Table Of ContentRESTRICTION BOUNDS FOR THE FREE RESOLVENT AND
RESONANCES IN LOSSY SCATTERING
4 JEFFREY GALKOWSKIANDHARTSMITH
1
0
2 Abstract. WeestablishhighenergyL2 estimatesfortherestrictionofthefreeGreen’sfunction
n to hypersurfaces in Rd. As an application, we estimate the size of a logarithmic resonance free
a region for scattering by potentials of the form V ⊗δΓ, where Γ ⊂ Rd is a finite union of com-
J
pact subsets of embedded hypersurfaces. In odd dimensions we prove a resonance expansion for
4 solutions to thewave equation with such a potential.
2
]
P
A
. 1. Introduction
h
t
a Scattering by potentials is used in math and physics to study waves in many physical systems
m
(see for example [7], [14], [22], [23] and the references therein). Examples include acoustics in
[
concert halls and scattering of light by black holes. One case of recent interest is scattering in
1 quantum corrals that are constructed using scanning tunneling microscopes [4] [10]. One model
v for this system is that of a delta function potential on the boundary of a domain Ω Rd (see for
3 ⊂
example [2], [4], [10]). In this paper, we study scattering by such a delta function potential on
4
2 hypersurfaces Γ Rd.
6 ⊂
We assume that Γ Rd is a finite union of compact subsets of embedded C1,1 hypersurfaces;
.
1 that is, it is a union ⊂of compact subsets of graphs of C1,1 functions. The Bunimovich stadium
0
4 is an example of a domain in two dimensions which has boundary that is C1,1, but not C2. We
1 let δ denote the surface measure on Γ, considered as a distribution on Rd, and take V to be a
Γ
v: bounded,self-adjoint operator on L2(Γ). For u∈ Hl1oc(Rd), we then define(V ⊗δΓ)u:= (Vu|Γ)δΓ.
i
X ResonancesaredefinedaspolesofthemeromorphiccontinuationfromImλ 0oftheresolvent
≫
ar RV(λ) = ( ∆V,Γ λ2)−1,
− −
where ∆ is the unbounded self-adjoint operator
V,Γ
−
∆ := ∆+V δ
V,Γ Γ
− − ⊗
(See Section 2.1 for the formal definition of ∆ ). If the dimension d is odd, R (λ) admits a
V,Γ V
−
meromorphic continuation to the entire complex plane, and to the logarithmic covering space of
C 0 if d is even (see Section 7).
\{ }
Theimaginary part of a resonance gives the decay rate of the associated resonant states. Thus,
resonancesclosetotherealaxisgiveinformationaboutlongtermbehaviorofwaves. Inparticular,
since the seminal work of Lax-Phillips [14] and Vainberg [21], resonance free regions near the real
axis have been used to understand decay of waves.
1
2 J.GALKOWSKIANDH.SMITH
In this paper, we demonstrate the existence of a resonance free region for delta function poten-
tials on a very general class of Γ.
Theorem 1. Let Γ Rd be a finite union of compact subsets of embedded C1,1 hypersurfaces,
⊂
and suppose V is a self-adjoint operator on L2(Γ).
Then for all ǫ > 0 there exists R > 0 such that, if λ is a resonance for ∆ , then
V,Γ
−
(1.1) Imλ 1d−1 ǫ log(Reλ ) if Reλ R,
≤ − 2 Γ − | | | |≥
(cid:16) (cid:17)
where d is the diameter of the convex hull of Γ. If Γ can be written as a finite union of strictly
Γ
convex C2,1 hypersurfaces, then we can replace 1 by 2 in (1.1).
2 3
Remarks:
Theseboundsonthesizeoftheresonancefreeregionarenotgenerallyoptimal,forexample
• in the case that Γ = ∂B(0,1) R2. In [12], the first author uses a microlocal analysis
⊂
of the transmission problem (1.7) to obtain sharp bounds in the case that Γ = ∂Ω is C∞
with Ω strictly convex.
In the smooth, strictly convex case, scattering in other types of transmission problems
•
was considered in [9] and [15].
Let R (λ) be the analytic continuation of the outgoing free resolvent ( ∆ λ2)−1, defined
0
− −
initially forImλ> 0. Theorem 1 follows fromboundson an operator related tothe freeresolvent.
In particular, we study the restriction of R (λ) to hypersurfaces Γ Rd. Let γ denote restriction
0
⊂
to Γ, and γ∗ the inclusion map f fδ . Let G(λ) : L2(Γ) L2(Γ) be obtained by restricting
Γ
7→ →
the kernel G (λ,x,y) of R (λ) to Γ,
0 0
G(λ) := γR (λ)γ∗.
0
Theorem 1 will follow as a consequence of the following theorem,
Theorem 2. Let Γ Rd be a finite union of compact subsets of embedded C1,1 hypersurfaces.
⊂
Then G(λ) is a compact operator on L2(Γ), and
(1.2) G(λ) L2(Γ)→L2(Γ) C λ −21 log λ edΓ(Imλ)−,
k k ≤ h i h i
where d is the diameter of the convex hull of Γ. Moreover, if Γ is a finite union of compact
Γ
subsets of strictly convex C2,1 hypersurfaces, then
(1.3) G(λ) L2(Γ)→L2(Γ) C λ −23 log λ edΓ(Imλ)−.
k k ≤ h i h i
1
Here we set λ = (2+ λ 2)2, and (Imλ)− = max(0, Imλ). Compactness follows easily by
h i | | −
Rellich’s embedding theorem, or the bounds on G (λ,x,y) in Section 2.2. The powers 1 and
0 2
2 in (1.2) and (1.3), respectively, are in general optimal. This follows from the fact that the
3
corresponding estimates for the restriction of eigenfunctions in Section 4 are the best possible.
However, it is likely that the factor of log λ is not needed. In Section 3 we prove estimate (1.2)
h i
in dimension two without it. Also, for the flat case in general dimensions, the estimate (1.2) holds
without it. We also expect that estimate (1.3) holds for C1,1 strictly convex hypersurfaces, but
do not pursue that here.
RESTRICTION BOUNDS FOR THE FREE RESOLVENT 3
1 2
In the case that Imλ λ 2, respectively Imλ λ 3, the above bounds can be improved
≥ | | ≥ | |
upon.
Theorem 3. Let Γ Rd be a finite union of compact subsets of embedded C1,1 hypersurfaces.
⊂
Then for Imλ> 0,
G(λ) L2(Γ)→L2(Γ) C Imλ −1.
k k ≤ h i
We next use the results above to analyze the long term behavior of waves scattered by the
potentialV δ . Theorem1impliesinparticularthatthereareonlyafinitenumberofresonances
Γ
⊗
in the set Imλ> A, for any A< . We give a resonance expansion for the wave equation
− ∞
(1.4) ∂2 ∆+V δ u= 0, u(0,x) = 0, ∂ u(0,x) = g L2 ,
t − ⊗ Γ t ∈ comp
with wave propa(cid:0)gator U(t) define(cid:1)d using the functional calculus for ∆V,Γ. Let mR(λ) be the
−
multiplicity of the pole of R (λ) at λ, that is the dimension of the set of resonant states with
V
resonance λ, and let be the domain of ( ∆ )N.
N V,Γ
D −
Theorem 4. Let d > 0 be odd, and assume that Γ Rd is a finite union of compact subsets of
⊂
embedded C1,1 hypersurfaces, and that V is a self-adjoint operator on L2(Γ).
Let 0 > µ2 > > µ2 and 0 < ν2 < < ν2 be the nonzero eigenvalues of ∆ , and
− 1 ··· − N 1 ··· M − V,Γ
λ the resonances with Imλ < 0. Then for any A > 0 and g L2 , the solution U(t)g to
{ j} ∈ comp
(1.4) admits an expansion
N M
(1.5) U(t)g = (2µk)−1etµkΠµkg+tΠ0g+P0g+ (2νk)−1sin(tνk)Πνkg
k=1 k=1
X X
mR(λj)−1
+ e−itλj tk g+E (t)g,
Pλj,k A
ImλXj>−A kX=0
where Π and Π respectively denote the projections onto the µ2 and ν2 eigenspaces. The
µk νk − k k
maps are bounded from L2 , and is a symmetric map to the 0-resonances.
Pλj,k comp → Dloc P0
The operator E (t) : L2 L2 has the following property: for any χ C∞(Rd) equal to 1
A comp → loc ∈ c
on a neighborhood of Γ, and any N 0, there exists T < so that
A,χ,N
≥ ∞
kχEA(t)χkL2→DN ≤ CA,χ,Ne−At, t > TA,χ,N .
Under the assumption that Γ = ∂Ω for a bounded open domain Ω Rd, and that V and ∂Ω
⊂
satisfy higher regularity assumptions, we obtain estimates for χE (t)χg in the spaces
A
:= H1(Rd) (HN(Ω) HN(Rd Ω)), N 1.
N
E ∩ ⊕ \ ≥
1
If ∂Ω is of C1,1 regularity, and V is bounded on H2(∂Ω), then we show 1 = 2, and
D D ⊂ E
convergence in follows from Theorem 4. For smooth boundaries we show the following.
2
E
Theorem 5. Suppose that Γ = ∂Ω is C∞ and that V is bounded on Hs(∂Ω) for all s. Then the
operator E (t) defined in (1.5) has the following property: for any χ C∞(Rd) equal to 1 on a
A c
∈
neighborhood of Ω, and integer N 1, there exists T < so that
A,χ,N
≥ ∞
kχEA(t)χkL2→EN ≤ CA,χ,N e−At, t > TA,χ,N .
4 J.GALKOWSKIANDH.SMITH
In addition to describing resonances as poles of the meromorphic continuation of the resolvent,
we will give a more concrete description of resonances in Sections 6 and 7. We show that λ is a
resonance of the system if and only if there is a nontrivial λ-outgoing solution u to the
loc
∈ D
equation
(1.6) ( ∆ λ2+V δ )u= 0,
Γ
− − ⊗
where we define
= u : χu whenever χ C∞(Rd) and χ = 1 on a neighborhood of Γ .
loc c
D { ∈ D ∈
Here we say that u is λ-outgoing if for some R < , and some compactly supported d(cid:9)istribution
∞
g, we can write
u(x) = R (λ)g (x) for x R.
0
| | ≥
Moreover, if we assume that Γ = ∂Ω(cid:0) for a C(cid:1) 1,1 domain Ω, and that V : H21(∂Ω) H12(∂Ω),
we show this is equivalent to solving the following transmission problem with u H→1 (Rd), and
∈ loc
with u = u H2(Ω), u = u H2 (Rd Ω),
|Ω 1 ∈ |Rd\Ω 2 ∈ loc \
( ∆ λ2)u = 0 in Ω
1
− −
( ∆ λ2)u2 = 0 in Rd Ω
(1.7) u∂−1u=−+u2∂ u +Vu =0 oonn ∂∂ΩΩ\
ν 1 ν′ 2 1
Here, ∂ν and ∂ν′ are respectiveluy2tλh-eoiunttgeoriinorgand exterior normal derivatives of u at ∂Ω.
The outline of this paper is as follows. In Section 2 we present the definition of ∆ and
V,Ω
−
its domain, as well as some preliminary bounds on the outgoing Green’s function G (λ,x,y). In
0
Section 3 we give a simple proof of Theorem 2 for d= 2. In Section 4 we establish Theorem 2 for
Imλ 0 in all dimensions, deriving the estimates from restriction estimates for eigenfunctions
≥
of the Laplacian. We include a proof of the desired restriction estimate for hypersurfaces of
regularity C1,1, since the result appears new, and also provide the proof of Theorem 3. In Section
5 we complete the proof of Theorem 2 for Imλ < 0 using the Phragm´en-Lindel¨of theorem. In
Section 6 we demonstrate the meromorphic continuation of R (λ), give the proof of Theorem 1,
V
and relate resonances to solvability of an equation on Γ, and for Γ = ∂Ω to solvability of (1.7). In
Section7wegivemoredetailed structureofthemeromorphiccontinuation ofR (λ). Weestablish
V
mapping bounds for compact cutoffs of R (λ), and use these to prove Theorems 4 and 5 by a
V
contour integration argument. In Section 8 we prove a needed transmission property estimate for
boundaries of regularity C1,1.
Acknowledgements. The authors would like to thank Maciej Zworski for valuable guidance
and discussions and Semyon Dyatlov for many useful conversations. This material is based upon
work supported by the National Science Foundation under Grants DGE-1106400, DMS-1201417,
and DMS-1161283. This work was partially supported by a grant from the Simons Foundation
(#266371 to Hart Smith).
RESTRICTION BOUNDS FOR THE FREE RESOLVENT 5
2. Preliminaries
2.1. Determination of ∆ and its domain. We define the operator ∆ using the sym-
V,Γ V,Γ
− −
metric, densely defined quadratic form
QV,Γ(u,w) := h∇u,∇wiL2(Rd)+hVu,wiL2(Γ)
with domain H1(Rd) L2(Rd).
⊂
For Γ a finite union of compact subsets of C1,1 hypersurfaces (indeed Lipschitz hypersurfaces
suffice), we can bound
1 1
kukL2(Γ) ≤ CkukL22kukH21 ≤ CǫkukH1 +Cǫ−1kukL2.
It follows that there exist c,C > 0 such that
|QV,Γ(u,w)| ≤ kukH1kwkH1 and ckuk2H1 ≤QV,Γ(u,u)+Ckuk2L2.
By Reed-Simon [16, Theorem VIII.15], Q (u,w) is determined by a uniqueself-adjoint operator
V,Γ
∆V,Γ, with domain consisting of u H1 for which QV,Γ(u,w) C w L2.
− D ∈ ≤ k k
For u , by the Riesz representation theorem, we have Q (u,w) = f,w for some f
V,Γ
∈ D h i ∈
L2(Rd), and taking w C∞(Rd) shows that in the sense of distributions
c
∈
(2.1) ∆u+(Vu )δ = f .
Γ Γ
− |
Conversely, if u H1(Rd) and (2.1) holds for some f L2(Rd), then by density of C∞ H1 we
c
have Q (u,w)∈= f,w for w H1(Rd), hence u ∈, and ∆ u is given by the left h⊂and side
V,Γ V,Γ
h i ∈ ∈ D −
of (2.1). We thus can write (up to a constant of proportionality)
u D = u H1 + ∆V,Γu L2,
k k k k k k
where finiteness of the second term carries the assumption that ∆ u L2.
V,Γ
∈
The domain is defined for N 1 by the condition ∆ u , and we will
N V,Γ N−1
D ⊂ D ≥ ∈ D
recursively define (consistent up to constants with the definition using the functional calculus)
kukDN = kukH1 +k∆V,ΓukDN−1, N ≥ 1.
Suppose that χ C∞(Rd Γ) and that u H1 solves (2.1). Then,
c
∈ \ ∈
∆(χu)= χf +2 χ u+(∆χ)u L2(Rd).
∇ ·∇ ∈
Hence,
χu H2 Cχ u D.
k k ≤ k k
That is, H1(Rd) H2 (Rd Γ), with continuous inclusion. Similar arguments show that
D ⊂ ∩ loc \
H1(Rd) H2N(Rd Γ).
DN ⊂ ∩ loc \
The behavior of u near Γ may be more singular. For general V acting on L2(Γ), from (2.1)
and the fact that (Vu Γ)δΓ H−12−ǫ(Rd) for all ǫ > 0, we conclude that u H23−ǫ(Rd). However,
| ∈ ∈
under additional assumptions on V and Γ we can give a full description of near Γ.
D
For the purposes of the remainder of this section we assume that Γ = ∂Ω for some bounded
opendomain Ω Rd, andthat ∂Ω is aC1,1 hypersurface; that is, locally ∂Ω can bewritten as the
⊂
6 J.GALKOWSKIANDH.SMITH
graph of a C1,1 function. We assume also that V : H12(∂Ω) H21(∂Ω). Then since u H1(Rd),
1 → ∈
Vu ∂Ω H2(∂Ω). Hence, Lemma 8.2 combined with (2.1) shows that
| ∈
= H1(Rd) (H2(Ω) H2(Rd Ω)),
2
D ⊂ E ∩ ⊕ \
with continuous inclusion. We remark that H2(Ω)and H2(Rd Ω)can beidentified as restrictions
ofH2(Rd)functions; see[8]and[17, TheoremVI.5]. Thus,ifu\ bothuandits firstderivatives
3 ∈ D 1
have well defined traces on ∂Ω, respectively of regularity H2(∂Ω) and H2(∂Ω).
For w H1(Rd) and u H1(Rd) (H2(Ω) H2(Rd Ω)), it follows from Green’s identities
∈ ∈ ∩ ⊕ \
that
Q (u,w) = ∆u,w + ∆u,w + ∂ u+∂ u+Vu,w ,
V,∂Ω h− iΩ h− iRd\Ω h ν ν′ i∂Ω
where ∂ and ∂ denote the exterior normal derivatives from Ω and Rd Ω. Thus, in the case
ν ν′
1 1 \
that V : H2(∂Ω) H2(∂Ω), we can completely characterize the domain of the self-adjoint
→ D
operator ∆ as
V,∂Ω
−
(2.2) = u H1(Rd) H2(Ω) H2(Rd Ω) such that ∂ u+∂ u+Vu = 0 ,
ν ν′
D ∈ ∩ ⊕ \
in which case ∆(cid:8) u= ∆u (cid:0)∆u . (cid:1) (cid:9)
V,∂Ω |Ω ⊕ |Rd\Ω
2.2. Bounds on Green’s function. We conclude this section by reviewing bounds on the con-
volution kernel G (λ,x,y) associated to the operator R (λ). It can be written in terms of the
0 0
Hankel functions of the first kind,
G (λ,x,y) = C λd−2 λ x y −d−22 H(1) λ x y ,
0 d | − | d2−1 | − |
(cid:0) (cid:1) (cid:0) (cid:1)
for some constant C . If d is odd, this can be written as a finite expansion
d
d−2 c
G (λ,x,y) = λd−2eiλ|x−y| d,j .
0 j
λ x y
j=Xd−21 | − |
For x= y this form extends to λ C, and defines the anal(cid:0)ytic exte(cid:1)nsion of R (λ). In particular,
0
6 ∈
we have the upper bounds
x y 2−d, x y λ −1,
(2.3) G (λ,x,y) . | − | | − |≤ | |
| 0 | e−Imλ|x−y| λ d−23 x y 1−2d , x y λ −1.
| | | − | | − |≥ | |
For d even, and d = 2, the bounds (2.3) hold for Imλ > 0, as well as for the analytic extension
6
(1)
to π argλ 2π. For π < argλ < 2π this follows by the asymptotics of H (z); see for
n
− ≤ ≤ −
example [1, (9.2.3)]. To see that it extends to the closed region, we use the relation (valid in all
dimensions)
G (eiπλ,x,y) G (λ,x,y) = iλd−2 eiλhx−y,ωidω = C λd−2 λ x y −d−22 J λ x y
0 − 0 (2π)d−1 Sd−1 d | − | d2−1 | − |
Z
(cid:0) (cid:1) (cid:0) (cid:1)
wheredω is surfacemeasureon theunitsphereSd−1 Rd, andeiπ indicates analytic continuation
⊂
through positive angle π. The bounds (2.3) then follow from the asymptotics of J (z) and the
n
RESTRICTION BOUNDS FOR THE FREE RESOLVENT 7
bounds for Imλ 0. We also note as a consequence of the above that, for λ R 0 , and any
≥ ∈ \{ }
sheet of the continuation in even dimensions,
(2.4) G0(eiπλ,x,y) G0(λ,x,y) = 2πi(sgnλ)d λ −1δ[Sd−1(x y),
− | | λ −
where δSd−1 denotes surface measure on the sphere ξ = λ in Rd.
λ | | | |
In the case that d = 2, in (2.3) one need replace x y 2−d by ln x y in the bounds for
| − | − | − |
x y λ −1. However, for our purposes we use only the following global bound in case d= 2,
| − | ≤ | |
G0(λ,x,y) . e−Imλ|x−y| λ −21 x y −12 , d = 2, π argλ 2π.
| | | | | − | − ≤ ≤
Finally, weobservethatsinceG (λ,x,y)issmoothaway fromthediagonal, withthesingularity
0
at x= y integrable over Γ, it follows that G(λ) is a compact operator on L2(Γ) for every λ. This
also follows from (7.7).
3. Estimates for d =2
In this section we give an elementary proof of estimate (1.2) of Theorem 2 for d = 2. Indeed,
we can prove the following stronger result,
Theorem 6. Under the conditions of Theorem 2 the following holds, for π argλ 2π,
− ≤ ≤
1 1
C λ −2 Imλ −2 f L2(Γ), Imλ 0,
kG(λ)fkL2(Γ) ≤ Chhλii−21he−dΓIimλkkfkkL2(Γ), Imλ ≥≤ 0.
Proof. We use the kernel bounds, for x,y in a bounded set,
G0(λ,x,y) Ce−Imλ|x−y| λ −12 x y −12 .
| | ≤ h i | − |
By the Schur test and symmetry of the kernel, the operator norm is bounded by the following
sup G (λ,x,y) dσ(y).
0
x ZΓ| |
First consider Imλ 0. Then e−Imλ|x−y| e−dΓImλ for x,y Γ, and since Γ is a finite union of
≤ ≤ ∈
subsets of C1,1 hypersurfaces the desired bound follows from the following, which holds if Γ is a
bounded subset of the graph of a Lipschitz function,
1
sup x y −2 dσ(y) C.
x ZΓ| − | ≤
For Imλ 0, we use instead the bound
≥
sup e−Imλ|x−y| x y −12 dσ(y) C Imλ −12 .
x ZΓ | − | ≤ h i
(cid:3)
8 J.GALKOWSKIANDH.SMITH
4. Resolvent Bounds in the Upper Half Plane
In this section, we prove Theorems 2 and 3 for Imλ> 0.
We assume that Γ is a finite union of compact subsets of embedded C1,1 hypersurfaces, with
induced surface measure. For f L2(Γ) we use fδ = γ∗f to denote the induced compactly
Γ
∈
supported distribution.
For Imλ > 0 let R (λ) = ( ∆ λ2)−1 be the operator with Fourier multiplier (ξ 2 λ2)−1.
0
− − | | −
For the proof of both Theorem 2 and Theorem 3 we will estimate
(4.1) Q (f,g) := R (λ)(fδ )gδ .
λ 0 Γ Γ
Z
ForImλ >0,therighthandside(4.1)agreeswiththedistributionalpairingofR0(λ)(fδΓ) H32−ǫ
with gδΓ H−12−ǫ, and hence by the Plancherel theorem ∈
∈
fδ (ξ)gδ (ξ)
Γ Γ
(4.2) Q (f,g) = dξ.
λ ξ 2 λ2
Z d| | −d
For λ 2, the uniform bounds
| | ≤
Qλ(f,g) C f L2(Γ) g L2(Γ), λ 2,
| | ≤ k k k k | | ≤
follow easily from fδΓ,gδΓ H−12−ǫ(Rd), so we focus on λ 2.
∈ | |≥
We start by showing that resolvent bounds for λ in the upper half plane can be deduced from
restrictionboundsforfδ . Indeed,thefollowingequivalenceholdswithδ replacedbyanyregular
Γ Γ
measure supported on a compact set.
d
Lemma 4.1. Suppose that for some α (0,1) the following estimate holds for r > 0,
∈
2
(4.3) fδ (ξ) δ(ξ r) C r α f 2 .
Γ | |− ≤ h i k kL2(Γ)
Z (cid:12) (cid:12)
Then, for λ in the upper half pl(cid:12)(cid:12)adne wit(cid:12)(cid:12)h λ 2,
| |≥
Qλ(f,g) C λ α−1log λ f L2(Γ) g L2(Γ),
| | ≤ | | | |k k k k
where Q is as in (4.1).
λ
Proof. Consider first the integral in (4.2) over ξ λ 1. Since ξ 2 λ2 ξ 2 λ 2 , by
| |−| | ≥ | | − ≥ | | −| |
the Schwartz inequality and (4.3) this piece of the integral is bounded by
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
f L2(Γ) g L2(Γ) r α r2 λ 2 −1dr C λ α−1log λ f L2(Γ) g L2(Γ).
k k k k Z|r−|λ||≥1h i −| | ≤ | | | |k k k k
(cid:12) (cid:12)
Next, if Imλ 1, then ξ 2 λ2 λ(cid:12), and by (cid:12)(4.3)
≥ | | − ≥ | |
(cid:12) (cid:12)
(cid:12) fδ(cid:12) (ξ)gδ (ξ)
(cid:12)(cid:12) Z||ξ|−|λ||≤1 d|Γξ|2−dλΓ2 dξ (cid:12)(cid:12) ≤ C|λ|α−1kfkL2(Γ)kgkL2(Γ).
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
RESTRICTION BOUNDS FOR THE FREE RESOLVENT 9
Thus, we may restrict our attention to 0 Imλ 1 and ξ λ 1. For this piece we use
≤ ≤ | |−| | ≤
that (4.3) implies
(cid:12) (cid:12)
(cid:12) (cid:12)
2
(4.4) fδ (ξ) δ(ξ r) C r α f 2 ,
∇ξ Γ | |− ≤ h i k kL2(Γ)
Z (cid:12) (cid:12)
due to the compact support o(cid:12)(cid:12)f fδΓd. (cid:12)(cid:12)
We consider Reλ 0, the other case following similarly, and write
≥
1 1 ξ
= log(ξ λ),
ξ 2 λ2 ξ +λ ξ ·∇ξ | |−
| | − | | | |
where the logarithm is well defined since Im(ξ λ) < 0. Let χ(r) = 1 for r 1 and vanish for
| |− | | ≤
r 3. We then use integration by parts, together with (4.3) and (4.4) to bound
| |≥ 2
1 ξ
(cid:12) χ(|ξ|−|λ|) ξ +λfδΓ(ξ)gδΓ(ξ) ξ ·∇ξlog(|ξ|−λ)dξ (cid:12) ≤ C|λ|α−1kfkL2(Γ)kgkL2(Γ).
(cid:12) Z | | | | (cid:12)
(cid:12) d d (cid:12)
(cid:12) (cid:12) (cid:3)
(cid:12) (cid:12)
To conclude the proof of Theorem 2, we need to show that (4.3) holds with α = 1 when Γ is a
2
compact subset of a C1,1 hypersurface, and with α= 1 for a compact subset of a strictly convex
3
C2,1 hypersurface. Since we work locally we assume that Γ is given by the graph x = F(x′),
n
where by an extension argument we assume that F is a C1,1 function (respectively C2,1 function)
defined on Rn, and we replace surface measure on Γ by dx′. By scaling we may assume that
F 1 . We may also assume that F(0) = 0.
|∇ | ≤ 20
Let r 1, and let δSd−1 =δ(ξ r) be surface measure on the sphere Sdr−1 of radius r. Assume
≥ r | |−
that g(ξ) is a function belonging to L2(Sd−1), and define
r
Tg(x) = eihx,ξig(ξ)δ(ξ r).
| |−
Z
Let χ(x′) C∞(Rd−1) be supported in the unit ball. By duality, (4.3) with α = 1 is equivalent
∈ c 2
to the following estimate
1
(4.5) (Tg)(x′,F(x′)) 2χ(x′)dx′ 2 ≤ Cr14 kgkL2(Srd−1),
(cid:18)Z (cid:19)
(cid:12) (cid:12)
and for α = 1 is equivalen(cid:12)t to the same e(cid:12)stimate with r14 replaced by r16.
3
The estimate (4.5) is known as a restriction estimate for eigenfunctions of the Laplacian. Lp
generalizations in the setting of a smooth Riemannian manifold, with restriction to a smooth
submanifold, were studied by Burq, G´erard and Tzvetkov in [6]. The L2 estimates, again in the
smooth setting, were noted by Tataru [19] as being a corollary of an estimate of Greenleaf and
Seeger [13]. These estimates were generalized to the setting of restriction to smooth submanifolds
in Riemannian manifolds with metrics of C1,1 regularity by Blair [3]. In making a change of
coordinates to flatten a submanifold the resulting metric has one lower order of regularity, thus
the estimates of [3] do not apply directly to C1,1 submanifolds, and so we include here the proof
of the L2 estimate on C1,1 hypersurfacesof Euclidean space. Theestimate for strictly convex C2,1
10 J.GALKOWSKIANDH.SMITH
hypersurfacesdoes follow from [3], so we consider herejustthe case of a general C1,1 hypersurface
and α = 1.
2
We derive (4.5) from the following square function estimate for solutions to the wave equation.
Lemma 4.2. Suppose that f L2(Rd) and fˆ(ξ) is supported in the region 3r ξ 3r. Then
∈ 4 ≤ | | ≤ 2
1
1 4 4 1
(4.6) (cid:18)Z0 (cid:13)(cid:16)cos(t√−∆)f(cid:17)(x′,F(x′))(cid:13)L2(Rd−1,dx′)dt(cid:19) ≤ Cr4kfkL2(Rd).
(cid:13) (cid:13)
(cid:13) (cid:13)
The reduction of (4.5) to Lemma 4.2 is attained by letting f = ψTg, where ψ C∞(Rd)
c
∈
equals 1 on the ball of radius 3. Then cos(t√ ∆)f = cos(tr)Tg for x < 2 and t < 1. On the
− | | | |
other hand, fˆ= ψˆ gδSd−1 is rapidly decreasing away from the sphere ξ = r, so the difference
∗ r | |
between fˆand its tru(cid:0)ncation(cid:1) to 3r ξ 3r is easily handled. Also, a simple calculation shows
4 ≤ | |≤ 2
that, uniformly over r,
kψˆ∗(gδSrd−1)kL2(Rd) ≤ CkgkL2(Srd−1).
Proof of Lemma 4.2. Given a function F such that sup F (x′) F(x′) r−1, then (4.6) holds
r x′ r
| − |≤
if we can show that
1
1 4 4 1
(4.7) (cid:18)Z0 (cid:13)(cid:16)cos(t√−∆)f(cid:17)(x′,Fr(x′))(cid:13)L2(Rd−1,dx′)dt(cid:19) ≤ Cr4kfkL2(Rd).
(cid:13) (cid:13)
This follows from th(cid:13)e fact that (4.7), together(cid:13)with the frequency localization of f, implies the
gradient bound, uniformly over s,
1
1 4 4 5
(cid:18)Z0 (cid:13)∂s(cid:16)cos(t√−∆)f(cid:17)(x′,Fr(x′)+s)(cid:13)L2(Rd−1,dx′)dt(cid:19) ≤Cr4kfkL2(Rd).
(cid:13) (cid:13)
(cid:13) (cid:13)
1
We will take Fr to be a mollification of the C1,1 function F on the r−2 spatial scale. Precisely,
d−1 1
let Fr = φr1/2 ∗F, where φr1/2 =r 2 φ(r2x), with φ a Schwartz function of integral 1. Then
1
sup Fr(x) F(x) Cr−1, sup Fr(x) F(x) Cr−2 ,
x | − | ≤ x |∇ −∇ | ≤
and F is a smooth function with derivative bounds
r
(4.8) sup ∂xαFr(x) Cr|α|2−2 , α 2.
x | | ≤ | | ≥
Inestablishing (4.7) wemay replace cos(t√ ∆) by exp(it√ ∆). We then usea TT∗ argument
− −
toreducetoprovingmappingpropertiesforanoperatoronΓ [0,1]. Precisely, letK (t s,x y)
r r
× − −
denote the kernel of the operator
ρ r−1D)exp i(t s)√ ∆ , D := i∂,
− − −
(cid:0) (cid:0) (cid:1)
