Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials

0.41 MB·English

by  Haruya Mizutani

Strichartz estimates for Schr¨odinger equations with variable
coefficients and unbounded potentials
Haruya Mizutani∗
2
1 Abstract
0
The present paper is concerned with Schr¨odinger equations with variable coefficients
2
and unbounded electromagnetic potentials, where the kinetic energy part is a long-range
r
a perturbation of the flat Laplacian and the electric (resp. magnetic) potential can grow sub-
M
quadratically(resp. sublinearly)atspatialinfinity. Weprovesharp(local-in-time)Strichartz
estimates, outside a large compact ball centered at origin, for any admissible pair includ-
0
1 ing the endpoint. Under the nontrapping condition on the Hamilton flow generated by the
kinetic energy, global-in-space estimates are also studied. Finally, under the nontrapping
]
P condition, we prove Strichartz estimates with an arbitrarily small derivative loss without
A asymptotic flatness on the coefficients.
.
h
t
a 1 Introduction
m
[ Inthis paper,westudysharp(local-in-time) Strichartz estimates for Schro¨dingerequations with

2 variable coefficients and unbounded electromagnetic potentials. More precisely, we consider the
v
following Schro¨dinger operator:
1
0
2 1 d
5 H = ( i∂ A (x))gjk(x)( i∂ A (x))+V(x), x Rd,
j j k k
. 2 − − − − ∈
2 j,k=1
X
0
2 where d 1 is the spatial dimension. Throughout the paper we assume that gjk,V and A are
j
1 ≥
: smooth and real-valued functions on Rd and that (gjk(x))j,k is symmetric and positive definite:
v
i
X d
r gjk(x)ξjξk cξ 2, x,ξ Rd,
a ≥ | | ∈
j,k=1
X
with some c> 0. Moreover, we suppose the following condition:
2010 Mathematics Subject Classification. Primary 35Q41,35B45; Secondary 35S30, 81Q20.
Key words and phrases. Schr¨odinger equation, Strichartz estimates, asymptotically flat metric, unbounded
electromagnetic potentials.
∗Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan. E-mail:
[email protected]. The author is supported by GCOE ‘Fostering top leaders in mathematics’,
KyotoUniversity.
1
Assumption 1.1. There exists µ 0 such that for any α Zd,
≥ ∈ +
∂α(gjk(x) δ ) C x −µ−|α|,
| x − jk | ≤ αh i
∂αA (x) C x 1−µ−|α|,
| x j | ≤ αh i
∂αV(x) C x 2−µ−|α|, x Rd.
| x | ≤ αh i ∈
Then, it is well known that H admits a unique self-adjoint realization on L2(Rd), which we
denote by the same symbol H. By the Stone theorem, H generates a unique unitary propagator
e−itH on L2(Rd) such that the solution to the Schro¨dinger equation:
i∂ u(t) = Hu(t), t R; u = ϕ L2(Rd),
t t=0
∈ | ∈
is given by u(t) = e−itHϕ.
In order to explain the purpose of the paper more precisely, we recall some known results.
Let us first recall well known properties of the free propagator e−itH0, where H = ∆/2. The
0
−
distributionkernelofe−itH0 isgiven explicitly by(2πit)−d/2ei|x−y|2/(2t) ande−itH0ϕthussatisfies
the dispersive estimate:
e−itH0ϕ C t −d/2 ϕ , t = 0.
|| ||L∞(Rd) ≤ | | || ||L1(Rd) 6
Moreover, e−itH0 enjoys the following (global-in-time) Strichartz estimates:
e−itH0ϕ C ϕ ,
|| ||Lp(R;Lq(Rd)) ≤ || ||L2(Rd)
where (p,q) satisfies the following admissible condition:
2 1 1
p 2, = d , (d,p,q) = (2,2, ). (1.1)
≥ p 2 − q 6 ∞
(cid:18) (cid:19)
Strichartz estimates imply that, for any ϕ L2, e−itH0ϕ Lq for a.e. t R, where
∈ ∈ q∈Qd ∈
Q = [2, ], Q = [2, ) and Q = [2,2d/(d 2)] for d 3. These estimates hence can be
1 ∞ 2 ∞ d − ≥ T
regarded as Lp-type smoothing properties of Schro¨dinger equations, and have been widely used
in the study of nonlinear Schro¨dinger equations (see, e.g., [8]). Strichartz estimates for e−itH0
were first proved by Strichartz [32] for a restricted pair of (p,q) with p = q = 2(d + 2)/d,
and have been generalized for (p,q) satisfying (1.1) and p = 2 by [15]. The endpoint estimate
6
(p,q) = (2,2d/(d 2)) for d 3 was obtained by [20].
− ≥
For Schro¨dinger operators with electromagnetic potentials, i.e., H = 1( i∂ A)2 + V,
2 − x −
(short-time) dispersive and (local-in-time) Strichartz estimates have been extended with poten-
tials decaying at infinity [34] or growing at infinity [13, 35]. In particular, it was shown by
[13, 35] that if gjk = δ , V and A satisfy Assumption 1.1 with µ 0 and all derivatives of
jk
≥
the magnetic field B = dA are of short-range type, then e−itHϕ satisfies (short-time) dispersive
estimates:
e−itHϕ C t −d/2 ϕ ,
|| ||L∞(Rd) ≤ | | || ||L1(Rd)
for sufficiently small t = 0. Local-in-time Strichartz estimates, which have the forms
6
e−itHϕ C ϕ , T > 0,
|| ||Lp([−T,T];Lq(Rd)) ≤ T|| ||L2(Rd)
2
are immediate consequences of this estimate and the TT∗-argument due to Ginibre-Velo [12]
(see Keel-Tao [15] for the endpoint estimate). For the case with singular electric potentials or
with supercriticalelectromagnetic potentials, werefer to [34,36,38,9]. We mention that global-
in-time dispersive and Strichartz estimates for scattering states have been also studied under
suitabledecayingconditionsonpotentialsandassumptionsforzeroenergy;see[19,37,30,12,10]
and reference therein. We also mention that there is no result on sharp global-in-time dispersive
estimates for magnetic Schro¨dinger equations.
On the other hand, the influence of the geometry on the behavior of solutions to linear
and nonlinear partial differential equations has been extensively studied. From this geometric
viewpoint, sharp local-in-time Strichartz estimates for Schro¨dinger equations with variable co-
efficients (or, more generally, on manifolds) have recently been investigated by many authors
under several conditions on the geometry ; see, e.g., [31, 6, 26, 16, 4, 3, 7, 24] and reference
therein. In [31], [26], [4] authors studied the case on the Euclidean space with nontrapping
asymptotically flat metrics. The case on the nontrapping asymptotically conic manifold was
studied by [16] and [24]. In [3] the author considered the case of nontrapping asymptotically
hyperbolic manifold. For the trapping case, it was shown in [6] that Strichartz estimates with a
loss of derivative 1/p hold on any compact manifolds withoutboundaries. They also proved that
the loss 1/p is optimal in the case on Sd. In [4], [3] and [24], authors proved sharp Strichartz
estimates, outsidea large compact set, withoutthenontrappingcondition. Morerecently, it was
shown in [7] that sharp Strichartz estimates still hold for the case with hyperbolic trapped tra-
jectories of sufficiently small fractal dimension. We mention that there are also several works on
global-in-time Strichartz estimates in the case of long-range perturbations of the flat Laplacian
on Rd([5, 33, 23]).
While (local-in-time) Strichartz estimates are well studied subjects for both of these two
cases (at least under the nontrapping condition), the literature is more sparse for the mixed
case. In this paper we give a unified approach to a combination of these two kinds of results.
More precisely, under Assumption 1.1 with µ > 0, we prove (1) sharp local-in-time Strichartz
estimates, outside a large compact set centered at origin, without the nontrapping condition;
(2) the global-in-space estimates with the nontrapping condition. Under the nontrapping con-
dition and Assumption 1.1 with µ 0, we also show local-in-time Strichartz estimates with
≥
an arbitrarily small derivative loss. We mention that all results include the endpoint estimates
(p,q) = (2,2d/(d 2)) for d 3. This is a natural continuation of author’s previous work [25],
− ≥
which was concerned with the non-endpointestimates for the case with at most linearly growing
potentials.
F( ) denotes the characteristic function designated by ( ). We now state the main result.
∗ ∗
Theorem 1.2 (Strichartz estimates near infinity). Suppose that H satisfies Assumption 1.1
with µ > 0. Then, for any T > 0, p 2, q < and 2/p = d(1/2 1/q) and for sufficiently
≥ ∞ −
large R >0, we have
F(x > R)e−itHϕ C ϕ , (1.2)
|| | | ||Lp([−T,T];Lq(Rd)) ≤ T|| ||L2(Rd)
where C > 0 may be taken uniformly with respect to R.
T
To state the result on global-in-space estimates, we recall the nontrapping condition. Let us
3
denote by k(x,ξ) denotes the classical kinetic energy:
d
1
k(x,ξ) = gjk(x)ξ ξ ,
j k
2
j,k=1
X
and by (y (t,x,ξ),η (t,x,ξ)) the Hamilton flow generated by k(x,ξ):
0 0
y˙ (t)= ∂ k(y (t),η (t)), η˙ (t)= ∂ k(y (t),η (t)); (y (0),η (0)) = (x,ξ).
0 ξ 0 0 0 x 0 0 0 0
−
NotethattheHamiltonianvectorfieldH ,generatedbyk,iscompleteonR2d since(gjk)satisfies
k
the uniform elliptic condition. Hence, (y (t,x,ξ),η (t,x,ξ)) exists for all t R.
0 0
∈
Definition 1.3. We say that k(x,ξ) satisfies the nontrapping condition if for any (x,ξ) R2d
∈
with ξ = 0,
6
y (t,x,ξ) + as t . (1.3)
0
| | → ∞ → ±∞
The second result is the following.
Theorem 1.4 (Global-in-space Strichartz estimates). Suppose that H satisfies Assumption 1.1
with µ 0. Let T > 0, p 2, q < and 2/p = d(1/2 1/q). Then, for any r > 0, there exists
≥ ≥ ∞ −
C > 0 such that
T,r
F(x < r)e−itHϕ C H 1/pϕ . (1.4)
|| | | ||Lp([−T,T];Lq(Rd)) ≤ T,r||h i ||L2(Rd)
Moreover if we assume in addition that k(x,ξ) satisfies the nontrapping condition (1.3), then
F(x < r)e−itHϕ C ϕ . (1.5)
|| | | ||Lp([−T,T];Lq(Rd)) ≤ T,r|| ||L2(Rd)
In particular, combining with Theorem 1.2, we have (global-in-space) Strichartz estimates
e−itHϕ C ϕ ,
|| ||Lp([−T,T];Lq(Rd)) ≤ T|| ||L2(Rd)
under the nontrapping condition (1.3), provided that µ > 0.
When µ 0 we have the following partial result.
≥
Theorem 1.5 (Near sharp estimates without asymptotic flatness). Suppose that H satisfies
Assumption 1.1 with µ 0 and k(x,ξ) satisfies the nontrapping condition (1.3). Let T > 0,
≥
p 2, q < and 2/p = d(1/2 1/q). Then, for any ε > 0, there exists C > 0 such that
T,ε
≥ ∞ −
e−itHϕ C H εϕ .
|| ||Lp([−T,T];Lq(Rd)) ≤ T,ε||h i ||L2(Rd)
There are some remarks.
Remark 1.6. (1) Theestimates of forms(1.2), (1.4)and (1.5)have peenproved by [31,4]when
A 0 and V is of long-range type. Theorems 1.2 and 1.4 hence are regarded as generalizations
≡
of their results for the case with growing electromagnetic potential perturbations.
(2) Theonly restriction for admissible pairs, in comparison to theflat case, is to exclude (p,q) =
(4, ) for d= 1, which is due to the use of the Littlewood-Paley decomposition.
∞
4
(3) The missing derivative loss H ε in Theorem 1.5 is due to the use of the following local
h i
smoothing effect (due to Doi [11]):
x −1/2−ε D 1/2e−itHϕ C ϕ .
||h i h i ||L2([−T,T];L2(Rd)) ≤ T,ε|| ||L2(Rd)
It is well known that this estimate does not holds when ε = 0 even for H = H . We would
0
expect that Theorem 1.2 still holds true for the case with critical electromagnetic potentials in
the following sense:
x −1 ∂αA (x) + x −2 ∂αV(x) C x −|α|,
h i | x j | h i | x | ≤ αβh i
(at least if gjk satisfies the bounds in Assumption 1.1 with µ > 0). However, this is beyond our
techniques (see, also remark 4.2).
The rest of the paper is devoted to the proofs of Theorems 1.2, 1.4 and 1.5. Throughout the
paper we use the following notations: x stands for 1+ x 2. We write Lq = Lq(Rd) if there
h i | |
is no confusion. For Banach spaces X and Y, we denote by the operator norm from
p ||·||X→Y
X to Y. We write Z = N 0 . We denote the set of multi-indices by Zd. We denote by K
+ ∪{ } +
the kinetic energy part of H and by H the free Schro¨dinger operator:
0
d
1 1 1
K = ∂ gjk(x)∂ , H = ∆ = ∂2.
−2 j k 0 −2 −2 j
j,k j=1
X X
p(x,ξ) denotes the classical total energy (modulo lower order terms):
d
1
p(x,ξ) = gjk(x)(ξ A (x))(ξ A (x))+V(x).
j j k k
2 − −
j,k=1
X
For h (0,1] we consider Hh := h2H as a semiclassical Schro¨dinger operator with h-dependent
∈
electromagnetic potentials h2V and hA . We denote the corresponding total energy by p (x,ξ):
j h
d
1
p (x,ξ) = gjk(x)(ξ hA (x))(ξ hA (x))+h2V(x).
h j j k k
2 − −
j,k=1
X
Before starting the details of the proofs, we here describe the main ideas. At first we
remark that, since our Hamiltonian H is not bounded below in general, the Littlewood-Paley
decomposition associated with H does not hold for any p > 2. To overcome this difficulty, we
consider the following partition of unity on the phase space R2d:
ψ (x,ξ)+χ (x,ξ) = 1,
ε ε
where ψ is supported in (x,ξ); x < εξ for some sufficiently small constant ε > 0. It is easy
ε
{ h i | |}
to see that the total energy p(x,ξ) is elliptic on suppψ :
ε
C−1 ξ 2 p(x,ξ) C ξ 2, (x,ξ) suppψ ,
ε
| | ≤ ≤ | | ∈
and we hence can prove a Littlewood-Paley type decomposition of the following form:
1/2
Op(ψ )u C u +C Op (a )f(h2H)u 2 ,
|| ε ||Lq ≤ q|| ||L2 q || h h ||Lq
(cid:18)h=2X−j,j≥0 (cid:19)
5
where 2 q < , f(h2 );h = 2−j,j 0 is a 4-adic partition of unity on [1, ) and a is
h
≤ ∞ { · ≥ } ∞
an appropriate h-dependent symbol supported in x < 1/h, ξ I for some open interval
{| | | | ∈ }
I ⋐ (0, ), Op(ψ ) and Op (a ) denote the corresponding pseudodifferential and semiclassical
∞ ε h h
pseudodifferential operators, respectively.
Then, the idea of the proof of Theorem 1.2 is as follows. In view of the above Littlewood-
Paley estimate, the proof is reduced to that of Strichartz estimates for F(x > R)Op (a )e−itH
| | h h
and Op(χ )e−itH. In order to prove Strichartz estimates for F(x > R)Op (a )e−itH, we
ε | | h h
use semiclassical approximations of Isozaki-Kitada type. We however note that because of
the unboundedness of potentials with respect to x, it is difficult to construct directly such
approximations. To overcome this difficulty, we introduce a modified Hamiltonian H due to [38]
so that H = H for x L/h and H = K for x 2L/h for some constant L 1. Then,
| | ≤ | | ≥ ≥
Hh = h2H can be regarded as a “long-range perturbation” of the semiclassical freeeSchro¨dinger
operatoreHh = h2H . We also introduece thecorrespondingclassical total energy p (x,ξ) so that
0 0 h
pe(x,ξ) =ep (x,ξ) for x L/h and p (x,ξ) = k(x,ξ) for x 2L/h. Let a± be supported
h h | | ≤ h | | ≥ h
in outgoing and incoming regions R < x < 1/h, ξ I, xˆ ξˆ> 1/2 , respeectively, so that
{ | | | | ∈ ± · }
Fe(x > R)a = a++a−, where xˆ = x/ex . Rescaling t th, we first construct the semiclassical
app|r|oximatiohns fohr e−ihtHeh/hOp (a±)∗ |of|the following7→forms
h h
e−itHeh/hOp (a±)∗ = J (S±,b±)e−itH0h/hJ (S±,c±)∗+O(hN), 0 t 1/h,
h h h h h h h h ≤ ± ≤
respectively,whereS±solvetheEikonalequationassociatedtop andJ (S±,b±)andJ (S±,c±)
h h h h h h h h
areassociated semiclassical Fourierintegral operators. Themethodoftheconstructionissimilar
to as that of Robert [28]. On the other hand, we will see that ief L 1 is large enough, then the
≥
Hamiltonflowgeneratedbyp withinitialconditionsinsuppa± cannotescapefrom x L/h
h h {| | ≤ }
for 0 < t 1/h, respectively, i.e.,
± ≤
e
πx exptHpeh(suppa±h) ⊂ {|x| ≤ L/h}, 0< ±t ≤ 1/h.
(cid:0) (cid:1)
Since p = p for x L/h, we have
h h
| | ≤
e exptHpeh(suppa±h) = exptHph(suppa±h), 0< ±t ≤ 1/h.
We thus can expect (at least formally) that the corresponding two quantum evolutions are
approximatelyequivalentmodulosomesmoothingoperator. Wewillprovethefollowingrigorous
justification of this formal consideration:
(e−itHh/h e−itHeh/h)Op (a±)∗ C hM, 0 t 1/h, M 0,
|| − h h ||L2→L2 ≤ M ≤ ± ≤ ≥
where Hh = h2H. By using such approximations for e−itHh/hOp (a±)∗, we prove local-in-time
h h
dispersive estimates for Op (a±)e−itH Op (a±)∗:
h h h h
Op (a±)e−itH Op (a±)∗ C t −d/2, 0 < h 1, 0 < t < 1.
|| h h h h ||L1→L∞ ≤ | | ≪ | |
Strichartz estimates follow from these estimates and the abstract Theorem dueto Keel-Tao [20].
StrichartzestimatesforOp(χ )e−itH followfromthefollowingshort-timedispersiveestimate:
ε
Op(χ )e−itH Op(χ )∗ C t −d/2, 0 < t < t 1.
|| ε ε ||L1→L∞ ≤ ε| | | | ε ≪
6
To prove this, we construct an approximation for Op(χ )e−itH Op(χ )∗ of the following form:
ε ε
Op(χ )e−itH Op(χ )∗ = J(Ψ,a)+O (1), t < t ,
ε ε H−γ→Hγ ε
| |
where the phase function Ψ = Ψ(t,x,ξ) is a solution to a time-dependent Hamilton-Jacobi
equation associated to p(x,ξ) and J(Ψ,a) is the corresponding Fourier integral operator. In the
construction, the following fact plays an important rule:
∂α∂βp(x,ξ) C , (x,ξ) suppχ , α+β 2.
| x ξ | ≤ αβ ∈ ε | | ≥
We note that if (gjk) Id = 0 depends on x then these bounds do not hold without such
jk d
− 6
a restriction of the initial condition. Using these bounds, we can follow a classical argument
due to [21] and construct an approximation for e−itH Op(χ )∗ of the form J(Ψ,b) modulo some
ε
smoothingterm. Next, usinganEgorov typelemma, wewillprove thatOp(χ )(e−itH Op(χ )∗
ε ε
−
J(Ψ,b)) still can be considered as an “error” term.
The proof of Theorem 1.4 is based on a standard idea by [31, 6, 4]. Strichartz estimates with
loss follow from semiclassical Strichartz estimates up to time scales of order h, which can be
verified by the standard argument. Moreover, under the nontrapping condition, we will prove
that the missing 1/p derivative loss can berecovered by using local smoothing effects dueto Doi
[11].
Theproof of Theorem 1.5 is based on a slight modification of that of Theorem 1.4. By virtue
of the Strichartz estimates for Op(χ )e−itH and the Littlewood-Paley decomposition, it suffices
ε
to show
Op (a )e−itHϕ h−ε ϕ , 0 < h 1.
|| h h ||Lp([−T,T];Lq) ≤ || ||L2 ≪
To prove this estimate, we first prove semiclassical Strichartz estimates for Op (a )e−itH up to
h h
time scales of order hinf x . The proof is based on a refinement of the standard WKB approx-
| |
imation for the semiclassical propagator Op (a )e−itHh/h. Combining semiclassical Strichartz
h h
estimates with a partition of unity argument with respect to x, we will obtain the following
Strichartz estimate with an inhomogeneous error term:
Op (a )e−itHϕ C ϕ +C x −1/2−εh−1/2−εOp (a )e−itHϕ ,
|| h h ||Lp([−T,T];Lq) ≤ T|| ||L2 ||h i h h ||L2([−T,T];L2)
for any ε >0, which, combined with local smoothing effects, implies Theorem 1.5.
The paper is organized as follows. We first record some known results on the semiclassical
pseudodifferential calculus and prove the above Littlewood-Paley decomposition in Section 2.
Using dispersive estimates, which will be studied in Sections 4 and 5, we shall prove Theorem
1.2 in Section 3. We construct approximations of Isozaki-Kitada type and prove dispersive
estimates for Op (a±)e−itH Op (a±)∗ in Section 4. Section 5 discuss the dispersive estimates
h h h h
for Op(χ )e−itH Op(χ )∗. The proof of Theorem 1.4 and Theorem 1.5 are given in Section 6
ε ε
and Section 7, respectively.
Acknowledgements. We would like to express his sincere thanks to Erik Skibsted for valuable
discussionsandforhospitality atInstitutforMatematiske Fag, AarhusUniversitet, whereapart
of this work was carried out.
7
2 Semiclassical functional calculus
Throughout this section we assume Assumption 1.1 with µ 0, i.e.,
≥
∂αgjk(x) + x −1 ∂αA (x) + x −2 ∂αV(x) C x −|α|. (2.1)
| x | h i | x j | h i | x | ≤ αβh i
The goal of this section is to prove a Littlewood-Paley type decomposition under suitable re-
striction on the initial data. At first we record (without proof) some known results on the
pseudodifferential calculus which will be used throughout the paper. We refer to [27, 22] for the
details of the proof.
2.1 Pseudodifferential calculus
For the metric g = dx2/ x 2 +dξ2/ ξ 2 and a weight function m(x,ξ) on the phase space R2d,
h i h i
we use H¨ormander’s symbol class notation S(m,g), i.e., a S(m,g) if and only if a C∞(R2d)
∈ ∈
and
∂α∂βa(x,ξ) C m(x,ξ) x −|α| ξ −|β|, α,β Zd.
| x ξ | ≤ αβ h i h i ∈ +
To a symbol a C∞(R2d) and h (0,1], we associate the semiclassical pseudodifferential
∈ ∈
operator (h-PDO for short) defined by Op (a):
h
1
Op (a)f(x) = ei(x−y)·ξ/ha(x,ξ)f(y)dydξ, f S(Rd).
h (2πh)d ∈
Z
When h = 1 we write Op(a) = Op (a) for simplicity. The Calder´on-Vaillancourt theorem shows
h
that for any symbol a C∞(R2d) satisfying ∂α∂βa(x,ξ) C , Op (a) is extended to a
∈ | x ξ | ≤ αβ h
bounded operator on L2(Rd) uniformly with respect to h (0,1]. Moreover, for any symbol a
∈
satisfying
∂α∂βa(x,ξ) C ξ −γ, γ > d,
| x ξ | ≤ αβh i
Op(a ) is extended to a bounded operator from Lq(Rd) to Lr(Rd) with the following bounds:
h
Op (a) C h−d(1/q−1/r), 1 q r , (2.2)
|| h ||Lq→Lr ≤ qr ≤ ≤ ≤ ∞
where C > 0 is independent of h (0,1]. These bounds follow from the Schur lemma and an
qr
∈
interpolation (see, e.g., [4, Proposition 2.4]).
For two symbols a S(m ,g) and b S(m ,g), the composition Op (a)Op (b) is also a
∈ 1 ∈ 2 h h
h-PDO and written in the form Op (c) = Op (a)Op (b) with a symbol c S(m m ,g) given
h h h ∈ 1 2
by c(x,ξ) = eihDηDza(x,η)(z,ξ) . Moreover, c(x,ξ) has the following expansion
z=x,η=ξ
|
N−1 h|α|
c= ∂αa ∂αb+hNr with r S( x −N ξ −Nm m ,g). (2.3)
i|α|α! ξ · x N N ∈ h i h i 1 2
|α|=0
X
The symbol of the adjoint Op (a)∗ is given by a∗(x,ξ) = eihDηDza(z,η) S(m ,g)
h |z=x,η=ξ ∈ 1
which has the expansion
N−1 h|α|
a∗ = ∂α∂αa+hNr∗ with r∗ S( x −N ξ −Nm ,g). (2.4)
i|α|α! ξ x N N ∈ h i h i 1
|α|=0
X
8
2.2 Littlewood-Paley decomposition
As we mentioned in the outline of the paper, H is not bounded below in general and we hence
cannot expect that the Littlewood-Paley decomposition associated with H, which is of the form
∞ 1/2
u C u +C f(2−2jH)u 2 ,
|| ||Lq ≤ q|| ||L2 q || ||Lq
(cid:18)j=0 (cid:19)
X
holds if q = 2. The standard Littlewood-Paley decomposition associated with H also does
0
6
not work well in our case, since the commutator of H with the Littlewood-Paley projection
f(2−2jH ) can be grow at spatial infinity. To overcome this difficulty, let us introduce an
0
additional localization as follows. Given a parameter ε > 0 and a cut-off function ϕ C∞(R)
∈ 0
such that ϕ 1 on [0,1/2] and suppϕ [0,1], we define ψ (x,ξ) by
ε
≡ ⊂
x
ψ (x,ξ) = ϕ h i .
ε
εξ
(cid:18) | |(cid:19)
Itis easy to see that ψ is boundedin S(1,g) and supportedin (x,ξ) R2d; x < εξ .
ε 0<ε≤1
{ } { ∈ h i | |}
Moreover, forsufficiently smallε> 0, thetotal energy p(x,ξ)isuniformlyellipticon thesupport
of ψ and Op(ψ )H thus is essentially bounded below.
ε ε
In this subsection we prove a Littlewood-Paley type decomposition on the range of Op(ψ ).
ε
We begin with the following proposition which tells us that, for any f C∞(R) and h (0,1],
∈ 0 ∈
Op(ψ )f(h2H) is approximated in terms of the h-PDO.
ε
Proposition 2.1. There exists ε> 0 such that, for any f C∞(R), we can construct bounded
∈ 0
families a S( x −j ξ −j,g), j 0, such that
h,j h∈(0,1]
{ } ⊂ h i h i ≥
(1) a is given explicitly by a (x,ξ) = ψ (x,ξ/h)f(p (x,ξ)). Moreover,
h,0 h,0 ε h
suppa suppψ (, /h) suppf(p )
h,j ε h
⊂ · · ∩
(x,ξ) R2d; x < 1/h, ξ I ,
⊂ { ∈ h i | | ∈ }
for some open interval I ⋐ (0, ). In particular, we have
∞
Op (a ) C h−d(1/q′−1/q), 1 q′ q ,
|| h h,j ||Lq′→Lq ≤ jqq′ ≤ ≤ ≤ ∞
uniformly in h (0,1].
∈
(2) For any integer N > 2d, we set a = N−1hja . Then,
h j=0 h,j
P
Op(ψ )f(h2H) Op (a ) C hN/2, 2 q ,
|| ε − h h ||L2→Lq ≤ qN ≤ ≤ ∞
uniformly in h (0,1].
∈
The following is an immediate consequence of this proposition.
Corollary 2.2. For any 2 q and h (0,1], Op(ψ )f(h2H) is bounded from L2(Rd) to
ε
≤ ≤ ∞ ∈
Lq(Rd) and satisfies
Op(ψ )f(h2H) C h−d(1/2−1/q),
|| ε ||L2→Lq ≤ q
where C > 0 is independent of h (0,1].
q
∈
9
Remark 2.3. If V,A 0, then Proposition 2.1 and Corollary 2.2 hold without the additional
≡
term Op(ψ ). We refer to [6] (for the case on compact manifolds without boundary) and to [4]
ε
(for the case with metric perturbations on Rd). For more general cases with Laplace-Beltrami
operators non-compact manifolds with ends, we refer to [2, 1].
Proof of Proposition 2.1. We begin with the well-known Helffer-Sjo¨strand formula [17]:
1 ∂f
f(h2H) = (z)(h2H z)−1dz dz¯,
−2πi C ∂z¯ − ∧
Z
e
where f(z) is an almost analytic extension of f(λ). Since f C∞(R), f(z) is also compactly
∈ 0
supported and satisfies ∂ f(z) = O( Imz M) for any M > 0. We shall construct a semiclassical
z¯
| |
approxiemation of Op(ψ )(h2H z)−1 for z C [0, ). Although the meethod is based on the
ε
− ∈ \ ∞
e
standardsemiclassical parametrixconstruction (see, e.g.,[27,6]), wegivethedetails oftheproof
since we consider the composition of the PDO, Op(ψ ), which is not in the semiclassical regime,
ε
with the semiclassical resolvent (h2H z)−1.
−
p(x,ξ) and p (x,ξ) denote the principal symbol and the subsymbol of H, respectively, i.e.,
1
H = p(x,D)+p (x,D). (2.1) and the support property of ψ imply
1 ε
∂α∂βp(x,ξ) C x −|α| ξ 2−|β|,
| x ξ | ≤ αβh i h i (2.5)
∂α∂βp (x,ξ) C x −1−|α| ξ 1−|β|, (x,ξ) suppψ .
| x ξ 1 | ≤ αβh i h i ∈ ε
Moreover, we obtain
(gjk(x)ξ A (x) + gjk(x)A (x)A (x))+ V(x) Cεξ 2, (x,ξ) suppψ ,
j k j k ε
| | | | | | ≤ | | ∈
j,k
X
where C > 0 is independent of x,ξ and ε. This estimate and the uniform ellipticity of k imply
that p(x,ξ) is also uniformly elliptic on suppψ :
ε
C−1 ξ 2 p(x,ξ) C ξ 2, (x,ξ) suppψ ,
1 | | ≤ ≤ 1| | ∈ ε
provided that ε > 0 is small enough. Then, for any integer N 0, we can find symbols
≥
q (z,x,ξ), j = 0,1,...,N 1, and r (z,x,ξ), depending holomorphically on z C R, such
h,j h,N
− ∈ \
that
N−1
Op(ψ ) = Op(q (z))(h2H z)+Op(r (z)).
ε h,j h,N
−
j=0
X
More precisely, q is given explicitly by
h,0
ψ (x,ξ)
ε
q (z,x,ξ) = .
h,0 h2p(x,ξ) z
−
Using (2.5) and the fact that ψ S(1,g), we obtain
ε
∈
x −|α| ξ 2l−|β|h2l
∂α∂βq (z,x,ξ) C h i h i .
| x ξ h,0 | ≤ αβ h2p(x,ξ) z l+1
0≤l≤|β|+|α| | − |
X
10