Instructions for use
A uthor(s ) C ho,Y onggeun; Ozawa,T ohru
C itation Hokkaido University Preprint S eries in Mathematics, 792: 1-10
Is s ue D ate 2006
D O I 10.14943/83942
D oc UR L http://hdl.handle.net/2115/69600
T ype bulletin (article)
F ile Information pre792.pdf
ON RADIAL SOLUTIONS OF SEMI-RELATIVISTIC HARTREE EQUATIONS
YONGGEUN CHO AND TOHRU OZAWA
Abstract. We consider the semi-relativistic Hartree type equation with non-local nonlinearityF(u) =λ(|x|−γ∗ |u|2
)u,0< γ < n, n≥1. In [2], the global well-posedness (GWP) was shown for the value ofγ∈(0, 2n
n+1), n≥2 with large data andγ∈ (2, n), n ≥3 with small data. In this paper, we extend the previous GWP result to the case forγ ∈(1,2n−1
n ), n ≥2 with radially
symmetric large data. Solutions in a weighted Sobolev space are also studied.
1. Introduction
We consider the following Cauchy problem describing boson stars
½
i∂tu= √
1−∆u+F(u) in Rn×R, n≥1
u(0) =ϕ∈Hs(Rn).
(1)
Here F(u) = (Vγ∗ |u|2)uis Hartree type nonlinearity, where∗ denotes the
convo-lution inRn andVγ(x) =λ|x|−γ withλ∈R, 0< γ < n. Hs= (1−∆)− s
2L2is the
Sobolev space of orders∈R. We consider (1) in the form of the integral equation
u(t) =U(t)ϕ−i
Z t
0
U(t−t′)F(u)(t′)dt′,
(2)
where
(U(t)ϕ)(x) = (e−it√1−∆ϕ)(x)≡ 1
(2π)n
Z
Rn
ei(x·ξ−t√1+|ξ|2)ϕb(ξ)dξ
andϕbdenotes the Fourier transform defined asRRne−ix·ξϕ(x)dx.
If the solutionuof (1) or (2) has sufficient decay at infinity and smoothness, it satisfies two conservation laws:
ku(t)kL2 =kϕkL2, E(u)≡K(u) +V(u) =E(ϕ),
(3)
where K(u) = h√1−∆u, ui, V(u) = 1
4hF(u), ui and h,i is the complex inner product inL2.
Recently, the equation (1) has been extensively studied. Elgart and Schlein [5] and Fr¨ohlich and Lenzmann [6] considered the mean field limit problem of boson stars with Coulomb potential. The finite time blowup solution with negative energy was studied by Fr¨ohlich and Lenzmann [6]. Fr¨ohlich, Jonsson and Lenzmann [7] proved the existence of traveling solitary waves inR3.
2000Mathematics Subject Classification. 35Q40, 35Q55, 47J35.
Key words and phrases. semi-relativistic Hartree type equation, global well-posedness, radially symmetric solution.
The first author is supported by Japan Society for the Promotion of Science under JSPS Postdoctoral Fellowship For Foreign Researchers.
For the well-posedness results, we refer the readers to the papers [10, 2]. Lenz-mann in [10] established the global well-posedness inH12(R3) forγ= 1. In [2], we
considered the general case 0< γ < n, n≥1 and showed the local and global exis-tence by utiliizing the Strichartz estimates. In particular, we showed the global ex-istence for 0< γ < n2+1n inH12 with large data and for 2< γ < ninHs, s > γ
2−
n−2 2n
with small data.
The first result of this paper is on the global existence of radially symmetric solutions of (2) for 2n
n+1 ≤γ < 2n−1
n , n≥2.
Theorem 1.1. Let γ satisfy 1 < γ < 2nn−1, n≥2, s≥ 1
2. If λ >0, then for any radially symmetric function ϕ∈Hs,(2) has a unique radially symmetric solution
u∈C(R;Hs)∩Lq locHe
1 2,12−σ
2n n−1 for
q= n2−n1+εandσ= 12+ε′ with sufficiently small
ε, ε′ >0. For all time the energy and L2 norm of u(t) are conserved. If λ < 0, then there exists ρ >0 such that the same conclusion holds for ϕ with kϕkL2 ≤ρ.
Moreover,
ku(t)kHs.kϕkHsexp
³
C|t|(1 +kϕkL22+|E(ϕ)|) q q−2
´
.
(4)
HereHs
r= (1−∆)−s/2Lr and
e
Hrs,s′ ={v:kvkHes,s′
r ≡ k(−∆) s 2P
≤1vkLr +k(1−∆) s′
2P1>1vkLr <∞}
are the usual Sobolev space and a hibrid Sobolev space, where P≤1 and P>1 are frequency projection over frequency less than 1 and greater than 1. We meanHs
byHs
2 and ˙Hsby ˙H2s. Hereafter, we denote the spaceL
q
T(B) byLq(−T, T;B) and
its norm byk · kLqTB for some Banach spaceB, and alsoLq(B) with normk · kLqB
byLq(R;B), 1≤q≤ ∞.
In order to prove Theorem 1.1, we pursue the contraction mapping argument. For this purpose, we use the energy and L2 conservation laws and the Strichartz estimate for radial functions. By the Strichartz estimate we mean (see [11, 12]):
kU(t)ϕkLq0 TH
s−σ0
r0 .kϕkH s0,
° ° ° °
Z t
0
U(t−t′)F(t′)dt′
° ° ° °Lq1
TH s1−σ1 r1
.kFkL1 THs1,
(5)
where (qi, ri), i= 0,1, satisfy that for anyθ∈[0,1]
2
qi= (n−1 +θ)
µ1
2 − 1
ri
¶
, 2σi= (n+ 1 +θ)
µ1
2 − 1
ri
¶
,
2≤qi, ri≤ ∞, (qi, ri)6= (2,∞).
(6)
IfϕandF are radially symmetric, then by the well-known decay property of the Fourier transform of measure on unit sphere the estimate (5) can be extended as:
kU(t)ϕk LpTHe
1 2,s−σ p
.kϕkHs,
° ° ° °
Z t
0
U(t−t′)F(t′)dt′
° ° ° °Lp
THe 1 2,s−σ p
.kFkL1 THs,
(7)
wheres∈Rand n2−n1 < p <∞, σ= n2 −n+1p . The second estimate does not follow
SEMIRELATIVISTIC EQUATION 3
Interpolating (5) and (7)1, we get wider range of pairs (q, r). For Theorem 1.1, we need only the pairs (q,n2−n1) withqslightly larger than n2−n1. To put is another way, givenε >0 we can findqandσsuch that n2−n1 < q <n2−n1+ε, 21n < σ < 21n+ε
and
kU(t)ϕk LqTHe
1 2,12−σ
2n n−1
.kϕkH1 2,
° ° ° °
Z t
0
U(t−t′)F(t′)dt′
° ° ° °Lq
THe 1 2,12−σ
2n n−1
.kFk L1
TH 1 2,
(8)
With these pairs we can make the value ofσclose to 1
2n and the value γto
2n−1
n .
Next we consider a radial solution in weighted Sobolev spaceHs,r={v ∈Hs:
kvkHs,r ≡ k(1 +|x|2) r
2(1−∆) s
2vkL2 <∞}.
Theorem 1.2. Let n≥2 and1 < γ < 2nn−1. Letϕ andube as in Theorem 1.1.
If in additionϕ∈H1,1, thenu∈C(R;H1,1)∩LqlocHe 1 2,
1 2−σ 2n n−1
, whereq andσare the numbers as stated in Theorem 1.1. Moreover, if n≥3, then
ku(t)kH1,1.kϕkH1,1exp
³
C|t|(1 +kϕkL22+|E(ϕ)|) q q−2
´
.
(9)
The essential parts of the proof for the global existence inH1,1are the estimate (4) and the following estimates
kx√1−∆F(u)(t)kL2.(ku(t)k2H1+ku(t)k2
e
H 1 2,12−σ 4
)ku(t)kH1,1 for n= 2,
(10)
kx√1−∆F(u)(t)kL2.(ku(t)k2
H12 +ku(t)k
2
e
H 1 2,12−σ
2n n−1
)ku(t)kH1,1 for n≥3,
(11)
kx√1−∆u(t)kL2 .kϕkH1,1+
Z t
0
(k∇ukL2+kx √
1−∆F(u)kL2)dt′.
(12)
For the inequality (12), we do not need the radial symmetry. To obtain (10) and (11), we need to estimate k|∇Vγ(u)||x|ukL2 for which we establish the pointwise
estimate of fractional integral of radial function forn≥3. See Lemma 3.1 below. We can estimatek|∇Vγ(u)||x|ukL2 forn≥2 without using Lemma 3.1. For
ex-ample, k|∇Vγ(u)||x|ukL2 ≤ k∇Vγ(u)kLnkxuk
Ln2−n2 .k∇ukL2kuk
2
L 2n
n−2(γ−1)kukH 1,1.
Hence combining this with (4), we have at leastku(t)kH1,1 .exp(C|t|exp(C|t|)).
If 0 < γ ≤1, then in view of GWP in Hs, s≥ 1
2 of [10, 2], from the estimate
k|∇Vγ(u)||x|ukL2 .k∇ukL2kuk
H12kukH1,1we deduce the GWP inH
1,1without
ra-dial symmetry condition. One can also prove the global existence of rara-dial solutions
Hk,lwith integers k, l >1 by the same method as in Section 3.
If not specified, the notationA.B andA&BdenoteA≤CBandA≥C−1B, respectively. Different positive constants possibly depending on n, λ and γ might be denoted by the same letterC. A∼B means that both A.B andA&B.
2. Proof of Theorem 1.1
In this section, we prove Theorem 1.1. For simplicity, we only consider the positive time because the proof for negative time can be treated in the same way.
1We can proceed the complex interpolation after changingU(t) an operator mapping from
Let us first define a complete metric spaceXT,ρwith metricd(u, v) =ku−vkXT,
whereXT =C([0, T];Hs)∩LqTHe 1 2,
1 2−σ 2n n−1
by
XT,ρ≡ {v∈XT : v is radially symmetric and kvkXT ≤ρ}.
As stated in the introduction, givenε >0 we can findqandσsatisfying (8). Now we define a mappingN :u7→N(u) onXT,ρby
N(u)(t) =U(t)ϕ−i
Z t
0
U(t−t′)F(u)(t′)dt′.
(13)
For anyu∈XT,ρ,N(u) is radially symmetric. By Strichartz estimate (8), we have kN(u)kXT .kϕkHs+kF(u)kL1THs.
(14)
To estimate of the second term on the RHS of (14), let us introduce a generalized Leibniz rule (see Lemma A1∼Lemma A4 in Appendix of [8]).
Lemma 2.1. For any s≥0we have
k(−∆)s2(uv)k
Lr .k(−∆) s 2uk
Lr1kvkLq1 +kukLq2k(−∆) s 2vk
Lr2,
1
r =
1
ri
+ 1
qi
, ri∈(1,∞), qi ∈(1,∞], i= 1,2.
Sinceγ≤2, we use Lemma 2.1 with (r1, q1) = (∞,2), (r2, q2) = (2γn,n2−nγ) and (r1, q1) = (2,n2−nγ) = (q2, r2), and Hardy-Littlewood-Sobolev inequality to obtain
kF(u)kL1
THs.kVγ(u)kL1TL∞kukL1THs+kVγ(u)kLqT′Hs2n γ
kuk LqTLn2−nγ
.T1−2qkuk LqTL
2n
n−γ−ε0kukLqTL 2n n−γ+ε0kukL
∞
THs
+T1−2qk|u|2k LqTHs
2n 2n−γ−ε0
kuk LqTL
2n n−γ+ε0
.T1−2q
µ
kuk LqTL
2n
n−γ−ε0kukLqTL 2n n−γ+ε0
¶
kukL∞
THs,
(15)
where 0< ε0< n−γ and we have used the inequality that for anyx∈Rn
|Vγ(u)(x)|.kuk L
2n n−γ−ε0kukL
2n n−γ+ε0.
(16)
Now if we chooseεandε0 so small thatγ <2−2σand 2n
n−1 ≤
2n n−(γ−ε0)
< 2n n−(γ+ε0) ≤
2n
n−1−2(12 −σ),
then we have from (14), (15) and embeddingsH12 ∩He 1 2,
1 2−σ 2n
n−1 ֒→L
r for any 2n n−1 ≤
r≤ 2n n−1−2(1
2−σ)<
2n
n−2 that kukLn−(2γn±ε0 )
.kukH1
2 +kukHe12,12−σ 2n n−1
and
kN(u)kXT ≤C(kϕkHs+ (T+T 1−2
q)kuk3
XT)≤C(kϕkHs+ (T+T 1−2
q)ρ3)
for some constantC. Thus if we chooseρand T so thatCkϕkHs ≤ ρ
2 and C(T+
T1−2q)ρ3≤ ρ
2, then we conclude thatN maps from XT,ρ to itself. For anyu, v∈XT,ρ, we have
d(N(u), N(v)).kF(u)−F(v)kL1 THs
.k(| · |−γ∗(|u|2− |v|2))ukL1
THs+k(| · | −γ
∗ |v|2)(u−v)kL1 THs.
SEMIRELATIVISTIC EQUATION 5
By (16) and H¨older’s inequality, we have for sufficiently smallε0>0
k(| · |−γ∗(|u|2− |v|2))ukL1 THs
.T12k| · |−γ∗(|u|2− |v|2)k L2
TL∞kukL
∞
THs
+k| · |−γ∗(|u|2− |v|2)kLq′ THs2n
γ+ε0 kuk
LqTL 2n n−(γ+ε0 )
.T12ρk|u|2− |v|2k 1 2 L1
TL n n−(γ+ε0 )k|
u|2− |v|2k12 L1
TL n n−(γ−ε0 )
+T1−2qρku−vk L∞
THs(kukLq TL
2n
n−(γ−ε0 ) +kvkLqTL 2n n−(γ−ε0))
+T1−2qρku−vk LqTL
2n
n−(γ−ε0)(kukL
∞
THs+kvkL∞THs).
(18)
Now by another H¨older’s inequality in time, we have
k(| · |−γ∗(|u|2− |v|2))ukL1
THs .(T+T 1−2
q)ρ2d(u, v).
Similarly,
k(| · |−γ∗ |v|2)(u−v)kL1 THs
.k(| · |−γ∗ |v|2)kL1
TL∞ku−vkL
∞
THs
+T1−2qk(| · |−γ∗ |v|2)k LqTL
2n
γ+ε0ku−vkLqTL 2n n−(γ+ε0 )
.T1−2qkvk LqTL
2n
n−(γ−ε0 )kvkLqTL 2n
n−(γ+ε0 )d(u, v)
+T1−2qkvk L∞
THskvkLq TL
2n
n−(γ−ε0 )ku−vkLq TL
2n n−(γ+ε0 ).
(19)
Hence by Sobolev embedding we get
k(| · |−γ∗ |v|2)(u−v)kL1
THs.(T+T 1−2
q)ρ2d(u, v).
Substituting these two estimates into (17) and then using the factC(T+T1−2q)ρ2≤ 1
2 for smallT, we conclude thatN is a contraction mapping onXT,ρ. The energy
andL2 conservations follow by the method in [13].
Now we show that the local solutions can be extended globally in time. For this purpose we prove an a priori estimate inXT for anyT >0. FixingT, sinceγ <2,
from the energy conservation we see that at anyt≤T, the solutionusatisfies that forλ >0,
1 2ku(t)k
2
H12 ≤
1 2ku(t)k
2
L2+E(u) =
1 2kϕk
2
L2+E(ϕ)
and forλ <0 1 2ku(t)k
2
H12 ≤
1 2ku(t)k
2
L2+|E(u)|+|V(u)|
≤1
2kϕk 2
L2+|E(ϕ)|+Ckuk2
Ln−2γn+1kuk
2
H12
≤12kϕk2L2+|E(ϕ)|+Ckuk2L−2γkuk
1+γ H12
=1 2kϕk
2
L2+|E(ϕ)|+Ckϕk2L−2γkuk
and hence by Young’s inequality and the smallness ofkϕkL2
ku(t)k2H1
2 ≤C(kϕk
2
L2+|E(ϕ)|).
(20)
From the estimates (20) and (16), we have forδ >0
kuk LqδHe
1 2,12−σ
2n n−1
.(kϕk2L2+|E(ϕ)|) 1
2 +δ(kϕk2
L2+|E(ϕ)|) 3 2
+δ1−2q(kϕk2
L2+|E(ϕ)|) 1 2kuk2
Lq δHe
1 2,12−σ
2n n−1
.
Thus for sufficiently smallδbut equivalent to the value (1 +kϕk2
L2+|E(ϕ)|)− q q−2,
kuk Lq(T
j−1,Tj;He 1 2,12−σ
2n n−1
)≤C(kϕk 2
L2+|E(ϕ)|) 1 2,
whereTj−Tj−1=δforj≤k−1,Tk=T andTk−Tk−1∼δ. This implies that
kukq Lq(0,T;He
1 2,12−σ
2n n−1
)≤
X
1≤j≤k kukq
Lq(T j−1,Tj;He
1 2,12−σ
2n n−1
)
.kδ(1 +kϕk2L2+|E(ϕ)|) q2
2q−4 .T(1 +kϕk2
L2+|E(ϕ)|) q2 2q−4.
(21)
Finally, we have from (15)
ku(t)kHs ≤ kϕkHs+kF(u)kL1
tHs .kϕkHs+
Z t
0 (kuk2
H12 +kuk
2
e
H 1 2,12−σ
2n n−1
)kukHsdt′.
Hence by Gronwall’s inequality and (21),
ku(t)kHs .kϕkHsexp
Ct(kϕk2L2+|E(ϕ)|) +Ct1− 2 qkuk2
LqtHe
1 2,12−σ
2n n−1
.kϕkHsexp
³
Ct(1 +kϕk2
L2+|E(ϕ)|) q q−2
´
.
(22)
This completes the proof of Theorem 1.1.
Proof of Strichartz estimate (7)of radial functions. For the first inequality, we fol-low the proof of Proposition 6.3 in [14]. By the spherical coordinate,
(U(t)ϕ)(x) =cn
Z ∞
0
e−it√1+ρ2dσc(rρ)ϕb(ρ)ρn−1dρ,
wherer=|x|,ρ=|ξ|and
c
dσ(rρ) =
Z
Sn−1
e−ix·ξdσ=
Z
Sn−1
eix·ξdσ.
Let us define a one-dimensional functionf byf(ρ) =w(ρ)ϕb(ρ)ρn−21 for some
posi-tive function to be chosen later and an operatorW(t) by (W(t)f)(r) =rn−p1(U(t)ϕ)(r).
Then since the spaceHe12,12−σ 2n
n−1 is equivalent to the space (−∆)
−1
4(1−∆) 1 4H
1 2,12−σ
2n n−1 ,
we have only to show that forw=ρ−12(1 +ρ2)14(1 +ρ2) n 4−
n+1 2p ,
kW(·)fkLp((0,T)×(0,∞)).kfkL2.
SEMIRELATIVISTIC EQUATION 7
By the change of variablep1 +ρ27→ρ,W(t)f is written as
(W(t)f)(r) =cnr n−1
p
Z
R
e−itρχ(1,∞)(ρ)dσc(r
p
ρ2−1) f(
p
ρ2−1)
w(pρ2−1)2
ρ(ρ2−1)n−41
p
ρ2−1 dρ.
Using Sobolev embedding ˙H12− 1
p(R) ֒→ Lp(R) and Plancherel theorem, it follows
from change of variablepρ2−17→ρthat
k(W(·)f)(r)k2
Lp(0,T)
.r2(np−1)
Z ∞
1
ρ2(32− 1
p)(ρ2−1)n− 3 2
¯ ¯
¯cdσ(rpρ2−1)¯¯¯2|f(
p
ρ2−1)|2
w(pρ2−1)2 dρ
.r2(np−1)
Z ∞
0
(1 +ρ2)1−1pρn−2
¯ ¯
¯cdσ(rρ)¯¯¯2|f(ρ)| 2
w(ρ)2 dρ. (24)
Hence by takingLp2 norm inr-variable to (24), we have
kW(·)fk2Lp((0,T)×(0,∞)).
Z ∞
0
(1 +ρ2)1−p1ρn−2|f(ρ)| 2
w(ρ)2 A(ρ) 2dρ, (25)
where
A(ρ) =
µZ ∞
0
rn−1¯¯¯cdσ(rρ)¯¯¯p dr
¶1 p .
From the well-known decay of Fourier transform of measure on the unit sphere (see [16]), we have for p > n2−n1
A(ρ)p. Z ∞
0
rn−1(1 +rρ)−p(n2−1)dr.ρ−n.
Substituting this into (25), we finally have
kW(·)fk2
Lp((0,T)×(0,∞)).
Z ∞
0
(1 +ρ2)1−1pρn−2− 2n
p |f(ρ)| 2
w(ρ)2 dρ
. Z ∞
0
(1 +ρ2)n2− n+1
p ρ−1p1 +ρ2|f(ρ)| 2
w(ρ)2 dρ,
sincep > n2−n1. Therefore, if we choose w(ρ) =ρ−12(1 +ρ2) 1
4(1 +ρ2) n 4−
n+1 2p , then
we prove the claim.
Now we prove the second inequality. For this purpose, it suffices to show that
° ° ° °
Z t
0
W(t−t′)G(t′)dt′
° ° ° °
Lp((0,T)×(0,∞))
.kGkL1 TL2,
(26)
whereG(ρ, t′) =w(ρ)Fb(ρ, t)ρn−21 and
W(t−t′)G(t′) =cnr n−1
p
Z ∞
0
e−i(t−t′) √
1+ρ2
c
dσ(rρ)G(ρ, t′)
w(ρ) dρ
=rn−p1U(t−t′)F(r, t′).
Lemma 2.2. Let A and B be Banach spaces. Let K be an operator such that
kKGkLqT(A) ≤CkGkLpT(B) for 1 ≤p≤q ≤ ∞ with kernel k defined by KG(t) =
RT
0 k(t−t′)G(t′)dt′, whereCdoes not depend onT. Ifp < q, then the low-diagonal operator Ke defined by KGe = R0tk(t −t′)G(t′)dt′ satisfies that kKGe kLqT(A) ≤
e
CkGkLq
T(B), whereCe does not depend on T.
By Lemma 2.2 with kernelk(t) =W(t),A=L2 andB=Lp , it suffices to show that
k
Z T
0
W(t−t′)G(t′)dt′kLp((0,T)×(0,∞)).kGkL1 TL2.
(27)
In fact, since for any G ∈ L1
T(L2(0,∞)) we can find a unique radial function F ∈L1T(Hn2−
n+1
p (Rn)) such thatG(ρ, t) =w(ρ)Fb(ρ, t)ρn−21 and hence
k
Z T
0
W(t−t′)G(t′)dt′kLp((0,T)×(0,∞))=kL(∆)
Z T
0
U(t−t′)F(t′)dt′kLp((0,T)×Rn),
whereL(∆) = (−∆)14(1−∆)− 1
4. Then by the Strichartz estimate (7), we have
kL(∆)
Z T
0
U(t−t′)F(t′)dt′kLp((0,T)×Rn)
=kL(∆)U(t)
Z T
0
U(−t′)F(t′)dt′kLp((0,T)×Rn).kFk L1
TH n 2−
n+1
p =kGkL 1 TL2.
This proves (27) and thus the claim (26). ¤
3. Proof of Theorem 1.2
We proceed a similar line to the proof of Theorem 1.1(contraction scheme for local existence, energy and L2 conservation for global time extension) except for
H1,1 estimate. For this purpose, we will prove only a priori estimates (10), (11) and (12).
Let us begin with proof of (11). Using the commutator relation
[x,√1−∆ ] =∇(1−∆)−12,
(28)
we have
kx√1−∆F(u)kL2 ≤ k∇(1−∆)− 1 2Fk
L2+k √
1−∆(xF(u))kL2
.kFkL2+kxF(u)kL2+k∇(Vγ(u)xu)kL2.
By using the estimate (15), we have
kFkL2+kxF(u)kL2 .(kuk2
H12 +kuk
2
e
H 1 2,12−σ
2n n−1
)kukH1,1
and
k∇(Vγ(u)xu)kL2≤ kVγ(u)kL∞k∇(xu)kL2+k|∇Vγ(u)||x|ukL2
.(kuk2
H12 +kuk
2
e
H 1 2,12−σ
2n n−1
)kukH1,1+k|∇Vγ(u)||x|ukL2.
SEMIRELATIVISTIC EQUATION 9
Lemma 3.1. Let γ satisfy 0< γ < n−1 forn≥3. If f ∈Hα and g∈Hβ are radially symmetric functions withα, β≥0andα+β ≤γ, then for any x6= 0
Z
Rn
|f(y)||g(y)|
|x−y|γ dy.|x| α+β−γ
kfkHαkgkHβ.
Now lettingf =|∇u|andg=|u|, by Lemma 3.1 withα= 0, β= 0 for 1< γ≤3 2 and α = 0, β = 12 for 32 < γ < 2nn−1, we get |∇Vγ(u)| . |x|−γkukH1kukL2 for
1< γ≤ 3
2 and|∇Vγ(u)|.|x|−
γ+1 2kuk
H1kuk H12 for
3 2 < γ <
2n−1
n and hence
k|∇Vγ(u)||x|ukL2.kukH1kuk H12
° ° ° °|
u| |x|θ
° ° ° °
L2
for some θ ∈ (0,12). Therefore the Hardy inequality yields k|∇Vγ(u)||x|ukL2 . kukH1kuk2
H12. This proves the estimate (11). Combining the argument in the
introduction and the above one, we have (10).
Now we prove (12). From the identity (28) we have
x√1−∆u(t) =U(t)x√1−∆ϕ−i
Z t
0
x√1−∆F(u(t′))dt′
−itU(t)∇ϕ−
Z t
0
U(t−t′)(t−t′)∇F(u(t′))dt′,
where the last two terms are rewritten as
−i
Z t
0
U(t)∇ϕ dt′−
Z t
0
Z t′
0
U(t−t′′)∇F(u(t′′))dt′dt′′
=−i
Z t
0
U(t−t′)∇(U(t′)ϕ−i
Z t′
0
U(t−t′)F(u(t′′))dt′′)dt′
=−i
Z t
0
U(t−t′)∇u(t′)dt′.
Hence
x√1−∆u(t) =U(t)x√1−∆ϕ−i
Z t
0
U(t−t′)(−∇u+x√1−∆F(u))dt′.
This implies the estimate (12).
Proof of Lemma 3.1. Fixingx, we divide the integration into three parts as follows.
Z
Rn
|f(y)||g(y)| |x−y|γ dy=
Z
|y|>2|x| +
Z
|x|
2 ≤|y|≤2|x|
+
Z
|y|<|x2| ≡
I+II+III.
ForI, since|x−y| ≥ |y2| for|y|>2|x| andα+β≤γ, we have
I. Z
|y|>2|x||
y|α+β−γ|f(y)| |y|α
|g(y)|
|y|β dy.|x| α+β−γ
kfkHαkgkHβ.
Since uis radially symmetric, we may assume that x=re1 =r(0,0,· · · ,0,1). Using the spherical coordinates (ρ, θ1, θ2,· · ·, θn−1)∈(0,∞)×[0, π]×[0, π]× · · · × [0,2π] fory variable, the integralsII andIII are converted into
II+III=
ÃZ 2r
r 2
+
Z r 2
0
!
ρα+β+n−1|f(ρ)| ρα
|g(ρ)|
where
Ω(r, ρ) =
Z 2π
0
Z π
0 · · ·
Z π
0
(ρ2sin2θ1+ (ρcosθn−1−r)2)− γ 2
×sinn−2θ1sinn−3θ2· · ·sinθn−2dθ1· · ·dθn−2dθn−1. If r
2 ≤ρ≤2r, then by the fact
2n−2−γ >−1
Ω(r, ρ).ρ−γ
Z π
0
sinn−2−γθ1dθ1.ρ−γ.
Ifρ < r2, then Ω(r, ρ).r−γ, since |ρcosθ
n−1−r| ≥ r2. Therefore by H¨older and Hardy inequalities we have
II+III.rα+β−γ
Z ∞
0
ρn−1|f(ρ)| ρα
|g(ρ)| ρβ dρ.r
α+β−γ
kfkHαkgkHβ.
This completes the proof of the lemma. ¤
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Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan E-mail address:[email protected]
E-mail address:[email protected]
