
then (2.4)1rn−2∫B(x,r)V(y)dy=1. Moreover,  (2.5)1rn−2∫B(x,r)V(y)dy~1,iffr~1m(x,V).Lemma 2.4.
There existsl0>0 such that for any x and y in ℝn,(2.6)1C(1+m(x,V)|x−y|)−l0≤m(x,V)m(y,V)≤C(1+m(x,V)|x−y|)l0/(l0+1). In particular, m(x,V)~m(y,V),
if |x−y|<C/m(x,V).Lemma 2.5.
There existsl1>0 such that (2.7)∫B(x,R)V(y)|x−y|n−2dy≤CRn−2∫B(x,R)V(y)dy≤C(1+Rm(x,V))l1.Lemma 2.6.
There existsC>0,c>0,
and l0>0 such that, for any x,y∈ℝn,(2.8)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.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.
Letq>s≥0,q≥max{1,sn/α},α>0,
and k be sufficiently
large, then there are positive constants k0,C,
and Ck such that (2.9)∫|x−y|<rV(y)s|x−y|n−αdy≤Crα−2s{1+rm(x,V)}sk0,∫ℝnV(y)s{1+m(x,V)|x−y|}k|x−y|n−αdy≤Ckm(x,V)2s−α for any r>0,x∈ℝn,
and V∈Bq.In order to prove Theorem1.1, 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.
AssumeV∈Bn/2 and let ΓH(x,y) be the fundamental solution of H.
For any positive integer N,
there exists a constant CN such that (2.10)0≤ΓH(x,y)≤CN(1+m(x,V)|x−y|)N1|x−y|n−4.


## 3. Proof of the Main Results

In this
section, we will prove Theorems1.1 and 1.2.Theorem 3.1.
SupposeV∈Bq1,q1≥n/2.
Then for 1<p≤q1/2 there exists a positive constant Cp such that (3.1)∥V2H−1f∥Lp(ℝn)≤Cp∥f∥Lp(ℝn).Proof.
Letf∈Lp(ℝn) and (3.2)u(x)=∫ℝnΓH(x,y)f(y)dy. We need to show
that (3.3)∥V2u∥Lp(ℝn)≤C∥f∥Lp(ℝn). Write (3.4)u(x)=∫|x−y|<rΓ(x,y)f(y)dy+∫|x−y|≥rΓ(x,y)f(y)dy=u1(x)+u2(x), where r=1/m(x,V).
Because of the self-improvement of theBq1 class, V∈Bq0 for some q0>q1,
we have (3.5)|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′=Cr4−2n/q0(∫|x−y|<r|f(y)|q0/2dy)2/q0, where 1/q′+2/q0=1.
Thus,(3.6)∫ℝn|V2(y)u1(y)|q0/2dy≤C∫ℝn(∫|x−y|<r|f(y)|q0/2dy)V(x)q0m(x,V)n−2q0dx=C∫ℝn|f(y)|q0/2(∫|x−y|<1/m(x,V)V(x)q0m(x,V)n−2q0dx)dy. Now, let R=1/m(y,V).
Then (3.7)∫|x−y|<1/m(x,V)V(x)q0m(x,V)n−2q0dx≤CR2q0−n∫|x−y|<CRV(x)q0dx≤CR2q0(R−n∫|x−y|<CRV(x)dx)q0=C(1Rn−2∫|x−y|<CRV(x)dx)q0≤C, where we used (1.2),
Lemmas 2.3 and 2.4.
Hence, we have proved that for someq0>q1≥n/2, (3.8)∫ℝn|V2(x)u1(x)|q0/2dx≤C∫ℝn|f(x)|q0/2dx.By choosing s=2,α=4,
and r=1/m(x,V) in Lemma 2.7, we immediately
have (3.9)∫|x−y|<1/m(x,V)V2(x)|x−y|n−4dx≤4k0. Thus, (3.10)∫ℝn|V2(x)u1(x)|dx≤C∫ℝn|f(y)|(∫|x−y|<1/m(x,V)V2(x)|x−y|n−4dx)dy≤Ck0∫ℝn|f(y)|dy. Therefore, by using
interpolation we have (3.11)∥V2u1∥Lp1(ℝn)≤C∥f∥Lp1(ℝn)for1≤p1≤q02. Then we deal with u2.
For1<p≤q0/2,
by the Hölder  inequality, (3.12)|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|≥rdy(1+|x−y|m(x,V))N|x−y|n−4)1/p′=Cr4/p′(∫|x−y|≥r|f(y)|pdy(1+|x−y|m(x,V))N|x−y|n−4)1/p, 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, for1≤p≤q0/2, (3.13)∫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. Fix y∈ℝn and let R=1/m(y,V).
By Lemmas 2.4, 2.6, and 2.7, (3.14)∫|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)|2pdxR4−4p(1+|x−y|R−1)N1|x−y|n−4(N1=N−4(p−1)l0l0+1)≤Ck1R4−4pm(y,V)4p−4≤C if we choose N large enough.
From this, we have(3.15)∫ℝn|V2(x)u2(x)|pdx≤∫ℝn|f(x)|pdxfor1≤p≤q02. Thus the theorem is proved.Now we give the proof of Theorem1.1.Proof of Theorem1.1.
SupposeV∈Bq1 for some q1≥n/2.
By Theorem 3.1, we have(3.16)∥V2((−Δ)2+V2)−1f∥Lp(ℝn)≤C∥f∥Lp(ℝn)for1≤p≤q12. It follows that (3.17)∥(−Δ)2((−Δ)2+V2)−1f∥Lp(ℝn)≤C∥f∥Lp(ℝn)for1≤p≤q12. Because ∇4(−Δ)−2 is a Calderón-Zygmund operator, for 1<p≤q1/2,
we have (3.18)∥∇4((−Δ)2+V2)−1f∥Lp(ℝn)≤Cp∥(−Δ)2((−Δ)2+V2)−1f∥Lp(ℝn)≤Cp∥f∥Lp(ℝn).Proof of Theorem1.2.
Note that∇4(−Δ)−2 satisfies (3.19)|{x∈ℝn:|∇4(−Δ)−2f(x)|≥λ}|≤C1λ∥f∥L1(ℝn).
Thus, by  the proof of Theorem1.1, (3.20)|{x∈ℝn:|∇4((−Δ)2+V2)−1f(x)|≥λ}|≤C1λ∥(−Δ)2((−Δ)2+V2)−1f∥L1(ℝn)≤C1λ∥f∥L1(ℝn).



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