Fourth-order Schr\"odinger type operator with singular potentials

0.17 MB·

by  Federica Gregorio

FOURTH-ORDER SCHRO¨DINGER TYPE OPERATOR WITH SINGULAR
POTENTIALS
FEDERICAGREGORIOANDSEBASTIANMILDNER
Abstract. In this paper, we study the biharmonic operator perturbed by an inverse
fourth-order potential. In particular, we consider the operator A=∆2−c|x|−4 where c
6 is any constant such that c < (cid:0)N(N−4)(cid:1)2. The semigroup generated by −A in L2(RN),
4
1 N ≥5, extrapolates to a bounded holomorphic C0-semigroup on Lp(RN) for p∈[p′0,p0]
20 wRhieesrzetpr0an=sfoN2rm−N4∆aAnd−1p/′02isonitsLdpu(RalNe)xpfoornaenllt.pM∈o(pre′o,v2e].r,weobtaintheboundednessofthe
0
y
a
M
1. Introduction
6
1
Let us consider the biharmonic operator

] A =∆2.
P 0
A In this paper, we want to study the perturbation of A with the singular potential V(x) =
0
. c|x|−4 a.e. where c<C∗ := N(N−4) 2. More precisely, we consider the operator
h 4
at (cid:0) A=(cid:1)A −V =∆2− c .
m 0 |x|4
ThebiharmonicoperatorA isincludedinaclassofhigherorderelliptic operatorsstudied
[ 0
by Davies in [7]. In particular, he proves that for N < 4, (e−tA0)t≥0 induces a bounded
2 C -semigroup on Lp(RN) for all 1 ≤ p < ∞ and that Gaussian-type estimates for the heat
v 0
kernel hold. Denoting by K the heat kernel associated to the operator A , he proves that
3 0
4 there exist c1,c2,k >0 such that
52 |K(t,x,y)|≤c1t−N/4e−c2|x−t1y/|34/3+kt
0
. for all t>0 and x,y ∈RN.
1
The result is different for N > 4. In this case he proves that the semigroup (e−tA0)
0 t≥0
extends to a bounded holomorphic C -semigroup on Lp(RN) for all p ∈ [p′,p ], where p =
6 0 0 0 0
1 N2N−4 andp′0 isitsdualexponent. AnanalogoussituationholdswhenreplacingA0 byA,which
: wasremarkedforexamplein[9,Section6]byLiskevich,SobolandVogt(seealsoRemark3.6
v
i and Proposition 3.7 below).
X
We can define the Riesz transform associated to A by
r
a 1 ∞
∆A−1/2 := t−1/2∆e−tAdt.
Γ(1/2)
Z0
2010 Mathematics Subject Classification. 47D06,35J10, 42B20,47F05.
Key words and phrases. Singular potentials, holomorphic C0-semigroup, biharmonic operator, Rellich
inequality,sesquilinearforms,off-diagonal estimates,Riesztransforms.
The first author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilita` e le loro
Applicazioni(GNAMPA)oftheIstitutoNazionalediAltaMatematica(INdAM).
1
2 F.GREGORIOANDS.MILDNER
TheboundednessoftheRiesztransformonLp(RN)impliesthatthedomainofA1/2isincluded
in the Sobolev space W2,p(RN). Thus, we obtain W2,p-regularity of the solution to the
evolution equation with initial datum in Lp(RN). The boundedness of the Riesz transforms
forSchr¨odingeroperatorshaswidelybeenstudiedinharmonicanalysis. Severalauthorshave
generalized the results for elliptic operators L of order 2m or for Riemannian manifolds, see
for example [4], [3], [6] and the references therein. Blunck and Kunstmann in [6] apply the
Caldero´n-Zygmundtheoryfornon-integraloperatorsto obtainestimates on∆L−1/2 since, in
general, operators of order 2m do not satisfy Gaussian bounds if 2m < N. More precisely,
they prove an abstract criterion for estimates of the type
kBL−αfk ≤C kfk , p∈(q ,2],
Lp(Ω) p Lp(Ω) 0
whereB,Larelinearoperators,α∈[0,1),q ∈[1,2)andΩisameasurespace. Wewillapply
0
this criterion (Theorem 4.1 below) to our situation (B,L,Ω)=(∆,A,RN) with q =p′.
0 0
We will treat the operator A=∆2−V in L2(RN) as the associated operator to the form
a(u,v)=(∆u,∆v) −(Vu,v)
2 2
with D(a)={u∈H2(RN):k|V|1/2uk <∞}. As a consequence of the Rellich inequality,
2
N(N −4) 2 |u(x)|2
dx≤ |∆u(x)|2dx (1.1)
(cid:16) 4 (cid:17) ZRN |x|4 ZRN
for all u∈H2(RN) with N ≥5 (cf. [12]), one obtains D(a)=H2(RN) and
a(u):= |∆u(x)|2dx− V(x)|u(x)|2dx≥η |∆u(x)|2dx (1.2)
ZRN ZRN ZRN
for some η ∈ (0,1), i.e., a is densely defined and positive semi-definite. Moreover, thanks to
(1.1)and(1.2), the normsk·ka andk·kH2 areequivalent,therefore(D(a),k·ka)iscomplete,
i.e., a is closed. Consequently, see for example [10, Theorem 1.54], −A is the generator of a
C -semigroup(e−tA) onL2(RN)thatiscontractiveandholomorphiconthesectorΣ(π/2).
0 t≥0
In the following, making use of multiplication operators and off-diagonal estimates, we
prove that, for N ≥ 5, the semigroup (e−tA) extrapolates to a bounded holomorphic C -
t≥0 0
semigroup on Lp(RN) for all p ∈ [p′,p ] and that the Riesz transform associated to A is
0 0
bounded on Lp(RN) for all p∈(p′,2].
0
Notation. Throughout the paper, we assume N ≥ 5 and denote by k·k the norm
p→q
of operators acting from Lp(RN) into Lq(RN) and p′ is the dual exponent of p, p′ = p .
p−1
Further, we set Σ(θ) := {z ∈ C\{0} : |argz| < θ} for θ ∈ (0,π/2]. Finally, χ denotes the
E
characteristic function of a set E and B(x,r) the ball around x of radius r.
2. The twisted semigroup
Inordertoshowtheboundedness oftheRiesz transformandobtainoff-diagonalestimates
for the semigroup generated by −A we use the classical Davies perturbation technique, and
estimatethe twistedsemigroup. Therefore,denotingbyαamulti-indexwith|α|=α +···+
1
α and Dα the corresponding partial differential operator on C∞(RN), we define E :={φ∈
N
C∞(RN;R) bounded:|Dαφ|≤1 for all 1≤|α|≤2} and the twisted forms
a (u,v):=a(e−λφu,eλφv)
λφ
with D(a )=H2(RN), λ∈R and φ∈E. A simple computation shows that
λφ
A :=eλφAe−λφ
λφ
FOURTH-ORDER SCHRO¨DINGER TYPE OPERATOR 3
with D(A ) = {u ∈ L2(RN) : e−λφu ∈ D(A)} is the associated operator to the form a .
λφ λφ
Moreover,there exist 0<γ <1 and k >1 such that the inequality
|a (u)−a(u)|≤γa(u)+k(1+λ4)kuk2 (2.1)
λφ 2
holds for all u∈H2(RN), λ∈R and φ∈E. Indeed, we have
a (u)=a(u)+λ4 |∇φ|4|u|2dx−λ2 |∆φ|2|u|2dx
λφ
ZRN ZRN
+4λ3iIm |∇φ|2∇φ·∇uudx+2λ2Re |∇φ|2u∆udx
ZRN ZRN
−4λ2Re ∆φ∇φ·∇uudx+2λiIm ∆φu∆udx
ZRN ZRN
−4λ2 |∇φ·∇u|2dx+4λiIm ∇φ·∇u∆udx.
ZRN ZRN
Now, the application of (1.2), the Gagliardo-Nirenberg inequality
k∇uk2 ≤kuk k∆uk , u∈H2(RN)
2 2 2
and Young’s inequality yields for 0<ε<1
|a (u)−a(u)|≤N2(λ4+λ2)kuk2
λφ 2
+4(Nλ2kuk )(N1/2|λ|k∇uk )+2(Nλ2ε−1kuk )(εk∆uk )
2 2 2 2
+4(N|λ|kuk )(N1/2|λ|k∇uk )+2(N|λ|ε−1kuk )(εk∆uk )
2 2 2 2
+4Nλ2k∇uk2+4(N1/2|λ|ε−1k∇uk )(εk∆uk )
2 2 2
≤4ε2k∆uk2+4N2(λ2+λ4)ε−2kuk2+10Nλ2ε−2k∇uk2
2 2 2
≤4ε2k∆uk2+4N2(λ2+λ4)ε−2kuk2
2 2
+10(Nλ2ε−3kuk )(εk∆uk )
2 2
≤9ε2k∆uk2+9N2(λ2+λ4)ε−6kuk2
2 2
≤(9ε2/η)a(u)+18N2(1+λ4)ε−6kuk2.
2
Forthe restofthis article,we fix γ andk suchthat inequality (2.1) holds. Then the forms
a +2k(1+λ4)areclosedanduniformlysectorial(seeforexample[10,Theorem1.19]). Thus
λφ
theoperators−A −2k(1+λ4)generatecontractiveholomorphicC -semigroupsonL2(RN)
λφ 0
with a common sector of holomorphy Σ(Θ). Therewith, we can show the following lemma.
Lemma 2.1. (a) For all z ∈Σ(Θ), λ∈R and φ∈E the following inequality holds
ke−zAλφk2→2 ≤e2k(1+λ4)Rez. (2.2)
(b) There exists M >0 such that
Θ
k∆e−zAλφk ≤M |z|−1/2e2k(1+λ4)Rez (2.3)
2→2 Θ
holds for all z ∈Σ(Θ/2), λ∈R and φ∈E.
Proof. Let λ∈R and φ∈E. As mentioned before, we have
ke−z(Aλφ+2k(1+λ4))k2→2 ≤1, (2.4)
for all z ∈Σ(Θ), which implies (2.2). Moreover,by the Cauchy formula,
k(A +2k(1+λ4))e−z(Aλφ+2k(1+λ4))k ≤(|z|sin(Θ/4))−1 (2.5)
λφ 2→2
4 F.GREGORIOANDS.MILDNER
holds for all z ∈Σ(Θ/2). Further, (1.2) and (2.1) yield
(1−γ)ηk∆vk2 ≤(1−γ)a(v)≤Re(a (v)+2k(1+λ4)kvk2)
2 λφ 2
≤k(A +2k(1+λ4))vk kvk
λφ 2 2
for all v ∈ D(Aλφ). Taking v = e−z(Aλφ+2k(1+λ4))u and applying the estimates (2.4) and
(2.5), we conclude (2.3) with M =1/ (1−γ)ηsin(Θ/4). (cid:3)
Θ
Finally, we prove Lp−Lq estimatespfor the twisted semigroups.
Lemma 2.2. Let p′ ≤p≤2≤q ≤p . Then there exists M >0 such that
0 0 pq
ke−zAλφukq ≤Mpq|z|−N4(p1−q1)e2k(1+λ4)Rezkukp
holds for all z ∈Σ(Θ/2), λ∈R, φ∈E and u∈L2(RN)∩Lp(RN).
Proof. Let z ∈ Σ(Θ/2), λ ∈ R and φ ∈ E. Then, by Sobolev’s embedding theorem (cf. [1,
Theorem 4.31]) and Lemma 2.1, one obtains
ke−zAλφuk 2N ≤CSk∆e−zAλφuk2 ≤CSMΘ|z|−1/2e2k(1+λ4)Rezkuk2 (2.6)
N−4
for all u∈L2(RN). Applying the Riesz-Thorininterpolationtheorem to e−zAλφ with respect
to the bounds (2.2) and (2.6), we achieve the L2−Lq estimate
ke−zAλφk2→q ≤M2q|z|−N4(12−1q)e2k(1+λ4)Rez
with M2q =(CSMΘ)N2(12−1q). Then a duality argument yields the Lp−L2 estimate. Finally,
we only have to combine these two and use the semigroup property to conclude the Lp−Lq
estimate with Mpq =(2CSMΘ)N2(p1−q1). (cid:3)
3. Off-diagonal estimates
Inthissection,westudyoff-diagonalestimates,whichenableustoobtaintheextrapolation
of the semigroup (e−tA) and the boundedness of the Riesz transform ∆A−1/2.
t≥0
We say that a family (T(z)) , θ ∈ (0,π/2], of bounded linear operators on L2(RN)
z∈Σ(θ)
satisfies Lp−Lq off-diagonal estimates for 1≤p ≤q ≤ ∞ if there exist c ,c >0 such that
1 2
for eachconvex,compactsubsets E,F ofRN, for eachu∈L2(RN)∩Lp(RN) supported in E
and for all z ∈Σ(θ), such an inequality holds
d(E,F)4/3
kT(z)uk ≤c |z|−γpqexp −c kuk ,
Lq(F) 1 2 |z|1/3 p
(cid:16) (cid:17)
where γ = N 1 − 1 and
pq 4 p q
(cid:0) (cid:1)d(E,F)=sup[inf{φ(x)−φ(y):x∈E,y ∈F}].
φ∈E
Daviesprovedthatthis distance is equivalentto the Euclideanone if the sets E andF are
additionally disjoint, [7, Lemma 4]. We recall this result.
Lemma 3.1. If E and F are disjoint, convex, compact subsets of RN, then
d (E,F)≤d(E,F)≤N1/2d (E,F),
e e
where d (E,F) is the Euclidean distance between E and F.
e
FOURTH-ORDER SCHRO¨DINGER TYPE OPERATOR 5
Remark 3.2. (a) Since the distance d between non-disjoint sets is zero, we can drop the
assumption of disjointedness in the previous lemma without changing the statement.
(b) For E,F ⊂ RN compact, convex, x,y ∈ RN and r > 0 such that E ⊂ B(x,r) and
F ⊂B(y,r) we obtain
d(E,F)4/3 ≥2−1/3|x−y|4/3−(2r)4/3.
Indeed, we can estimate as follows
|x−y|≤2r+d (E,F)≤2r+d(E,F)≤21/4((2r)4/3+d(E,F)4/3)3/4.
e
The following proposition relates the results of the previous section with the notion of
off-diagonal estimates.
Proposition 3.3. Let θ ∈(0,π/2] and (T(z)) be a family in L(L2(RN)) that satisfies
z∈Σ(θ)
T(z)=D T(s4z)D , s∈(0,1), z ∈Σ(θ), (3.1)
s 1/s
where D is the dilation operator, i.e., D v(x)=v(sx) a.e. for all v ∈L1 (RN). Further let
s s loc
1≤p≤q <∞ and M,ω >0 such that
keλφT(z)e−λφuk ≤M|z|−γpqeω(1+λ4)|z|kuk
q p
holds for all z ∈ Σ(θ), λ > 0, φ ∈ E and u ∈ L2(RN)∩Lp(RN). Then (T(z)) satisfies
z∈Σ(θ)
Lp−Lq off-diagonal estimates.
Proof. Let z ∈ Σ(θ), E,F be convex, compact subsets of RN and u ∈ L2(RN)∩Lp(RN)
supported in E. Then the assumption yields
kT(z)uk ≤ke−λφχ k keλφT(z)e−λφeλφχ uk
Lq(F) F ∞ E q
≤e−λinfFφM|z|−γpqeω(1+λ4)|z|kχ eλφk kuk
E ∞ p
≤e−λ(infFφ−supEφ)M|z|−γpqeω(1+λ4)|z|kukp
for all λ > 0 and φ ∈ E. Minimising the right-hand side with respect to φ ∈ E and choosing
λ as d(E,F) 1/3 we obtain
4ω|z|
(cid:0) (cid:1) d(E,F)4/3
kT(z)uk ≤e2ω|z|M|z|−γpqexp −c kuk
Lq(F) ω |z|1/3 p
(cid:16) (cid:17)
with c = 3 . Now, we use the scaling property to get rid of the factor e2ω|z|. For
ω 4(4ω)1/3
s∈(0,1) we estimate
kT(z)uk =kD χ T(s4z)χ D uk
Lq(F) s sF sE 1/s q
=s−Nq kχsFT(s4z)χsED1/sukq
≤e2ωs4|z|M|z|−γpqexp −cω(d(sE|,zs|F1/)3/s)4/3 s−NpkD1/sukp
(cid:16) (cid:17)
≤e2ωs4|z|M|z|−γpqexp − cω d(E,F)4/3 kuk .
N2/3 |z|1/3 p
(cid:16) (cid:17)
Taking s→0, we get Lp−Lq off-diagonalestimates for (T(z)) . (cid:3)
z∈Σ(θ)
Now, since (e−zA) satisfies the scaling property (3.1) thanks to the invariance of
z∈Σ(Θ/2)
the Laplacian, we can infer from Lemma 2.2 the following statement.
Corollary 3.4. (e−zA) satisfies Lp−Lq off-diagonal estimates for all p∈[p′,2] and
z∈Σ(Θ/2) 0
q ∈[2,p ].
0
6 F.GREGORIOANDS.MILDNER
Finally, we are able to state the following theorem.
Theorem 3.5. The semigroup (e−tA) on L2(RN) extrapolates to a bounded holomorphic
t≥0
C -semigroup on Lp(RN) for all p∈[p′,p ].
0 0 0
Proof. It suffices to show that the family (e−zA) is uniformly bounded onLp(RN) to
z∈Σ(Θ/2)
infer the extrapolation to a bounded holomorphic C -semigroup on Lp(RN). Moreover, we
0
only have to treat the case p∈(2,p ].
0
Let p∈(2,p ], z ∈Σ(Θ/2) and C be the cube with centre n|z|1/4 and edge length |z|1/4
0 n
for all n ∈ ZN. Then, using the L2−Lp off-diagonal estimates for (e−zA) , Remark
z∈Σ(Θ/2)
3.2(b) and Ho¨lder’s inequality, we obtain
kχCne−zAχCmukp ≤c1e−c2|m−n|4/3|z|−γ2p|Cm|21−p1kχCmukp
=c e−c2|m−n|4/3kχ uk
1 Cm p
for all m,n ∈ ZN and u ∈ L2(RN)∩Lp(RN) with c ,c > 0 independent of z, u, m and n.
1 2
Since the operator B: ℓ1(ZN)→ℓ1(ZN) with
(Bx) =c e−c2|m−n|4/3x , m∈ZN, x∈ℓ1(ZN)
m 1 n
nX∈ZN
is bounded on ℓ1(ZN) as well as on ℓ∞(ZN), the Riesz-Thorin interpolation theorem yields
that B is also bounded on ℓp(ZN). Setting uˆ=(kχCnukp)n∈ZN we conclude
ke−zAuk ≤kBuˆk ≤kBk kuˆk ≤kBk kuk
p ℓp ℓp→ℓp ℓp ℓp→ℓp p
for all u∈L2(RN)∩Lp(RN). (cid:3)
Remark 3.6. We have provided this proof as an application of the previous results, which we
will also need in the next section to prove the boundedness of the Riesz transform. Actually,
wecould have also applied [9,Proposition6.1], which holds in ageneral settingofhigher order
operators defined by closed, sectorial sesquilinear forms. We recall this statement according
to the notations of our situation.
Proposition 3.7. Let a be a closed, sectorial sesquilinear form in L2(RN) with D(a) =
H2(RN) such that for some C,k >0
1k∆uk2 ≤Rea(u)≤C(k∆uk2+kuk2)
2 2 2 2
and
|a (u)−Rea(u)|≤ 1Rea(u)+k(1+λ4)kuk2
λφ 4 2
hold for all u ∈ H2(RN), λ ≥ 0 and φ ∈ E. Then the holomorphic C -semigroup (e−tA)
0 t≥0
on L2(RN), associated with a, extrapolates to a holomorphic C0-semigroup Tp = (e−tAp)t≥0
on Lp(RN) for all p ∈ [p′,p ]. The sector of holomorphy of T and the spectrum σ(A ) are
0 0 p p
p-independent.
4. Riesz transform
We show that ∆A−1/2 ∈L(Lp(RN)) for all p∈( 2N ,2]. We already know that the Riesz
N+4
transform of the operator A is bounded on L2(RN) thanks to the inequality
ηk∆uk2 ≤a(u)=kA1/2uk2, u∈H2(RN)
2 2
andtheselfadjointnessofA1/2. Then,provided∆A−1/2isofweaktype(p′,p′),wecanusethe
0 0
Marcinkiewicz interpolation theorem to obtain the boundedness on Lp(RN) for p′ <p≤2.
0
FOURTH-ORDER SCHRO¨DINGER TYPE OPERATOR 7
Let us recall the definition of weak type operators. Let (X,Σ,µ) be a measure space. An
operatorL isofweaktype (p,p)for 1≤p<∞, ifthere exists aconstantC suchthatfor any
measurable function f, such a relation holds
µ{x:|Lf(x)|≥λ}≤Cλ−pkfkp.
p
In order to prove that ∆A−1/2 is of weak type (p′,p′) we make use of [6, Theorem 1.1] in
0 0
the following adapted form.
Theorem 4.1. Let 1≤p<2<q ≤∞, q ∈(p,∞] and (e−tA) be a bounded holomorphic
0 t≥0
semigroup on L2(RN) such that A is injective and has dense range. Further, let α ∈ [0,1)
and B a linear operator satisfying D(Aα)⊂D(B) and the weighted norm estimates
|x−y|4/3
kχB(x,t1/4)e−tAχB(y,t1/4)kp→q ≤c1t−γpqexp −c2 t1/3 (4.1)
(cid:16) |x−y|4/3(cid:17)
kχB(x,t1/4)tαBe−eiσtAχB(y,t1/4)kp→q0 ≤c1t−γpq0 exp −c2 t1/3 (4.2)
hold for all x,y ∈RN, t>0, |σ|< π −θ, for some θ >0. Then(cid:16)BA−α is of we(cid:17)ak type (p,p)
2
provided BA−α is of weak type (2,2).
We can now state the main result of this section.
Theorem 4.2. The Riesz transform of the operator A is bounded on Lp(RN) for all p ∈
(p′,2].
0
Proof. We will show that the assumptions of Theorem 4.1 in the setting (B,α,p,q,q ) =
0
(∆,1/2,p′,p ,2) are satisfied to infer that ∆A−1/2 is of weak type (p′,p′).
0 0 0 0
First, we observe that A is injective and selfadjoint and has therefore dense range. More-
over,wehaveD(A1/2)=D(∆)and∆A−1/2 isboundedonL2(RN),henceofweaktype(2,2),
as waspointedoutabove. Now,it remains to showthat estimatesof the form(4.1)and(4.2)
are satisfied. Due to Remark 3.2(b), such estimates are direct consequences of Lp′0 −Lp0
off-diagonalestimates for (e−zA)z∈Σ(Θ/2), which we have already obtained, and Lp′0−L2 off-
diagonal estimates for the family (|z|1/2∆e−zA) . To achieve the last ones, we show
z∈Σ(Θ/2)
that
keλφ∆e−zAe−λφk ≤M|z|−1/2eω(1+λ4)|z| (4.3)
2→2
holds for all z ∈Σ(Θ/2), λ∈R and φ∈E with some M,ω >0. Indeed, we compute
eλφ∆e−zAe−λφu=eλφ∆e−λφe−zAλφu
=(λ2|∇φ|2−λ∆φ)e−zAλφu−2λ∇φ·∇e−zAλφu
+∆e−zAλφu,
which can be estimated, thanks to (2.2) and (2.3), in the following way
keλφ∆e−zAe−λφuk2 ≤8N2(1+λ4)ke−zAλφuk2
2 2
+16Nλ2k∇e−zAλφuk2+4k∆e−zAλφuk2
2 2
≤16N2(1+λ4)ke−zAλφuk2+12k∆e−zAλφuk2
2 2
≤16N2(1+λ4)e4k(1+λ4)|z|kuk2
2
+12M2|z|−1e4k(1+λ4)|z|kuk2
Θ 2
≤16(M2 +N2)|z|−1e5k(1+λ4)|z|kuk2.
Θ 2
8 F.GREGORIOANDS.MILDNER
Combining inequality (4.3) with the Lp′0 −L2 estimate of Lemma 2.2, we get
keλφ∆e−zAe−λφk ≤2M M|z|−1eω(1+λ4)|z|
p′0→2 p′02
for all z ∈ Σ(Θ/2), λ > 0 and φ ∈ E. Since the family (|z|1/2∆e−zA) satisfies the
z∈Σ(Θ/2)
scaling property (3.1), it also satisfies Lp′0 −L2 off-diagonal estimates by Propostion 3.3.
Thus, ∆A−1/2 is of weak type (p′,p′). Now, by the boundedness of ∆A−1/2 on L2(RN)
0 0
and the Marcinkiewicz interpolation theorem, we conclude that ∆A−1/2 ∈L(Lp(RN)) for all
p∈(p′,2]. (cid:3)
0
Finally, we obtain the following corollary.
Corollary 4.3. The parabolic problem associated to −A=−∆2+ c , c<C∗
|x|4
∂ u(t)=−Au(t) for t≥0,
t
( u(0)=f,
admits a unique solution for each initial datum f ∈ Lp(RN), p ∈ [p′,p ]. Moreover, if
0 0
f ∈Lp(RN) for p∈(p′,2], then the solution is in W2,p(RN).
0
Acknowledgements
One of the authors, F. G., would like to thank ProfessorEl Maati Ouhabaz and Professor
Abdelaziz Rhandi for their very helpful suggestions and discussions. She is also grateful to
the Institut de Math´ematiques of the University of Bordeaux for the kind hospitality during
her visit.
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FOURTH-ORDER SCHRO¨DINGER TYPE OPERATOR 9
Dipartimento di Fisica, Universita` degli Studi di Salerno, Via Ponte Don Melillo, 84084 FIS-
CIANO(Sa),Italy.
E-mail address: [email protected]
Technische Universita¨t Dresden, Institutfu¨r Analysis,01062Dresden, Germany
E-mail address: [email protected]