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SPECTRAL PROJECTORS, RESOLVENT, AND FOURIER RESTRICTION ON

THE HYPERBOLIC SPACE

PIERRE GERMAIN AND TRISTAN LÉGER

Abstract. We develop a unified approach to proving Lp−Lq boundedness of spectral projectors,

the resolvent of the Laplace-Beltrami operator and its derivative on Hd. In the case of spectralprojectors, and when p and q are in duality, the dependence of the implicit constant on p is shownto be sharp. We also give partial results on the question of Lp

− Lq boundedness of the Fourierextension operator. As an application, we prove smoothing estimates for the free Schrödingerequation on Hd and a limiting absorption principle for the electromagnetic Schrödinger equationwith small potentials.

Contents

1. Introduction 21.1. Fourier restriction, extension, and spectral projectors on the hyperbolic space 21.2. Background 21.3. Obtained results: boundedness of spectral projectors and resolvent 31.4. Obtained results: the Fourier extension problem 42. Harmonic analysis on the hyperbolic space 52.1. Analysis on Hd 52.2. Coordinate systems on Hd 63. Two examples 83.1. The radial example 83.2. The Knapp example 84. Boundedness of spectral projectors: proof of Theorem 1.1 94.1. Duality line 94.2. Optimality on the duality line 114.3. Off-duality line 125. Boundedness of the resolvent: proof of Theorem 1.2 125.1. Duality line 125.2. Off-duality line 155.3. Derivative of the resolvent 166. Boundedness of the Fourier extension operator 176.1. Isometries and Fourier transform 176.2. Lower bounds for the Fourier extension operator: proof of Proposition 1.3 187. Applications 197.1. Boundedness for small frequencies 197.2. Smoothing estimates 207.3. Limiting absorption principle for electromagnetic Schrödinger on Hd 21References 22

1

http://arxiv.org/abs/2104.04126v1

2 P. GERMAIN AND T. LÉGER

1. Introduction

1.1. Fourier restriction, extension, and spectral projectors on the hyperbolic space. Theaim of this section is to give a very succinct definition of the objects of interest in this paper. Amore thorough presentation can be found in Section 2.

We adopt the hyperboloid model for the hyperbolic space: denoting the Minkowski metric onRd+1 by

[x, y] = x0y0 − x1y1 − · · · − xdyd, x, y ∈ Rd+1,

we letHd = {x ∈ Rd+1, [x, x] = 1, x0 > 0},

and endow this manifold with the metric induced by Minkowski’s metric. The Laplace-Beltramioperator on Hd is denoted ∆Hd. It has spectrum [ρ2,∞), with

ρ =d− 1

2.

The Helgason Fourier transform is given by

f̃(λ, ω) =

∫

Hd

f(x)hλ,ω(x) dx, hλ,ω(x) = [x , (1, ω)]iλ−ρ, (λ, ω) ∈ R+ × Sd−1.

By analogy with the Euclidean case, we define the restriction Fourier operator to frequencies of sizeλ > 0 by

[Rλf ](ω) = |c(λ)|−1f̃(λ, ω), ω ∈ Sd−1

(mapping functions on Hd to functions on Sd−1), where c(λ) is the Harish-Chandra function. Itsdual with respect to the L2 scalar products on Hd and Sd−1 is the extension Fourier operator

[Eλf ](x) = |c(λ)|−1 1

ωd−1

∫

Sd−1

f(ω)hλ,ω(x) dω, x ∈ Hd (1.1)

(mapping functions on Sd−1 to functions on Hd). The spectral projectors Pλ for ∆Hd satisfy

−∆Hd = ρ2 +

∫λ2Pλ dλ,

and are given by

Pλ = EλRλ or equivalently Pλf(x) = |c(λ)|−2 1

ωd−1

∫

Sd−1

f̃(λ, ω)hλ,ω(x) dω.

Finally, we denote

D =√

−∆Hd − ρ2

and m(D) for the Fourier multiplier with symbol m(λ).

1.2. Background. The question of the boundedness of the spectral projectors from Lp′ → Lp isequivalent to that of the boundedness of the Fourier extension operator from L2 → Lp. This is trueon the hyperbolic space, considered here, as well as on the Euclidean space, where this question wasfirst considered. Optimal bounds were obtained by Tomas [27] except for the endpoint case, whichis due to Stein. These works of Tomas and Stein were the starting point of the Fourier restrictiontheory, which has flourished since, and proved to be related to a number of mathematical fields,from geometric combinatorics to number theory. We refer to the textbook by Demeter [9] for anaccount of these developments.

The question of extending these ideas to more general Riemannian manifolds was first investigatedby Sogge in the case of the sphere [20] and for general compact manifolds [21]. This ultimately ledto Theorem 5.1.1 in his textbook [22], which is optimal for general manifolds of finite geometry. Adifficult and interesting problem is to understand the relation between the geometry of the manifoldand the boundedness properties of the spectral projectors.

SPECTRAL PROJECTORS, RESOLVENT, AND FOURIER RESTRICTION ON THE HYPERBOLIC SPACE 3

A closely related question is that of the boundedness of resolvent operators. A foundationalpaper in this direction is due to Kenig, Ruiz and Sogge [17], who considered this question for secondorder, constant coefficient operators on the Euclidean space. Their results were later improvedby Gutiérrez [11] in the case of the Laplacian. The case of general compact manifolds was alsoconsidered, see in particular Dos Santos Ferreira-Kenig-Salo [10] and Bourgain-Shao-Sogge-Yao [4].

Turning to Lp harmonic analysis on the hyperbolic space, an important non-Euclidean feature isthe Kunze-Stein phenomenon, see Cowling [8] and Ionescu [14]. For instance, it plays a crucial rolein the derivation of dispersive estimates on Hd (Ionescu-Staffilani [16], Anker-Pierfelice [2]). We willheavily rely on it in the present paper, especially to deal with low frequencies. Another researchline on Hd has been the Lp boundedness of Fourier multipliers, for which we refer to Clerc-Stein [7],Taylor [26] and Anker [1]. The question of the boundedness of spectral projectors was considered byHuang-Sogge [13] and Chen-Hassell [6], who were able to identify Lebesgue spaces over which thespectral projector is bounded. In the present article, we go further and obtain optimal dependenceof the bounds on the Lebesgue exponents.

1.3. Obtained results: boundedness of spectral projectors and resolvent. As mentionedabove, this problem has already been considered in the literature; we improve existing results onseveral fronts. First, we show sharpness of the dependence on λ, p of the constant, in the case wherethe source and target exponents are in duality. As far as upper bounds go, it suffices to followcarefully the real and complex interpolation procedures ; regarding lower bounds, this is done byadapting the radial and Knapp examples to the hyperbolic setting. We also prove Lp → Lq estimatesfor a larger range of exponents than was previously known. Moreover, our method allows for a unifiedtreatment of restriction and resolvent estimates, which extend previously known results to a widerrange of exponents for d ≥ 3; we can also straightforwardly deduce estimates on the derivative of theflat resolvent. Finally we give simple applications of these results, and obtain smoothing estimatesfor the free Schrödinger equation, as well as a limiting absorption principle for the electromagneticSchrödinger equation on Hd.

Theorem 1.1 (Boundedness of the spectral projector). For λ > 1 and 2 ≤ p ≤ ∞,

‖Pλ‖Lp′→Lp .

λd−1− 2d

p if p ≥ pST = 2(d+1)d−1[

(p − 2)−1 + 1]λ(d−1)

(12− 1

p

)

if 2 < p ≤ pST

Furthermore, these bounds are optimal.

Moving away from the line of duality, we now choose the source Lebesgue exponent s ∈ [1, 2), and

the target Lebesgue exponent q ∈ (2,∞]. Then, there holds

‖Pλ‖Ls→Lq .

[λ

d−12

− dq +

[(q − 2)−1/2 + 1

]λ

(d−1)2

(12− 1

q

)] [λ

ds− d+1

2 +[(2− s)−1/2 + 1

]λ

(d−1)2 ( 1

s− 1

2)].

In [18], Li, motivated by the question of asymptotic stability of harmonic maps for wave mapsR × H2 → H2, showed Lp − Lq boundedness of the resolvent operator in dimension d = 2. Someestimates on the resolvent were also proved in general dimension in [13]. In the next theorem weobtain estimates on (D2 − τ − iε)−1 for a wider range of exponents, while keeping track of thedependence of the constant on τ.

Theorem 1.2 (Boundedness of the resolvent). Let z := τ + iε ∈ C, with τ > 1 and ǫ > 0. For

2 ≤ p <∞,

‖(D2 − z)−1‖Lp′→Lp .

{τ

d2(1− 2

p)−1

if pST < p 6 2dd−2

τd−12

( 12− 1

p)− 1

2 if 2 < p ≤ pST := 2(d+1)d−1

where the implicit constant does not depend on ε, but may depend on p.

1/s

1/q

•d−12d

◦I

IV

III

II

IIIIV

Figure 1. Boundedness of (D2−τ−iε)−1. The equations of the lines in this figureare as follows (using the convention that a line and its symmetric with respect tothe diagonal 1/q + 1/s = 1 have the same color). Yellow line: 1

q = d−12d ; Green line:

1q − 1

s = 2d ; Purple line: d−1

d+11q + 1

s = 1. This figure corresponds to the case d ≥ 3,

lines are arranged slightly differently for d = 2.

Moving away from the line of duality, it enjoys the bounds, for 1 ≤ s ≤ q ≤ ∞, for regions I, II,

III, IV defined in Figure 1, and for implicit constants independent of ǫ, but dependent on s, q :

• Region I : ‖(D2 − z)−1‖Ls→Lq . τρ

2

(1s− 1

q

)− 1

2 ,

• Region II : ‖(D2 − z)−1‖Ls→Lq . τρ

2

(1s− 1

q

)+ d

2

(12− 1

q

)− 3

4 ,

• Region III : ‖(D2 − z)−1‖Ls→Lq . τd2

(1s− 1

q

)−1.

• Region IV : ‖(D2 − z)−1‖Ls→Lq . τd2

(1s− 1

2

)− 3

4 .

Finally, we give applications of these estimates in Section 7 to smoothing estimates for theSchrödinger equation, and to resolvent estimates for electromagnetic perturbations of the Laplacian.

1.4. Obtained results: the Fourier extension problem. Since Pλ = EλRλ = Eλ(Eλ)∗, the

classical TT ∗ argument combined with Theorem 1.1 gives sharp estimates for the L2 → Lp operatornorm of Eλ. It also immediately implies that the Lp → Lq operator norm of Eλ is finite for anyp ≥ 2 and q > 2. This observation leads to asking what the operator norm of Eλ is in that range;and whether Eλ might be bounded outside of it. The following proposition provides part of theanswer.

Proposition 1.3 (Lower bounds for the operator norm of the Fourier extension operator). Recall

that Eλ is defined in (1.1).

(i) The operator Eλ is not bounded from Lp to Lq if p < 2 or q ≤ 2.(ii) If p ≥ 2 and q > 2, the operator norm of Eλ from Lp → Lq satisfies

‖Eλ‖Lp→Lq & (q − 2)−1/2λρ

p− ρ

q + λρ− d

q .


This proposition is proved by adapting the Knapp and radial examples to the hyperbolic setting,and by studying the effect of isometries of the hyperbolic space on the Fourier transform. It suggeststhe conjecture

if p ≥ 2, q > 2, ‖Eλ‖Lp→Lq . (q − 2)−1/2λρ

p− ρ

q + λρ−dq

which can also be formulated as

‖Eλ‖Lp→Lq .q

λρ

p− ρ

q if p ≥ 2, q > 2,d+ 1

d− 1

1

q≥ 1− 1

p

λρ−dq if p ≥ 2, q > 2,

d+ 1

d− 1

1

q≤ 1− 1

p

.

Acknowledgements. The authors would like to thank Alexandru Ionescu and Jean-PhilippeAnker for very helpful discussions.

P. Germain is supported by the NSF grant DMS-1501019, by the Simons collaborative grant onweak turbulence, and by the Center for Stability, Instability and Turbulence (NYUAD).

2. Harmonic analysis on the hyperbolic space

In this section, we recall basic facts about the space Hd as well as its Fourier theory. They canbe found in classical references such as the textbook by Helgason [12], the review by Bray [5], andthe nice and concise presentation in Ionescu-Staffilani [16].

2.1. Analysis on Hd.

2.1.1. Hyperboloid model. The Minkowski metric on Rd+1 is given by

[x, y] = x0y0 − x1y1 − · · · − xdyd if x, y ∈ Rd+1.

We define Hd as the hyperboloid (or to be more precise, the upper sheet of the hyperboloid)

Hd = {x ∈ Rd+1, [x, x] = 1, x0 > 0}and equip this space with the Riemannian metric induced by the Minkowski metric. This Riemann-ian metric induces in turn a measure, which will be denoted dx. We will further distinguish thepoint 0 = (1, 0, . . . , 0).

The group of isometries of the Minkowski space leaving Hd invariant is SO(d, 1), which we denoteG; it naturally acts on Hd. The isotropy group of 0 is naturally identified with isometries of theEuclidean space SO(d), which we denote K.

Normalizing the Haar measure on G so that

∫

G

f(g·0) dg =

∫

Hd

f(x) dx, we can define convolution

on Hd through

f ∗K(x) =

∫

G

f(g · 0)K(g−1 · x) dg.

Note that in the case where K is radial, we have

f ∗K(x) =

∫

Hd

f(x′)K(d(x, x′))dx′, (2.1)

where d(x, x′) denotes the geodesic distance between x and x′.We will also rely on a non-euclidean feature of Hd to deal with low frequencies, namely the Kunze-Stein phenomenon:

Lemma 2.1 ([3], Lemma 5.1). For every radial measurable function κ on Hd, every 2 6 q, q̃ < ∞and f ∈ Lq̃′(Hd),

‖f ∗ κ‖Lq .q ‖f‖Lq̃′

(∫ ∞

0(shr)d−1|κ(r)|Q(1 + r)µe−ρµrdr

)1/Q

,

where

µ =2min{q, q̃}q + q̃

, Q =qq̃

q + q̃.

2.1.2. Fourier Analysis. For ω ∈ Sd−1, let

b(ω) = (1, ω) ∈ Rd+1.

The analog of plane waves is provided by

hλ,ω(x) = [x , b(ω)]iλ−ρ, x ∈ Hd

(notice that [x, b(ω)] > 0 for x ∈ Hd). They satisfy

∆Hdhλ,ω(x) = −(λ2 + ρ2)hλ,ω(x).

The Helgason Fourier transform on Hd is defined as

[F̃f ](λ, ω) = f̃(λ, ω) =

∫

Hd

f(x)hλ,ω(x) dx, (λ, ω) ∈ R+ × Sd−1.

The inverse Fourier transform is then given by

f(x) =

∫ ∞

0

1

ωd−1

∫

Sd−1

f̃(λ, ω)hλ,ω(x)|c(λ)|−2 dλ dω,

for the Harish-Chandra function

c(λ) =22ρ−1Γ(ρ+ 1

2)

π1/2Γ(iλ)

Γ(ρ+ iλ)

whose asymptotics are as follows [15, 24]

c(λ)−1 =22ρ−1Γ(ρ+ 1

2)

π1/2(iλ)ρ +O(λρ−1).

We can deduce from the Fourier transform formula the expression of the spectral projectors for theLaplace-Beltrami operator ∆Hd :

−∆Hd − ρ2 =

∫λ2Pλ dλ, with Pλf(x) =

1

ωd−1

∫

Sd−1

f̃(λ, ω)hλ,ω(x)|c(λ)|−2 dω.

Writing D =√

−∆Hd − ρ2, radial Fourier multipliers are defined as follows:

m(D)f = F̃−1[m(λ)[F̃f ](λ, ω)

]=

∫m(λ)Pλf dλ.

Finally, The analog of Plancherel’s theorem holds: the Fourier transform is an isometry fromL2(Hd, dx) to L2(R+ × Sd−1, |c(λ)|−2 dλ dω

ωd−1).

2.2. Coordinate systems on Hd. We will use two coordinate systems on Hd, which we now define.

2.2.1. Polar coordinates. In this hyperboloid model for Hd, we can adopt polar coordinates

x = (chr, shr ω), (r, ω) ∈ R+ × Sd−1,

where r ≥ 0 is the geodesic distance to the origin 0. In these coordinates, the volume elementbecomes

dx = (shr)2ρ dr dω.

The spherical function is given by

Φλ(x) =1

ωd−1

∫

Sd−1

hλ,ω(x) dω.

It only depends on the distance r of x to the origin, and can be written

Φλ(r) =2ρ−1Γ(ρ+ 1

2)√πΓ(ρ)(shr)2ρ−1

∫ r

−reiλs(chr − chs)ρ−1 ds. (2.2)

Radial functions on Hd are invariant by K; in other words, they only depend on r. Therefore,

f̃(λ, ω) = f̃(λ) =

∫

Hd

f(x)Φλ(x) dx = ωd−1

∫ ∞

0f(r)Φλ(r)(shr)

2ρ dr.

It is not the case in general that the Fourier transform on Hd exchanges multiplication andconvolution; but it is true for radial functions. Namely, if K is radial, then

f̃ ∗K(λ, ω) = f̃(λ, ω)K̃(λ).

The convolution kernel K = F̃−1m(λ) associated to the even radial multiplier m is given by thefollowing formulas (see [25], Chapter 8, Section 5):

• If d is odd,

K(r) =1√2π

(−1

2π

1

shr

∂

∂r

)ρ

m̂(r). (2.3)

• If d is even,

K(r) =1√π

∫ ∞

r

(−1

2π

1

shs

∂

∂s

)d/2

m̂(s)(chs− chr)−1/2shs ds. (2.4)

We will need asymptotics for the spherical function:

Proposition 2.2. Assume λ > 1.

• If r < 1λ , Φλ(r) = O(1).

• If 1λ < r < 1, Φλ(r) = 2ρΓ(ρ+

1

2)cos(|λ|r − ρπ

2 )

λρ(shr)ρ+O

(1

λρ+1(shr)ρr

).

• If r > 1, Φλ(r) = 2ρΓ(ρ+1

2)cos(|λ|r − ρπ

2 )

λρ(shr)ρ+O

(1

λρ+1(shr)ρ

).

Proof. The first assertion follows immediately from (2.2); and the proof for the last two can befound in [24]. �

2.2.2. Iwasawa coordinates. Another global system of coordinates is deduced from the Iwasawadecomposition G = NAK, where A is the subgroup of G made up of Lorentz boosts in the firstvariable

A =

at =

cht sht 0sht cht 00 0 Id−1

, t ∈ R

and

N =

nv =

1 + |v|2/2 −|v|2/2 v⊤

|v|2/2 1− |v|2/2 v⊤

v −v Id−1

, v ∈ Rd−1

.

The coordinates s and v are then defined by x = nvas · 0. In other words,

x =

(chs+ e−s |v|2

2, shs+ e−s |v|2

2, e−sv1, ..., e

−svd−1

), (s, v) ∈ Rd.

Note that these coordinates are such that the orbits Nas · 0 are horocycles.The Riemannian metric becomes

e−2s((dv1)2 + · · ·+ (dvd−1)

2) + (ds)2,

and at the North Pole ωNP = (1, 0, . . . 0) ∈ Sd−1, the function hλ,ωNPbecomes

hλ,ωNP(s, v) = e(ρ−iλ)s. (2.5)

3. Two examples

3.1. The radial example.

Lemma 3.1. The spherical function Φλ satisfies

‖Φλ‖Lp ∼p

∞ if p ≤ 2

λ−ρ if 2 2dd−1

.

Proof. Using Proposition 2.2 we see that if r > 1λ then

|Φλ(r)| ∼1

λρ(shr)ρ.

Therefore we can compute

‖Φλ‖pLp ∼∫ 1/λ

0(shr)d−1dr +

∫ ∞

1/λ

1

λρp(shr)ρp(shr)d−1 dr ∼

∞ if p ≤ 2

λ−ρp if 2 2dd−1

.

�

3.2. The Knapp example.

Lemma 3.2. For δ < 1, let ϕδ be the characteristic function of the set {ω ∈ Sd−1, |ω −NP | < δ},where NP is the north pole (1, 0, . . . , 0) ∈ Sd−1. Assume that λδ2 ≪ 1 and λ > 1. Adopting the

Iwasawa coordinates from Section 2.2.2,

[Eλϕδ ](x) & λρδd−1eρs if −∞ < s < −12 log(λδ

2) and |v| ≪ 1δλ .

As a consequence,

‖Eλϕδ‖Lp &1

(p− 2)1/pλ

ρ2− ρ

p δρ.

Proof. The sphere Sd−1 has coordinates (ω1, . . . , ωd) in Rd. Close to the North Pole, we parameterizeSd−1 by ω2, . . . , ωd. Note that |∂ω2ω1| ∼ |ω2| < δ on the support of ϕδ.

In Iwasawa coordinates,

[x, b(ω)] = chs+ e−s |v|22

− ω1

(shs+ e−s |v|2

2

)− ω2e

−sv1 − · · · − ωde−svd−1,

so that in particular [x, b(ωNP )] = e−s.We want to find the set of (s, v) such that, if |ω −NP | < δ,

(1)

∣∣∣∣∂ω[x, b(ω)]

[x, b(ωNP )]

∣∣∣∣ ≪1

δ

(2) and

∣∣∣∣∂ωhλ,ω(x)

hλ,ω(x)

∣∣∣∣ ≪1

δ.

By symmetry, it suffices to replace above ∂ω by ∂ω2 . To address point (1), we can bound∣∣∣∣∣−(∂ω2ω1)(shs+ e−s |v|2

2 )− e−sv1

e−s

∣∣∣∣∣ . δ(1 + e2s) + δ|v|2 + |v1|.

Recalling that λδ2 ≪ 1, this is ≪ 1δ if

s < −logδ − C and |v| ≪ 1

δ. (3.1)

Under these conditions, and still assuming |ω −NP | < δ, we have [x, b(ω)] & [x, b(ωNP )] = e−s.Assuming these conditions are satisfied and turning to point (2),we estimate∣∣∣∣∂ω2hλ,ω(x)

hλ,ω(x)

∣∣∣∣ =∣∣∣∣(iλ− ρ)

∂ω2 [x, b(ω)]

[x, b(ω)]

∣∣∣∣ .∣∣∣∣(iλ− ρ)

∂ω2 [x, b(ω)]

e−s

∣∣∣∣ . λ[(1 + e2s)δ + δ|v|2 + |v1|

].

This is ≪ 1δ if

s < −1

2log(λδ2)− C, and |v| ≪ 1

λδ,

which in particular implies (3.1). Under these conditions, we have |hλ,ω(x)| & |hλ,ωNP(x)| = e−s,

which implies that

[Eλϕδ](x) & λρδd−1eρs.

We can now compute a lower bound for the Lp norm of Eλϕδ , using the expression for the metricin Iwasawa coordinates:

‖Eλϕδ‖pLp & (δd−1λρ)p∫ − 1

2log(λδ2)

−∞eρps

(e−s

λδ

)d−1

ds ∼ 1

p− 2λρ(

p

2−1)δρp.

�

4. Boundedness of spectral projectors: proof of Theorem 1.1

4.1. Duality line. Step 1: real interpolation The idea is to write PΛ = δΛ(D) and decompose δΛinto

δΛ(λ) = 2k0χ(2k0(λ− Λ)) +∑

k≥k0−1

2kψ(2k(λ− Λ)),

where

χ ∈ S,∫χ = 1, ψ = χ− 2χ(2·), 2k0 ∼ 1

Λ.

Furthermore, we choose χ such that ψ̂ is supported on an annulus. In order to use formulas (2.3)and (2.4), we need even multipliers, so that we will actually write

δΛ(λ)+δ−Λ(λ) = 2k0χ(2k0(λ− Λ)) + 2k0χ(2k0(−λ− Λ))︸︷︷︸JΛ,k0(λ)

+∑

k≥k0−1

2kψ(2k(λ− Λ)) + 2kψ(2k(−λ− Λ))︸︷︷︸KΛ,k0(λ)

.

We will denote the convolution kernels of JΛ,k(D) and KΛ,k(D) by

JΛ,k(r) and KΛ,k(r)

respectively.The operator norm of KΛ,k(D) can be bounded as follows:

• By Plancherel’s theorem, ‖KΛ,k(D)‖L2→L2 . 2k.• Turning to L1 → L∞ bounds, they can be obtained thanks to Lemma 4.1

‖KΛ,k(D)‖L1→L∞ . ‖KΛ,k(r)‖L∞ . Λρ(sh(c2k))−ρ.

Interpolating between these two estimates,

‖KΛ,k(D)‖Lp′→Lp . 22pk[Λ(sh(c2k))−1

](1− 2p

)ρ. (4.1)

Proceeding analogously,

‖JΛ,k0(D)‖Lp′→Lp . Λ− 2

pΛ(d−1)

(1− 2

p

).

Therefore, as long as p ∈ [2, pST ) ∪ (pST ,∞],

‖PΛ‖Lp′→Lp .∑

k≥k0

22pk[

Λ

sh(c2k)

](1− 2p

)ρ

.

Λd−1− 2d

p if p > pST = 2(d+1)d−1[

1 + (p − 2)−1]Λ(d−1)

(12− 1

p

)

if 2 0 xαe−βx =

(αβ

)αe−α where α, β > 0.

Step 2: complex interpolation Define the analytic family of operators

QΛ(s) =∑

k≥k0−1

2k2ks[ψ(2k(D − Λ)) + ψ(2k(−D − Λ))

]

It is such that

• QΛ(0) = PΛ − JΛ,k0 .• If Re(s) = −1, QΛ(s) maps boundedly L2 to L2 (uniformly in Ims), since

∥∥∥∥∥∥

∑

k≥k0−1

2k2ksψ(2k(λ− Λ))

∥∥∥∥∥∥L∞

.

∥∥∥∥∥∥

∑

k≥k0−1

|ψ(2k(λ− Λ))|

∥∥∥∥∥∥L∞

<∞.

• If Re(s) = ρ, QΛ(s) maps L1 to L∞ with operator norm O(Λρ) (uniformly in Ims) since,by Lemma 4.1 below,

∥∥∥∥∥∥

∑

k≥k0−1

2ksKΛ,k(r)

∥∥∥∥∥∥L∞

.

∥∥∥∥∥∥

∑

k≥k0−1

2ρk|KΛ,k(r)|

∥∥∥∥∥∥L∞

. Λρ.

The desired bound for p = pST follows by complex interpolation, see for example Theorem 4.1,Chapter 5 in [23].

Lemma 4.1 (Pointwise kernel bounds). For 2k & Λ−1, the following pointwise bounds hold:∣∣JΛ,k0(r)

∣∣ . Λd−1,

∣∣KΛ,k(r)∣∣ .

(Λ

sh(c2k)

)ρ [12kc6r62kC + 1r62kc(2

−kΛ−1)10],

for any r > 0 and some positive constants c, C.

Proof. If d is odd, we learn from (2.3) that

KΛ,k(r) =1√2π

(−1

2π

1

shr

∂

∂r

)ρ (ψ̂(2−kr)eirΛ + ψ̂(−2−kr)e−irΛ

),

from which the desired bound follows by inspection.If d is even, we learn from (2.4) that

KΛ,k(r) =1√π

∫ ∞

r

(−1

2π

1

shs

∂

∂s

)d/2 (ψ̂(2−ks)eisΛ + ψ̂(−2−ks)e−isΛ

)(chs− chr)−1/2shs ds.

If r ≫ 2k, the above is zero, while if r ≪ 2k, the desired bound follows easily by integrating byparts using the identity 1

iΛ∂seisΛ = eisΛ. There remains the case r ∼ 2k. Then, we split Kk,Λ(r)

into

KΛ,k(r) =

∫[φ(Λ(s − r)) + (1− φ(Λ(s − r))χ(s− r) + (1− φ(s− r))] . . . ds

= K(1)Λ,k(r) +K

(2)Λ,k(r) +K

(3)Λ,k(r)

(notice that K(3) = 0 if 2k ≪ 1). The desired bounds for K(1)Λ,k(r) follows by a crude estimate, using

the fact that chs − chr ∼ (s − r)shr on the support of its integrand. For K(2)Λ,k(r), one integration

by parts through the identity 1iΛ∂se

isΛ = eisΛ is necessary. Finally, K(3)k,Λ(r) is easier. Turning to

JΛ,k0 , we only give a brief justification in the case d odd. Since 2k0 ∼ Λ−1, the formula for JΛ,k0can be written

JΛ,k0(r) =

(1

shr

∂

∂r

)ρ

(F (rΛ) + F (−rΛ)) ,

for a function F with bounded derivatives. From this expression, it is not hard to derive the desiredbound. �

4.2. Optimality on the duality line. Since Pλ = (Rλ)∗Rλ, it follows that

‖(Rλ)∗‖L2→Lp′ = ‖Rλ‖Lp→L2 = ‖Pλ‖1/2Lp′→Lp

.

By Lemma 3.1, Φλ ∈ Lp for p > 2 only, so that Pλ can only be bounded from Ls to Lq if q > 2.Since Pλ is self-adjoint, this gives furthermore the condition s < 2.

We also learn from Lemma 3.1 that, if p > 2dd−1 ,

‖Eλ‖L2→Lp ≥|c(λ)|−1‖Φλ‖Lp(Hd)

‖1‖L2(Sd−1)

& λρ− d

p ,

which implies that, if p > 2dd−1 ,

‖Pλ‖Lp′→Lp & λd−1− 2d

p .

Finally, by Lemma 3.2, for any δ < λ−1/2,

‖Eλ‖L2→Lp ≥‖Eλϕδ‖Lp(Hd)

‖ϕδ‖L2(Sd−1)

& (p − 2)−1/2λρ

2− ρ

p ,

which implies

‖Pλ‖Lp′→Lp & (p− 2)−1λd−12

− d−1p .

4.3. Off-duality line. Since Pλ = EλRλ, R⋆λ = Eλ, we find ‖Pλ‖Lp′→Lp = ‖Eλ‖2L2→Lp = ‖Rλ‖2Lp′→L2

and

‖Pλ‖Lq→Ls . ‖Rλ‖Lq→L2‖Eλ‖L2→Ls

.[λ

d−12

− ds +

[(s− 2)−1/2 + 1

]λ

(d−1)2 ( 1

2− 1

s)] [λ

dq− d+1

2 +[(2− q)−1/2 + 1

]λ

(d−1)2

(1q− 1

2

)].

5. Boundedness of the resolvent: proof of Theorem 1.2

Notation: In this section, we denote

α(p, d) =

{d− 1− 2d

p if p ≥ pST = 2(d+1)d−1

(d− 1)(12 − 1

p

)if 2 1, ε > 0. We write

(D2 − τ − iε)−1 =D2 − τ

(D2 − τ)2 + ε2︸︷︷︸:=Rτ (D)

+iε

(D2 − τ)2 + ε2︸︷︷︸:=Iτ (D)

.

5.1.1. Treating Rτ (D). Let k0, k1 ∈ Z be such that 2k0 ∼ 1√τ, 2k1 ∼ 1

δ , δ > 0 to be chosen later.

Let χ : R → R be such that

χ(0) 6= 0, χ ∈ S,

and we require that the support of the Fourier transform of x 7→ χ(x)−χ(2x)x be contained in an

annulus. We write

D2 − τ

(D2 − τ)2 + ε2= χ(0)−1

[ (χ(0)− χ(2k0(D −

√τ)) D2 − τ

(D2 − τ)2 + ε2︸︷︷︸Aτ,k0(D)

+

k1∑

k=k0

[χ(2k(D −

√τ))− χ(2k+1(D −

√τ))

] D2 − τ

(D2 − τ)2 + ε2

︸︷︷︸Bτ,k0,k1(D)

+ χ(2k1+1(D −√τ))

D2 − τ

(D2 − τ)2 + ε2︸︷︷︸Cτ,k1(D)

].

As in the proof of Theorem 1.1, we are really considering the even extension of those multipliers.In the sequel we will abuse notations by still denoting χ its even part.Treating Aτ,k0(D): We introduce a second even cut-off φ : R → [0, 1] that is equal to 1 on [−1, 1],

whose support is included on [−2, 2]. We split the kernel into two pieces adding localizations φ( D10

√τ)

and 1− φ( D10

√τ) respectively. A

(1)τ,k0

and A(2)τ,k0

denote the corresponding kernels. We have

˜A

(1)τ,k0

(λ) =λ2 − τ

(λ2 − τ)2 + ε2φ( λ

10√τ

)(1− χ(2k0(λ−√

τ))

χ(0)

),

hence using the assumption on the support of φ we conclude

‖˜A(1)τ,k0

‖L∞

λ. τ−1. (5.1)

Note that to derive this bound we make crucial use of the χ factor.

Next we obtain a pointwise bound on the kernel A(1)τ,k0

.

Case 1: d is odd Then we see from (2.3) that A(1)τ,k0

is written as a sum of terms of the form F (r)∂lrm̂,

where l denotes an integer between 0 and ρ, and the function F satisfies the bound |F (r)| .

(shr)−2ρ+ler(ρ+l). Note that if r > 1/√τ , then |A(1)

τ,k0(r)| . τ

d2−1. Therefore from now on we assume

that r < 1/√τ .

Case 1.1: l is odd Then we can write

F (r)∂lrm̂

= F (r)∂lr

(∫

R

(cos(λr)− (−1)

l−12(rλ)l

l!

N∑

u=0

(−1)ur2uλ2u

(2u)!

)λ2 − τ

(λ2 − τ)2 + ε2φ(

λ

10√τ)(χ(0)− χ(2k0(λ−

√τ))

)dλ

)

= r2ρ−lF (r)

∫

R

(−1)l−12 λl

cos(λr)−∑Nu=0(−1)u (rλ)2u

(2u)!

r2ρ−l

λ2 − τ

(λ2 − τ)2 + ε2φ(

λ

10√τ)(χ(0)− χ(2k0(λ−

√τ))

)dλ,

where N denotes an integer we used the parity of χ, φ. We can deduce from the above that, remem-

bering the assumption r√τ < 1, the bound |A(1)

τ,k0(r)| . τ

d2−1 holds when N is taken large enough.

Case 1.2: l is even In this case we write

F (r)∂lrm̂ = shrF (r)∂r( 1

shr∂l−1r m̂

)− shrF (r)∂r

( 1

shr

)∂l−1r m̂.

We can apply a similar reasoning to ∂l−1r m̂ as in the previous case to both terms in the right-hand

side.Case 2: d is even The argument is similar. We start from (2.4) and combine the approach of Case1 for the differentiated part, and that of Lemma 4.1 for the square root term. Overall we obtain inall cases

‖A(1)τ,k0

‖L∞r

. τd2−1.

By interpolation with the L2 − L2 estimate deduced from (5.1), we conclude that

‖A(1)τ,k0

(D)‖Lp′→Lp . τd2

(1− 2

p

)−1.

For the second piece A(2)τ,k0

we have two cases to consider.

Case 1: d > 3 We use the pointwise estimate

∣∣A(2)τ,k0

(r)∣∣ . τ

ρ−12

(shr)ρ(r√τ)β

, β ∈ [ρ− 1, ρ+ 3], (5.2)

deduced from (2.3) and (2.4) by doing integration by parts in m̂. Note that we make crucial use ofthe localizer 1−φ at this stage, since it is necessary to be away from the principal value singularity.For the piece r 6 1 we use a local version of the Hardy-Littlewood-Sobolev inequality (which relies

on (2.1)) and for the piece r > 1 we use Lemma 2.1. We obtain that ‖A(2)τ,k0

(D)‖Lp′→Lp . τd2

(1− 2

p

)−1

for 1/p′ − 1/p 6 2/d.

Case 2: d = 2 We see from (2.4) that

If 0 < r 6 1,∣∣A(2)

τ,k0(r)

∣∣ . | ln r|, 1

rβ√τβ, for β ∈ (0, 1], (5.3)

If r > 1,∣∣A(2)

τ,k0(r)

∣∣ . 1

(rτ)2√shr

, (5.4)

hence using as above the Hardy-Littlewood-Sobolev inequality in the first region and Lemma 2.1 inthe second, we deduce that for 2 < p <∞

‖A(2)τ,k0

‖Lp′→Lp . τ− 2

p .

Treating Bτ,k0,k1(D): The idea here is that, for ǫ√τ≪ |D − τ | ≪ √

τ , one can approximate

D2 − τ

(D2 − τ)2 + ǫ2∼ 1

2√τ(D −√

τ).

To formalize this, we write

[χ(2k(D −

√τ ))− χ(2k+1(D −

√τ))

] D2 − τ

(D2 − τ)2 + ε2

=1

2√τ

[2kψ(2k(D −

√τ))− 2k+1ψ(2k+1(D −

√τ))

]+R

:=2k−1

√τΨ(2k(D −

√τ)) +R,

where ψ(x) = χ(x)−χ(0)x ,Ψ(x) = ψ(x)− 2ψ(2x) and

R := −[χ(2k(D −

√τ))− χ(2k+1(D −

√τ))

] (D2 − τ)(D −√τ)

2√τ[(D2 − τ)2 + ε2

]︸︷︷︸

:=R1

− 2kΨ(2k(D −√τ))

ε2

2√τ[(D2 − τ)2 + ε2

]︸︷︷︸

:=R2

.

To handle the main term, we use (4.1):

k1∑

k=k0

∥∥∥∥2k−1

√τΨ(2k(D −

√τ))

∥∥∥∥Lp′→Lp

. τα(p,d)−1

2 .

We can treat the remainder term R using functional calculus and the bounds from Theorem 1.1.More precisely,

‖R1‖Lp′→Lp .

∫ ∞

0

(|χ(2k0(λ−

√τ))|+ |χ(2k1(λ−

√τ))|

) 1√τ(λ+

√τ)

‖Pλ‖Lp′→ Lpdλ

. τα(p,d)−1

2 .

Similarly for the second piece,

‖R2‖Lp′→Lp .

∥∥∥∥(2k0ψ(2k0(D −

√τ))− 2k1ψ(2k1(D −

√τ))

) ε2

2√τ[(D2 − τ)2 + ε2

]∥∥∥∥Lp′→Lp

.ετ

α(p,d)−12√τδ

.

Treating Cτ,k1(D): Again, this lower order term can be bounded using functional calculus for D andTheorem 1.1:

‖Cτ,k1(D)‖Lp′→Lp .

∫ ∞

0

∣∣∣∣χ(2k1(λ−√

τ))

λ2 − τ + iε

∣∣∣∣‖Pλ‖Lp′→Lpdλ .δτα(p,d)/2

ε.

Putting all the estimates together and choosing δ = ε/√τ , the desired result follows.

5.1.2. Treating Iτ (D). Let φ : R+ → [0, 1], whose support is included in [1/2, 2] and equal to 1 on[2/3, 3/2]. Abusing notations, we denote φ its even extension to R. Next, we decompose the operator

Iτ (D) into I(1)τ (D) and I(2)

τ (D) by introducing localizations φ(D/10√τ) and 1− φ(D/10

√τ).

To deal with the singular part, we write that, using functional calculus for D, changing variablesand using Theorem 1.1,

‖I(1)τ (D)‖Lp′→Lp .

∫ ∞

0

εφ(

λ10

√τ

)

(λ2 − τ)2 + ε2‖Pλ‖Lp′→Lpdλ

.

∫ ∞

−√τ

φ( ελ+√

τ10

√τ

)

1 + λ2(ελ+√τ)2

(ελ+√τ)α(p,d)dλ

. τα(p,d)−1

2 .

The last piece is treated similarly, we obtain ‖I(2)τ (D)‖Lp′→Lp . τ

d2

(1− 2

p

)−1

for 1/p′ − 1/p 6 2/d.

5.2. Off-duality line. The proof is similar to the duality line case. Therefore we only detail theparts of the argument that are different. We use the same decomposition, and keep the notationsof the previous subsection. To bound the singular part, we write (see [11] for a similar reasoning inthe euclidean case) using Plancherel’s theorem and Theorem 1.1 that

‖2kΨ(2k(λ−√τ))f̃(λ, ω)‖2L2 = 22k

∫ ∞

0|Ψ(2k(λ−

√τ))|2‖Rλf‖2L2

ωdλ

. 2k‖f‖2Lp′

τ− d+1

2+ d

p′ if p ≥ pST = 2(d+1)d−1

τd−12

(1p′− 1

2

)if 2 < p ≤ pST

.

Next we interpolate this inequality with the (L1 − L∞) estimate that is a direct consequence ofLemma 4.1, and obtain that for 1 6 s 6 q

q−1 , 2 6 q,

‖2kΨ(2k(D −√τ))‖Ls→Lq .

2kq

(sh(c2k)

)ρ(1− 2

q

)

τd2

(1s− 1

2

)− 1

4 if 2s′

q ≥ pST

τρ

2

(1s− 1

q

)if 2 < 2s′

q ≤ pST.

To conclude we write as done above that if 2 < q 6 2dd−1 , then

‖Bτ,k0,k1(D)‖Ls→Lp .

k1∑

k>k0

2k−1

√τ

∥∥Ψ(2k(D −√τ))

∥∥Ls→Lq

.

τd2

(1s− 1

2

)− 3

4 if 2s′

q ≥ pST

τρ

2

(1s− 1

q

)− 1

2 if 2 < 2s′

q ≤ pST.

If q > 2dd−1 , then

‖Bτ,k0,k1(D)‖Ls→Lp .

τd2

(1s− 1

q

)−1 if 2s′

q ≥ pST

τρ

2

(1s− 1

q

)+ d

2

(12− 1

q

)− 3

4 if 2 < 2s′

q ≤ pST

.

The lower half of Figure 1 is obtained by duality.

Remark 5.1. This same reasoning can be applied to spectral projectors. It yields slightly differentbounds than above, where we relied on a duality trick. More precisely, we have, for the regionsdefined in Figure 1:

• Region I: ‖PΛ‖Ls→Lq . [(q − 2)−1 + 1][(s′/q − 1)−1 + 1]2/qΛρ(

1s− 1

q

)

• Region II: ‖PΛ‖Ls→Lq . [(s′/q − 1)−1 + 1]2/qΛρ(

1s− 1

q

)+d

(12− 1

q

)− 1

2

• Region III: ‖PΛ‖Ls→Lq . Λd(

1s− 1

q

)−1

• Region IV: ‖PΛ‖Ls→Lq . [(q − 2)−1 + 1]Λd(

1s− 1

2

)− 1

2 .

5.3. Derivative of the resolvent. Finally, we consider bounds on the derivative of the resolvent(they will be useful in the application to the electromagnetic perturbation of the Laplacian operatorin Section 7).

Theorem 5.2 (Boundedness of the derivative of the resolvent). Let z := τ + iε ∈ C \ {0}, with

ǫ > 0. For 1 ≤ s ≤ q ≤ ∞, for regions I, II, III, IV defined in Figure 2, and for implicit constants

independent of ǫ,

• Region I : ‖D(D2 − z)−1‖Ls→Lq . τρ

2

(1s− 1

q

),

• Region II : ‖D(D2 − z)−1‖Ls→Lq . τρ

2

(1s− 1

q

)+ d

2

(12− 1

q

)− 1

4 .

• Region IV : ‖D(D2 − z)−1‖Ls→Lq . τd2

(1s− 1

2

)− 1

4 .

In the particular case where the exponents are in duality, we have For 2 < p 6 2dd−1 ,

‖D(D2 − z)−1‖Lp′→Lp . τd−12

( 12− 1

p),

where the implicit constant does not depend on ε, but may depend on p.

Remark 5.3. Comparing the above with the statement of Theorem 1.2, there is no region III dueto the green line being shifted upward. This reflects the fact that such estimates are not availablein the Euclidean case.

Proof. The proof is similar to the above, therefore we only sketch the argument. The decompositionof the kernel is identical, and similar considerations lead to the following estimates for the nonsingular part of the kernel:

‖A(1)τ,k0

‖L∞

r. τρ, ‖˜A(1)

τ,k0‖L∞

λ. τ−

12 ,

which leads to

‖A(1)τ,k0

‖Lp′→Lp . τd2

(1− 2

p

)− 1

2 .

Using the pointwise estimate∣∣A(2)

τ,k0(r)

∣∣ . 1

(shr)ρrρ, (5.5)

we obtain that by the Hardy-Littlewood-Sobolev inequality ‖A(2)τ,k0

(D)‖Lp′→Lp . 1, for 1/p′−1/p 6

1/d in the region 0 < r < 1. When r > 1 we rely on Lemma 2.1, and we see that a similar estimate


1/s

1/q

•d−12d

◦I

IV

II

IV

Figure 2. Boundedness of D(D2 − τ − iε)−1. The equations of the lines in thisfigure are as follows (using the convention that a line and its symmetric with respectto the diagonal 1/q + 1/s = 1 have the same color). Yellow line: 1

q = d−12d ; Green

line: 1q − 1

s = 1d ; Purple line: d−1

d+11q +

1s = 1.

holds.To deal with the singular part, we simply write that

DD2 − τ

(D2 − τ)2 + ε2=

(D −√τ)2(D + τ)

(D2 − τ)2 + ε2+

√τ

D2 − τ

(D2 − τ)2 + ε2

where the first term is treated as a remainder term due to the cancellation of the singularity, and theboundedness for the second term directly follows from estimates proved above for the resolvent. �

6. Boundedness of the Fourier extension operator

Our aim in this section is to study the boundedness of the operator

Eλ : Lp(Sd−1) → Lq(Hd)

for p 6= 2 - as we saw, the case p = 2 is equivalent to that of the boundedness of the spectralprojectors, and was fully analyzed in Section 4.

6.1. Isometries and Fourier transform. For x ∈ Hd, ω ∈ Sd−1, let

〈x, ω〉 = −log[x, b(ω)],

so that

hλ,ω(x) = e(ρ−iλ)〈x,ω〉.


We can define an action of SO(d, 1) on Sd−1 as follows: first identify ω ∈ Sd−1 and b(ω) on the coneof Rd+1; then, if U ∈ SO(d, 1), set

Uω = ω′, with b(ω′) =1

(Ub(ω))0Ub(ω).

Using the identity

〈Uz,Uω〉 = 〈z, ω〉+ 〈U0, Uω〉from [5], equalities (2.1)-(2.3), we see that

f̃ ◦ U(λ, ω) =

∫

Hd

f(Ux)e(−iλ+ρ)〈x,ω〉dx

=

∫

Hd

f(x)e(−iλ+ρ)〈U−1x,ω〉dx

=

∫

Hd

f(x)e(−iλ+ρ)(〈x,Uω〉+〈U−1

0,ω〉)dx

= [U−10, b(ω)]iλ−ρf̃(λ,Uω).

6.2. Lower bounds for the Fourier extension operator: proof of Proposition 1.3. Wegather a few observations which together give the proof of this proposition. We use successively thesymmetry group of Hd and both examples from Section 3.

6.2.1. Lorentz boosts. Consider U =

cht sht 0sht cht 00 0 Id−1

. We can parametrize the sphere Sd−1 ⊂

Rd by (θ1, σ) ∈ [0, π]×Sd−2, and the formula ω = (cos θ1, sin θ1σ). A small computation shows that

Uω =1

cht+ sht cos θ1

(sht+ cht cos θ1

sin θ1σ

).

Introducing the new variable θ′1 defined as(cos θ′1sin θ′1

)=

1

cht+ sht cos θ1

(sht+ cht cos θ1

sin θ1

),

let

ω′ = Uω =

(cos θ′1sin θ′1σ

).

An elementary computation shows that

dθ′1dθ1

=1

cht+ sht cos θ1= [U−1

0, b(ω)]−1.

As a consequence, the volume element on the sphere becomes

dω = (sin θ1)d−2dθ1 dσ = [U−10, b(ω)]d−1(sin θ′1)

d−2dθ′1dσ = [U−10, b(ω)]d−1dω′.

We also note that

cos θ1 =sht− cos θ′1chtsht cos θ′1 − cht

,

which leads to the equivalent, as t→ ∞,

[U−10, b(ω)] = cht+ sht cos θ1 ∼ e−t 2

1− cos θ′1→ 0

except for θ′1 = 0.We learn from Section 6.1 that

(Eλg) ◦ U = Eλ([U−10, b(ω)]iλ−ρg ◦ U).

Therefore,

‖Eλ‖Lp→Lq ≥‖(Eλg) ◦ U‖Lq(Hd)

‖[U−10, b(ω)]−ρg ◦ U‖Lp(Sd−1)

.

Since U is an isometry, the numerator is independent of ‖U‖. As for the denominator we can write,changing variables from ω to ω′ that

‖[U−10, b(ω)]−ρg ◦ U‖pLp(Sd−1)

=

∫

Sd−1

[U−10, b(ω)]−ρp|g(Uω)|p dω

=

∫

Sd−1

[U−10, b(ω)]2ρ−ρp|g(ω′)|p dω′.

But as t→ ∞, [U−10, b(ω)] = cht+ sht cos θ → 0 almost everywhere in ω′, which implies that

‖[U−10, b(ω)]−ρg ◦ U‖Lp(Sd−1) → 0

if p < 2. This shows that Lp → Lq boundedness requires p ≥ 2.

6.2.2. The radial example. Testing Eλ on the test function 1 implies that

‖Eλ‖Lp→Lq ≥|c(λ)|−1‖Φλ‖Lq(Hd)

‖1‖Lp(Sd−1)

&

∞ if q ≤ 2

1 if 2 < q ≤ 2dd−1

λρ−dq if q > 2d

d−1

.

6.2.3. The Knapp example. Testing Eλ on the test function ϕδ implies that, if δ < λ−1/2,

‖Eλ‖Lp→Lq ≥‖Eλϕδ‖Lq(Hd)

‖ϕδ‖Lp(Sd−1)

& (q − 2)−1/2λρ

2− ρ

q δρ− d−1

p .

The right-hand side is maximal for δ = λ−1/2, which gives

‖Eλ‖Lp→Lq & (q − 2)−1/2λρ

p− ρ

q .

7. Applications

7.1. Boundedness for small frequencies. For our applications we will need a version of Theo-rems 1.1 and 1.2 for 0 < Λ, τ 6 1.

Lemma 7.1. Let p > 2,Λ > 0, Then

‖PΛ‖Lp′→Lp . Λ2.

Moreover if p <∞ and 1− 2p 6 2

d then

supε>0,τ∈[0,1]

‖(D2 − τ − iε)−1‖Lp′→Lp . 1.

Finally if 1− 2p 6 1

d then

supε>0,τ∈[0,1]

‖D(D2 − τ − iε)−1‖Lp′→Lp . 1.

Proof. For the first estimate, we see that the kernel is bounded pointwise by Λ2 in the region0 < s 6 1 by a power series expansion in (2.3), (2.4). In the region s > 1 the kernel is boundedpointwise by e−ρsΛs/〈Λs〉, and therefore we can use Lemma 2.1 to conclude.For the resolvent, we can handle the region far from the singularity (λ > 2

√τ) by appealing to the

pointwise bound (5.2) with β = ρ−1 if d > 3 and the log bound in (5.3) if d = 2. For the region thatis close to the singularity, we use the fact that the kernel is bounded pointwise by min{s2−d, e−ρs}.Then we appeal to the Hardy-Littlewood inequality for the region s 6 1 and Lemma 2.1 whens > 1.The boundedness of the derivative of the resolvent is proved similarly, relying on the pointwisebound (5.5) far from the singularity and

√τ min{s2−d, e−ρs} close to it. �

7.2. Smoothing estimates. We start with a smoothing effect for the homogenenous Schrödingerequation on Hd:

Theorem 7.2. Let 2 3. Let

γp =

1− d(12− 1

p

)if p > pST

1

2− d− 1

2

(12− 1

p

)if 2 < p 6 pST

.

We have

‖Dγpx e

it∆Hdf‖Lq

x(L2t (R))

. ‖u0‖L2x.

Moreover ∥∥∥∥∫ T

0D

γpx e

it∆Hdf(t)dt

∥∥∥∥L2x(H

d)

. ‖f‖Lp′

x (L2t (R))

and ∥∥∥∥D2γpx

∫ +∞

−∞ei(t−s)∆

Hdf(s)ds

∥∥∥∥Lpx(L

2t (R))

. ‖f‖Lp′

x (L2t (R))

.

Remark 7.3. Note that under the condition 1/p′ − 1/p 6 2/d, if d > 3 or d = 2, we have γp > 0.

Proof. This is a corollary of the restriction estimate from Theorem 1.1. Following [19], we changevariables, use the Plancherel theorem in t and then Minkowski’s inequality in x to obtain

∥∥∥∥∫ ∞

0eitλ

2λγp

[Pλf

]dλ

∥∥∥∥LpxL

2t

=1

2

∥∥∥∥∫ ∞

0eitλλ

γp

2[P√

λf]λ−1/2dλ

∥∥∥∥LpxL

2t

.∥∥λ

γp−1

2 P√λf

∥∥LpxL2

λ

.∥∥λ

γp−1

2 P√λf

∥∥L2λLpx.

Next we notice that a straightforward consequence of Theorem 1.1 and Lemma 7.1 is that ‖E√λ‖L2

ω→Lp .

min{λ

12 , λ−

2γp−1

4

}. Writing P√

λ = E√λR

√λ, we can therefore bound the above by a constant times

(∫ ∞

0‖R√

λf‖2L2ωλ−1/2dλ

)1/2

.

(∫ ∞

0‖f̃(λ, ω)‖2L2

ω|c(λ)|−2dλ

)1/2

. ‖f‖L2x.

For the first inequality we changed variables, and the last line comes from Plancherel’s Theorem forthe Fourier Helgason transform.The second inequality is the dual of the estimate just obtained.The third inequality of the theorem is proved using successively the first then second inequality ofthe theorem. �

Remark 7.4. Note that the proof also implies that the same result holds with the multiplier 〈Dx〉.

In the inhomogeneous case we have

Theorem 7.5. Let u solve {i∂tu+∆Hdu = f

u(x, 0) = 0.

Then for 2 3,

we have

‖u‖Lpx(L2

t (R)). ‖f‖

Lp′

x (L2t (R))

.

Proof. The proof is the same as in [19] using Theorems 1.2, 7.1 and the remark 7.4 above. �

Remark 7.6. In Theorems 7.2 and 7.5 we obtain a wider range of exponents than in the euclideancase.

7.3. Limiting absorption principle for electromagnetic Schrödinger on Hd. The resolventestimates obtained in Theorem 1.2 classically lead to a limiting absorption principle for small elec-tromagnetic potentials.Starting with the electric case, we have

Theorem 7.7. Let 2 3. Let V ∈ Lp

p−2x . Then there

exists δ > 0 such that if ‖V ‖L

pp−2x

< δ, then denoting z := τ + iε, ε > 0, we have

supτ,ε>0

‖(D2 + V − z)−1‖Lp′→Lp . 1.

Moreover the limit of (D2 + V − z)−1 as ε→ 0, denoted (D2 + V − τ − i0)−1, exists in the sense of

distributions and satisfies the same bound as above.

Proof. We first note that, using Theorems 1.2, 7.1

‖(D2 − z)−1V ‖Lp→Lp . ‖V ‖L

pp−2x

, (7.1)

where the implicit constants are independent of ε and τ since under the condition 1/p′ − 1/p 6 2/dif d > 3 or d = 2, the exponent of τ in Theorem 1.2 is negative.Therefore by a Neumann series argument, the operator

(I + (D − z)−1V

)is invertible in Lp if

‖V ‖L

pp−2x

is small enough. The desired estimates directly follow from the resolvent identity

(D2 + V − z)−1 =(I + (D2 − z)−1V

)−1(D2 − z)−1. (7.2)

To prove the second assertion we note that the limit exists in S ′ in the case where V = 0, as can beseen from the explicit formula of the kernel of the resolvent. Then using the estimate just proved,along with (7.2) and Fatou’s lemma, the result directly follows. �

In the magnetic case, we have a similar theorem, although we lose uniformity on τ in the boundsabove.

Theorem 7.8. Let p > 2 be such that 1/p′−1/p 6 1/d. Let A ∈(L

p

p−1x

)dbe such that ∇·A ∈ L

p

p−2x .

Let 1 > τ0 > 0. Then there exists δ(τ0) > 0 such that if ‖A‖(L

pp−2x

)d , ‖∇ ·A‖L

pp−2x

< δ, then denoting

z := τ + iε, ε > 0, we have

supε>0,τ∈(0,τ−1

0 ]

‖(D2 + i(A · ∇+∇ ·A

)− z)−1‖Lp′→Lp . 1.


Moreover the limit of (D2+i(A·∇+∇·A

)−z)−1 as ε→ 0, denoted (D2+i

(A·∇+∇·A

)−τ−i0)−1,

exists in the sense of distributions and satisfies the same bound as above.

Proof. The proof is very similar to the electric case, replacing (7.1) with∥∥(D2 − z)−1∇ ·A

∥∥Lp→Lp . ‖∇ ·A‖

Lp

p−2x

, ‖(D2 − z)−1A · ∇‖Lp→Lp .τ0 ‖A‖(L

pp−1x

)d ,

where the second inequality is proved by duality. Moreover the implicit constant does not dependon ε, but depends on τ0 in the second inequality. �

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Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New

York, NY 10012, USA

Email address: [email protected]

Princeton University, Mathematics Department, Fine Hall,Washington Road, Princeton, NJ 08544-

1000, USA

Email address: [email protected]

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