Strichartz estimates for the Schrodinger flow on compact Lie groups

Yunfeng Zhang

Abstract. We establish scale-invariant Strichartz estimates for the Schrodinger flow on any compact Lie

group equipped with canonical rational metrics. In particular, full Strichartz estimates without loss for

some non-rectangular tori are given. The highlights of this paper include estimates for some Weyl type

sums defined on rational lattices, different decompositions of the Schrodinger kernel that accommodate

different positions of the variable inside the maximal torus relative to the cell walls, and an application of

the BGG-Demazure operators or Harish-Chandra’s integral formula to the estimate of the difference between

characters.

Contents

1. Introduction 2

2. Statement of the Main Theorem 4

2.1. Rational Metric 4

2.2. Main Theorem 4

2.3. Organization of the Paper 5

3. First Reductions 6

3.1. Littlewood-Paley Theory 6

3.2. Reduction to a Finite Cover 7

3.3. Littlewood-Paley Projections of Product Type 8

4. Preliminaries on harmonic analysis on compact Lie groups 8

4.1. Fourier Transform 8

4.2. Root System and the Laplace-Beltrami Operator 9

5. The Schrodinger Kernel 11

6. The Stein-Tomas Argument 14

7. Dispersive Estimates on Major Arcs 22

7.1. Weyl Type Sums on Rational Lattices 22

7.2. From a Chamber to the Whole Weight Lattice 24

7.3. Pseudo-polynomial Behavior of Characters 27

7.3.1. Approach 1: via BGG-Demazure Operators 27

7.3.2. Approach 2: via Harish-Chandra’s Integral Formula 31

7.4. From the Weight Lattice to the Root Lattice 32

7.5. Root Subsystems 34

7.5.1. Identifying Root Subsystems and Rewriting the Character 34

7.5.2. Decomposition of the Weight Lattice 36

7.6. Lp Estimates 40

Acknowledgments 43

References 431

2 Y. ZHANG

1. Introduction

We start with a complete Riemannian manifold (M, g) of dimension d, associated to which are the Laplace-

Beltrami operator ∆g and the volume form measure µg. Then it is well known that ∆g is essentially

self-adjoint on L2(M) := L2(M,dµg); see [Str83] for a proof. This gives the functional calculus of ∆g,

and in particular gives the one-parameter unitary operators eit∆g which provides the solution to the linear

Schrodinger equation on (M, g). We refer to eit∆g as the Schrodinger flow. The functional calculus of ∆g

also gives the definition of the Bessel potentials, and thus the definition of the Sobolev space

Hs(M) := {u ∈ L2(M) | ‖u‖Hs(M) := ‖(I −∆)s/2u‖L2(M) <∞}.

We are interested in obtaining estimates of the form

(1.1) ‖eit∆gf‖LpLr(I×M) ≤ C‖f‖Hs(M)

where I ⊂ R is a fixed time interval, and LpLq(I ×M) is the space of Lp functions on I with values in

Lq(M). Such estimates are often called Strichartz estimates (for the Schrodinger flow), in honor of Robert

Strichartz [Str77] who first derived such estimates for the wave equation on Euclidean spaces.

The significance of Strichartz estimates is evident in many ways. Strichartz estimates have important

applications in the field of nonlinear Schrodinger equations, in the sense that many perturbative results

often require good control on the linear solution which is exactly provided by Strichartz estimates. Strichartz

estimates can also be interpreted as Fourier restriction estimates, which play a fundamental rule in the field

of classical harmonic analysis. Furthermore, the relevance of the distribution of eigenvalues and the norm

of eigenfunctions of ∆g in deriving the estimates makes Strichartz estimates also a subject in the field of

spectral geometry.

Many cases of Strichartz estimates for the Schrodinger flow are known in the literature. For noncompact

manifolds, first we have the sharp Strichartz estimates on the Euclidean spaces obtained in [GV95, KT98]:

(1.2) ‖eit∆f‖LpLq(R×Rd) ≤ C‖f‖L2(Rd)

where 2p + d

q = d2 , p, q ≥ 2, (p, q, d) 6= (2,∞, 2). Such pairs (p, q) are called admissible. This implies by

Sobolev embedding that

(1.3) ‖eit∆f‖LpLr(R×Rd) ≤ C‖f‖Hs(Rd)

where

(1.4) s =d

2− 2

p− d

r≥ 0,

p, q ≥ 2, (p, r, d) 6= (2,∞, 2). Note that the equality in (1.4) can be derived from a standard scaling

argument, and we call exponent triples (p, r, s) that satisfy (1.4) as well as the corresponding Strichartz es-

timates scale-invariant. Similar Strichartz estimates hold on many noncompact manifolds. For example, see

[AP09, Ban07, IS09, Pie06] for Strichartz estimates on the real hyperbolic spaces, [APV11, Pie08, BD07] for

Damek-Ricci spaces which include all rank-1 symmetric spaces of noncompact type, [Bou11] for asymptoti-

cally hyperbolic manifolds, [HTW06] for asymptotically conic manifolds, [BT08, ST02] for some perturbed

Schrodinger equations on Euclidean spaces, and [FMM15] for symmetric spaces G/K where G is complex.

For compact manifolds however, Strichartz estimates such as (1.2) are expected to fail. The Sobolev

exponent s in (1.1) is expected to be positive for (1.1) to possibly hold. And we also expect sharp Strichartz

STRICHARTZ ESTIMATES FOR THE SCHRODINGER FLOW ON COMPACT LIE GROUPS 3

estimates that are non-scale-invariant, in the sense that the exponents (p, r, s) in (1.1) satisfy

s >d

2− 2

p− d

r.

For example, from the results in [ST02, BGT04], we know that on a general compact Riemannian manifold

(M, g) it holds that for any finite interval I,

(1.5) ‖eit∆gf‖LpLr(I×M) ≤ C‖f‖H1/p(M)

for all admissible pairs (p, r). These estimates are non-scale-invariant, and the special case of which when

(p, r, s) = (2, 2dd−2 , 1/2) can be shown to be sharp on spheres of dimension d ≥ 3 equipped with canonical

Riemannian metrics. On the other hand, scale-invariant estimates are out of reach of the local methods

employed in [ST02, BGT04], and they are not well explored yet in the literature. To my best knowledge,

the only known results in the literature in this direction are on Zoll manifolds, which include all compact

symmetric spaces of rank 1, the standard sphere being a typical example; and on rectangular tori. We

summarize the results here. Consider the scale-invariant estimates

(1.6) ‖eit∆gf‖Lp(I×M) ≤ C‖f‖Hd2− d+2

p (M).

In the direction of Zoll manifolds, (1.6) is first proved in [BGT07] for the standard three-sphere for p = 6.

Then in [Her13], (1.6) is proved for all p > 4 for any three-dimensional Zoll manifold, but the methods

employed in that paper in fact prove (1.6) for p > 4 for any Zoll manifold with dimension d ≥ 3 and

for p ≥ 6 for any Zoll surface (d = 2). The paper crucially uses the property of Zoll manifolds that the

spectrum of the Laplace-Beltrami operator is clustered around a sequence of squares, and the spectral cluster

estimates ([Sog88]) which are optimal on spheres. In the direction of tori, (1.6) was first proved in [Bou93]

for p ≥ 2(d+4)d on square tori, by interpolating the distributional Strichartz estimate

(1.7) λ · µ{(t, x) ∈ I × Td | |eit∆gϕ(N−2∆g)f(x)| > λ}1/p ≤ CNd2−

d+2p ‖f‖L2(Td) ≤ C‖f‖

Hd2− d+2

p (Td).

for λ > Nd/4, p > 2(d+2)d , N ≥ 1, with the trivial subcritical Strichartz estimate

(1.8) ‖eit∆gf‖L2(I×Td) ≤ C‖f‖L2(Td).

The estimate (1.7) is a consequence of an arithmetic version of dispersive estimates:

(1.9) ‖eit∆gϕ(N−2∆g)‖L∞(Td) ≤ C

(N

√q(1 +N‖ tT −

aq ‖1/2)

)d‖f‖L1(Td)

where ‖ · ‖ stands for the distance from 0 on the standard circle with length 1, ‖ tT −aq ‖ <

1qN , a, q are

nonnegative integers with a < q and (a, q) = 1, and q < N . Here T is the period for the Schrodinger flow

eit∆g . Then in [Bou13], the author improved (1.8) into a stronger subcritical Strichartz estimate

(1.10) ‖eit∆gf‖L

2(d+1)d (I×Td)

≤ C‖f‖L2(Td)

which yields (1.6) for p ≥ 2(d+3)d . Eventually, (1.6) with an ε-loss is proved for the full range p > 2(d+2)

d in

[BD15], and (1.7) can be used to remove this ε-loss. Then authors in [GOW14, KV16] extended the results

to all rectangular tori. We will see in this paper that by a slight adaptation of the methods in [Bou93], we

may generalize (1.7) to all rational (not necessarily rectangular) tori Td = Rd/Γ where Γ ∼= Zd is a lattice

such that there exists some D 6= 0 for which 〈λ, µ〉 ∈ D−1Z for all λ, µ ∈ Γ, which can also be used for the

removal of the ε-loss of the results in [BD15] to yield (1.6) for the full range p > 2(d+2)d on such rational tori.

4 Y. ZHANG

The understanding of Strichartz estimates on compact manifolds is far from complete. It is not known

in general how the exponents (p, r, s) in the sharp Strichartz estimates are related to the geometry and

topology of the underlying manifold. Also, there still are important classes of compact manifolds on which

the Strichartz estimates have not been explored yet. Note that both standard tori and spheres on which

Strichartz estimates are known are special cases of compact globally symmetric spaces, and since all compact

globally symmetric spaces share the same behavior of geodesic dynamics as tori, from a semiclassical point

of view, it’s natural to conjecture that similar Strichartz estimates should hold on general compact globally

symmetric spaces. An important class of such spaces is the class of compact Lie groups. The goal of this paper

is to prove scale-invariant Strichartz estimates of the form (1.6) for M = G being any connected compact

Lie group equipped with a canonical rational metric in the sense that is described below, for all p ≥ 2(r+4)r ,

r being the rank of G. In particular, full Strichartz estimates without loss for some non-rectangular tori will

be given.

2. Statement of the Main Theorem

2.1. Rational Metric. Let G be a connected compact Lie group and g be its Lie algebra. By the classifi-

cation theorem of connected compact Lie groups, see Chapter 10, Section 7.2, Theorem 4 in [Pro07]), there

exists an exact sequence of Lie group homomorphisms

1→ A→ G ∼= Tn ×K → G→ 1

where Tn is the n-dimensional torus, K is a compact simply connected semisimple Lie group, and A is a

finite and central subgroup of the covering group G. As a compact simply connected semisimple Lie group,

K is a direct product K1 ×K2 × · · · ×Km of compact simply connected simple Lie groups.

Now each Ki is equipped with the canonical bi-invariant Riemannian metric gi that is induced from the

negative of the Cartan-Killing form. We use 〈 , 〉 to denote the Cartan-Killing form. Then we equip the torus

factor Tn with a flat metric g0 inherited from its representation as the quotient Rn/2πΓ and require that

there exists some D ∈ N such that 〈λ, µ〉 ∈ D−1Z for all λ, µ ∈ Γ. Then we equip G ∼= Tn ×K1 × · · · ×Km

the bi-invariant metric

(2.1) g = ⊗mj=0βjgj ,

βj > 0, j = 0, . . . ,m. Then g induces a bi-invariant metric g on G.

Definition 2.1. Let g be the bi-invariant metric induced from g in (2.1) as described above. We call g a

rational metric provided the numbers β0, · · · , βm are rational multiples of each other. If not, we call it an

irrational metric.

Provided the numbers β0, · · · , βm are rational multiples of each other, the periods of the Schrodinger flow

eit∆g on each factor of G are rational multiples of each other, which implies that the Schrodinger flow on G

as well as on G is also periodic (see Section 5).

2.2. Main Theorem. We define the rank of G to be the dimension of any of its maximal torus. This paper

mainly proves the following theorem.

Theorem 2.2. Let G be a connected compact Lie group equipped with a rational metric g. Let d be the

dimension of G and r the rank of G. Let I ⊂ R be a finite time interval. Consider the scale-invariant

STRICHARTZ ESTIMATES FOR THE SCHRODINGER FLOW ON COMPACT LIE GROUPS 5

Strichartz estimate

(2.2) ‖eit∆gf‖Lp(I×G) ≤ C‖f‖Hd2− d+2

p (G).

Then the following statements hold true.

(i) (2.2) holds for all p ≥ 2 + 8r .

(ii) Let G = Td be a flat torus equipped with a rational metric, that is, we can write Td = Rd/2πΓ such that

there exists some D ∈ R for which 〈λ, µ〉 ∈ D−1Z for all λ, µ ∈ Γ. Then (2.2) holds for all p > 2 + 4d .

The framework for the proof of this theorem will be based on [Bou93], in which the author proves some

Strichartz estimates for the case of square tori, based on a variant of the Hardy-Littlewood circle method.

We also refer to [Bou89] for applications of the circle method to Fourier restriction problems on tori. Note

that part (ii) of the above theorem provides full expected Strichartz estimates without loss for some non-

rectangular tori. We then have the following immediate corollary.

Corollary 2.3. Let d = 3, 4 and let Td be the flat torus equipped with a rational metric (not necessarily

rectangular). Then the nonlinear Schrodinger equation i∂ut = −∆u± |u|4d−2u is locally wellposed for initial

data in H1(Td). Furthermore, for d = 3, we have i∂ut = −∆u± |u|2u is locally wellposed for initial data in

H12 (Td).

We refer to [HTT11] and [KV16] for the definition of local well-posedness and a proof of this corollary.

Remark 2.4. To the best of my knowledge, the only known optimal range of p for (2.2) to hold is on square

tori Td, with p > 2 + 4d ([Bou93]); and on spheres Sd (d ≥ 3), with p > 4 ([BGT04, Her13]). For a general

compact Lie group, we do not yet have a conjecture about the optimal range. We will prove (Theorem 6.2)

the following distributional estimate: for any p > 2 + 4r ,

(2.3) λ · µ{(t, x) ∈ I ×G | |eit∆gϕ(N−2∆g)f(x)| > λ}1/p ≤ CNd2−

d+2p ‖f‖L2(G)

for all λ & Nd2−

r4 . It seems reasonable to conjecture that the above distributional estimate could be upgraded

to the estimate (2.2) for all p > 2 + 4r (which is the case for the tori). But this still will not be the optimal

range for a general compact Lie group, by looking at the example of the three sphere S3, which is isomorphic

to the group SU(2). The optimal range for S3 is p > 4, while Theorem 2.2 proves the range p ≥ 10, and the

above conjecture indicates the range p > 6. Estimate (2.2) for S3 on the optimal range p > 4 is proved in

[Her13] by crucially using the Lp-estimates of the spectral clusters for the Laplace-Beltrami operator ([Sog88])

which are optimal on spheres. On tori and more generally compact Lie groups with rank higher than 1, such

spectral cluster estimates fail to be optimal and do not help provide the desired Strichartz estimates. On the

other hand, the Stein-Tomas argument in our proof of Theorem 2.2 seems only sensitive to the L∞-estimate

of the Schrodinger kernel (Theorem 6.1) but not to the Lp-estimate (as in Proposition 7.28). This failure

of incorporating Lp-estimates for either the spectral clusters or the Schrodinger kernel may be one of the

reasons why Theorem 2.2 is still a step away from the optimal range.

2.3. Organization of the Paper. The organization of the paper is as follows. In Section 3, we will first

reduce the Strichartz estimates on G ∼= G/A to the spectrally localized Strichartz estimates with respect to

Littlewood-Paley projections of the product type on the covering group G. In Section 4, we will review the

basic facts of structures and harmonic analysis on compact Lie groups, including the Fourier transform, root

systems, structure of maximal tori, Weyl’s character and dimension formulas, and the functional calculus of

the Laplace-Beltrami operator. In Section 5 we will explicitly write down the Schrodinger kernel and interpret

6 Y. ZHANG

the Strichartz estimates as Fourier restriction estimates on the space-time, which then makes applicable the

argument of Stein-Tomas type in Section 6. Then comes the core of the paper, Section 7, in which we will

derive dispersive estimates for the Schrodinger kernel as the time variable lies in major arcs. In Section 7.1,

we will estimate some Weyl type exponential sums over the so-called rational lattices, which in particular

will imply the desired bound on the Schrodinger kernel for the non-rectangular rational tori. In Section 7.2,

we will rewrite the Schrodinger kernel for compact Lie groups into an exponential sum over the whole weight

lattice instead of just one chamber of the lattice, and will prove the desired bound on the kernel for the

case when the variable in the maximal torus stays away from all the cell walls by an application of the Weyl

type sum estimate established in Section 7.1. In Section 7.3, we will record two approaches to the pseudo-

polynomial behavior of characters, which will be applied to proving the desired bound on the Schrodinger

kernel when the variable in the maximal torus stays close to the identity. In Section 7.4, we further extend

the result to the case when the variable in the maximal torus stays close to some corner. Section 7.5 will

finally deal with the case when the variable in the maximal torus stays away from all the corners but close to

some cell walls. These cell walls will be identified as those of a root subsystem, and we will then decompose

the Schrodinger kernel into exponential sums over the root lattice of this root subsystem, thus reducing the

problem into one similar to those already discussed in previous sections. This will finish the proof of the

main theorem. In Section 7.6, we will derive Lp(G) estimates on the Schrodinger kernel as an upgrade of

the L∞(G)-estimate.

Throughout the paper:

• A . B means A ≤ CB for some constant C.

• A .a,b,··· B means A ≤ CB for some constant C that depends on a, b, · · · .• ∆, µ are short for the Laplace-Beltrami operator ∆g and the associated volume form measure µg

respectively when the underlying Riemannian metric g is clear from context.

• Lpx, Hsx, L

pt , L

ptL

qx, L

pt,x are short for Lp(M), Hs(M), Lp(I), LpLq(I×M), Lp(I×M) respectively when

the underlying manifold M and time interval I are clear from context.

• p′ denotes the number such that 1p + 1

p′ = 1.

3. First Reductions

3.1. Littlewood-Paley Theory. Let (M, g) be a compact Riemannian manifold and ∆ be the Laplace-

Beltrami operator. Let ϕ be a bump function on R. Then for N ≥ 1, PN := ϕ(N−2∆) defines a bounded

operator on L2(M) through the functional calculus of ∆. These operators PN are often called the Littlewood-

Paley projections. We reduce the problem of obtaining Strichartz estimates for eit∆ to those for PNeit∆.

Proposition 3.1. Fix p, q ≥ 2, s ≥ 0. Then the Strichartz estimate (1.1) is equivalent to the following

statement: Given any bump function ϕ,

‖PNeit∆f‖LpLq(I×M) . Ns‖f‖L2(M),

holds for all dyadic natural numbers N (that is, for N = 2m, m ∈ Z≥0). In particular, (2.2) reduces to

(3.1) ‖PNeit∆f‖Lp(I×G) ≤ Nd2−

d+2p ‖f‖L2(G).

This reduction is classical. We refer to [BGT04] for a proof.

We also record here the Bernstein type inequalities that will be useful in the sequel.

STRICHARTZ ESTIMATES FOR THE SCHRODINGER FLOW ON COMPACT LIE GROUPS 7

Proposition 3.2 (Corollary 2.2 in [BGT04]). Let d be the dimension of M . Then for all 1 ≤ p ≤ r ≤ ∞,

‖PNf‖Lr(M) . Nd( 1p−

1r )‖f‖Lp(M).(3.2)

Note that the above proposition in particular implies that (3.1) holds for N . 1 or p =∞.

3.2. Reduction to a Finite Cover.

Proposition 3.3. Let π : (M, g) → (M, g) be a Riemannian covering map between compact Riemannian

manifolds (then automatically with finite fibers). Let ∆g, ∆g be the Laplace-Beltrami operators on (M, g)

and (M, g) respectively and let µ and µ be the normalized volume form measures respectively, which define

the Lp spaces. Let π∗ be the pull back map. Define

C∞π (M) := π∗(C∞(M)),

and similarly define Cπ(M), Lpπ(M) and Hsπ(M). Then the following statements hold.

(i) π∗ : C(M)→ Cπ(M) and π∗ : C∞(M)→ C∞π (M) are well-defined and are linear isomorphisms.

(ii) π∗ : Lp(M)→ Lpπ(M) is well-defined and is an isometry.

(iii) ∆g maps C∞π (M) into C∞π (M) and the diagram

C∞(M)π∗//

∆g

��

C∞π (M)

∆g

��

C∞(M)π∗// C∞π (M)

commutes.

(iv) eit∆g maps L2π(M) into L2

π(M) and is an isometry, and the diagrams

L2(M)π∗//

eit∆g

��

L2π(M)

eit∆g

��

L2(M)π∗// L2π(M)

L2(M)π∗//

PN

��

L2π(M)

PN��

L2(M)π∗// L2π(M)

(3.3)

commute, where PN stands for both ϕ(N−2∆g) and ϕ(N−2∆g).

(v) π∗ : Hs(M)→ Hsπ(M) is well-defined and is an isometry.

Proof. Parts (i), (ii) and (iii) are direct consequences of the definition of a Riemannian covering map. For part

(iv), note that (i), (ii) and (iii) together imply that the triples (L2(M), C∞(M),∆g) and (L2π(M), C∞π (M),∆g)

are isometric as systems of essentially self-adjoint operators on Hilbert spaces, and thus have isometric func-

tional calculus. This implies (iv). Note that the Hs(M) and Hsπ(M) norms are also defined in terms of the

isometric functional calculus of (L2(M), C∞(M),∆g) and (L2π(M), C∞π (M),∆g) respectively, which implies

(v). �

Combining Proposition 3.1 and 3.3, Theorem 2.2 is reduced to the following.

Theorem 3.4. Let Ki’s be simply connected simple Lie groups and let G = Tn×K1× · · ·×Km be equipped

with a rational metric as in Definition 2.1. Then

(3.4) ‖PNeit∆f‖Lp(I×G) . Nd2−

d+2p ‖f‖L2(G)

holds for p ≥ 2 + 8r and N & 1.

8 Y. ZHANG

3.3. Littlewood-Paley Projections of Product Type. Let (M, g) be the Riemannian product of the

compact Riemannian manifolds (Mj , gj), j = 0, · · · ,m. Any eigenfunction of the Laplace-Beltrami operator

∆ on M with the eigenvalue λ ≤ 0 is of the form∏mj=0 ψλj , where each ψλj is an eigenfunction of ∆j on Mi

with eigenvalue λj ≤ 0, j = 0, · · · ,m, such that λ = λ0 + · · ·+ λn.

Given any bump function ϕ on R, there always exist bump functions ϕj , j = 0, · · · ,m, such that for all

(x0, · · · , xm) ∈ Rm+1≤0 with ϕ(x0 + · · ·+ xm) 6= 0, we have

∏mj=0 ϕj(xj) = 1. In particular,

ϕ ·m∏j=0

ϕj(xj) = ϕ.

For N ≥ 1, define

PN : = ϕ(N−2∆),

PN : = ϕ0(N−2∆0)⊗ · · · ⊗ ϕm(N−2∆m),

as bounded operators on L2(M). We call PN a Littlewood-Paley projection of the product type. We have

PN ◦ PN = PN .

This implies that we can further reduce Theorem 3.4 into the following.

Theorem 3.5. Let G = Tn × K1 × · · · × Km be equipped with a rational metric. Let ∆0,∆1, · · · ,∆m

be respectively the Laplace-Beltrami operators on Tn,K1, · · · ,Km. Let ϕj be any bump function for each

j = 0, · · · ,m. For N ≥ 1, let PN = ⊗mj=0ϕj(N−2∆j). Then

(3.5) ‖PNeit∆f‖Lp(I×G) . N

d2−

d+2p ‖f‖L2(G)

holds for p ≥ 2 + 8r and N & 1.

On the other hand, similarly, for each Littlewood-Paley projection PN of the product type, there exists a

bump function ϕ such that PN = ϕ(N−2∆) satisfies PN ◦PN = PN . Noting that ‖PNf‖L2 . ‖f‖L2 , (3.2)

then implies

‖PNf‖Lr(M) . Nd( 1

2−1r )‖f‖L2(M).(3.6)

for all 2 ≤ r ≤ ∞.

4. Preliminaries on harmonic analysis on compact Lie groups

4.1. Fourier Transform. Let G be a compact group and G be its Fourier dual, i.e. the set of equivalent

classes of irreducible unitary representations of G. For λ ∈ G, let πλ : Vλ → Vλ be the irreducible unitary

representation in the class λ, and let dλ = dim(Vλ). Let µ be the normalized Haar measure on G. Then for

f ∈ L2(G), define the Fourier transform

f(λ) =

∫G

f(x)πλ(x−1)dµ.

Then the inverse Fourier transform

f(x) =∑λ∈G

dλtr(f(λ)πλ(x))

converges in L2(G). We have the Plancherel identities

(4.1) ‖f‖L2(G) = (∑λ∈G

dλ‖f(λ)‖2HS)1/2,

STRICHARTZ ESTIMATES FOR THE SCHRODINGER FLOW ON COMPACT LIE GROUPS 9

(4.2) 〈f, g〉L2(G) =∑λ∈G

dλtr(f(λ)g(λ)∗).

Here ‖ · ‖HS denotes the Hilbert-Schmidt norm of endomorphisms.

For the convolution

(f ∗ g)(x) =

∫G

f(xy−1)g(y) dµ(y),

we have

(4.3) (f ∗ g)∧(j) = f(j)g(j),

If g(λ) = cλ · Iddλ×dλ , where cλ is a scalar, then

(4.4) ‖f ∗ g‖L2(G) ≤ supλ|cλ| · ‖f‖L2(G).

We also have the Hausdorff-Young inequality

‖f(λ)‖HS ≤ d1/2λ ‖f‖L1(G) for all λ ∈ G.(4.5)

4.2. Root System and the Laplace-Beltrami Operator. LetG be a compact simply connected semisim-

ple Lie group of dimension d and g be its Lie algebra, and let gC denote the complexification of g. Choose

a maximal torus B ⊂ G and let r be the dimension of B. Let b be the Lie algebra of B, which is a Cartan

subalgebra of g, and let bC denote its complexification. The Fourier dual B of B is isomorphic to a lattice

Λ ⊂ ib∗, which is the weight lattice, under the isomorphism

Λ∼−→ B, λ 7→ eλ.(4.6)

We have the root space decomposition gC = bC ⊕ (⊕α∈ΦgαC). Here Φ ⊂ ib∗,

gαC = {X ∈ gC : Adb(X) = eα(b)X for all b ∈ B},

and dimCgαC = 1. This implies

|Φ|+ r = d.(4.7)

The Cartan-Killing form 〈 , 〉 on ib∗ becomes a real inner product, and (Ψ, 〈 , 〉) becomes an integral root

system, that is, a finite set Φ in a finite dimensional real inner product space with the following requirements

(4.8)

(i) Φ = −Φ;

(ii) α ∈ Φ, k ∈ R, kα ∈ Φ⇒ k = ±1;

(iii) sαΦ = Φ for all α ∈ Φ;

(iv) 2 〈α,β〉〈α,α〉 ∈ Z for all α, β ∈ Φ.

Here sα is the reflection about the hyperplane α⊥ orthogonal to α, that is,

sα(x) := x− 2〈x, α〉〈α, α〉

α.

Let P be a system of positive roots such that Φ = P t −P . Then by (4.7), we have

|P | = d− r2

.(4.9)

We can describe the weight lattice Λ purely in terms of the root system

Λ = {λ ∈ ib∗ | 2〈λ, α〉〈α, α〉

∈ Z, for all α ∈ Φ}.

10 Y. ZHANG

The set Φ of roots generate the root lattice Γ and we have Γ ⊂ Λ and Λ/Γ is finite.

Let

Λ+ := {λ ∈ ib∗ | 2〈λ, α〉〈α, α〉

∈ Z≥0, for all α ∈ P}

be the set of dominant weights. We describe Λ,Λ+ in terms of a basis. Let {α1, · · · , αr} be the set of simple

roots in P . Let {w1, · · · , wr} be the corresponding fundamental weights, i.e. the dual basis to the coroot

basis { 2α1

〈α1,α1〉 , · · · ,2αr〈αr,αr〉}. Then

Λ = Zw1 + · · ·+ Zwr,

Λ+ = Z≥0w1 + · · ·+ Z≥0wr.

Let

C = R>0w1 + · · ·+ R>0wr(4.10)

be the fundamental Weyl chamber, and we have the decomposition

ib∗ = (⊔s∈W

sC) t (⋃α∈Φ

{λ ∈ ib∗ : 〈λ, α〉 = 0}),(4.11)

where W is the Weyl group. Here t stands for disjoint union.

Define

(4.12) ρ :=1

2

∑α∈P

α =

r∑i=1

wi.

Then we have

G ∼= Λ+

such that the irreducible representation πλ corresponding to λ ∈ Λ+ has the character χλ and dimension dλ

given by Weyl’s formulas

χλ|B =

∑s∈W (det s)es(λ+ρ)∑s∈W (det s)esρ

,(4.13)

dλ =

∏α∈P 〈α, λ+ ρ〉∏α∈P 〈α, ρ〉

.(4.14)

Let H ∈ b. We can think of −iH as a real linear functional on ib∗, and by the Cartan-Killing inner

product on ib∗, we thus get a correspondence between H ∈ b and an element in ib∗, still denoted as H.

Under this correspondence, eλ(H) = ei〈λ,H〉 and we rewrite Weyl’s character formula as

χλ(expH) =

∑s∈W det s ei〈s(λ+ρ),H〉∑s∈W det s ei〈ρ,H〉

.(4.15)

Also under this correspondence between b and ib∗, we have

B ∼= ib∗/2πΓ∨,

where

Γ∨ = Z2α1

〈α1, α1〉+ · · ·+ Z

2αr〈αr, αr〉

is the coroot lattice.

STRICHARTZ ESTIMATES FOR THE SCHRODINGER FLOW ON COMPACT LIE GROUPS 11

We define the cells to be the connected components of {H ∈ ib∗/2πΓ∨ | 〈α,H〉 /∈ 2πZ} and call {H ∈ib∗/2πΓ∨ | 〈α,H〉 ∈ 2πZ} the cell walls.

We also record here Weyl’s integral formula. Let f ∈ L1(G) be invariant under the adjoint action of G.

Then

(4.16)

∫G

f dµ =1

|W |

∫B

f(b)|DP (b)|2 db

Here dµ, db are respectively the normalized Haar measures of G and B, and

DP (H) =∑s∈W

det s ei〈ρ,H〉

is the Weyl denominator.

Finally we describe the functional calculus of the Laplace-Beltrami operator ∆. Given any irreducible

unitary representation (πλ, Vλ) of G in the class λ ∈ G ∼= Λ+, the operator ∆ acts on the space Mλ =

{tr(πλT ) | T ∈ End(Vλ)} of matrix coefficients by

∆f = −kλf, for all f ∈Mλ, λ ∈ G,

where

(4.17) kλ = |λ+ ρ|2 − |ρ|2.

Let f ∈ L2(G) and consider the inverse Fourier transform f(x) =∑λ∈Λ+ dλtr(πλ(x)f(λ)), then for any

bounded Borel function F : R→ C, we have

F (∆)f =∑λ∈Λ+

F (−kλ)dλtr(πλ(x)f(λ)).

In particular, we have

(4.18) eit∆f =∑λ∈Λ+

e−itkλdλtr(πλ(x)f(λ)),

(4.19) PNeit∆f =

∑λ∈Λ+

ϕ(− kλN2

)e−itkλdλtr(πλ(x)f(λ)).

Example 4.1. Let M = SU(2), which is of dimension 3 and rank 1. Let a ∼= R be the Cartan subalgebra

and A ∼= R/2πZ be the maximal torus. The root system is {±α}, where α acts on a by α(θ) = 2θ. The

fundamental weight is w = 12α. We normalize the Cartan-Killing form so that |w| = 1. The Weyl group W

is of order 2, and acts on a as well as a∗ through multiplication by ±1. For m ∈ Z≥0∼= Z≥0w = Λ+, we

have

dm = m+ 1,(4.20)

χm(θ) =ei(m+1)θ − e−i(m+1)θ

eiθ − e−iθ=

sin(m+ 1)θ

sin θ, θ ∈ R/2πZ,(4.21)

km = (m+ 1)2 − 1.(4.22)

5. The Schrodinger Kernel

Let f ∈ L2(G). Then (4.19) implies

(PNeit∆f)∧(λ) = ϕ(

kλN2

)e−itkλ f(λ).

12 Y. ZHANG

Define

(KN (t, ·))∧(λ) = ϕ(kλN2

)e−itkλIddλ×dλ ,

which implies

KN (t, x) =∑λ∈Λ+

ϕ(kλN2

)e−itkλdλχλ(x).(5.1)

Then we can write

PNeit∆f = KN (t, ·) ∗ f = f ∗KN (t, ·),

and we call KN (t, x) the Schrodinger kernel. Incorporating (4.14), (4.15) and (4.17) into (5.1), we get

KN (t, x) =∑λ∈Λ+

e−it(|λ+ρ|2−|ρ|2)ϕ(|λ+ ρ|2 − |ρ|2

N2)

∏α∈P 〈α, λ+ ρ〉∏α∈P 〈α, ρ〉

∑s∈W det (s)ei〈s(λ+ρ),H〉∑s∈W det (s)ei〈s(ρ),H〉

(5.2)

Example 5.1. Specializing the Schrodinger kernel (5.2) to G = SU(2), using (4.20), (4.21), and (4.22), we

have

KN (t, θ) =

∞∑m=0

ϕ((m+ 1)2 − 1

N2)(m+ 1)e−i((m+1)2−1)t e

i(m+1)θ − e−i(m+1)θ

eiθ − e−iθ, θ ∈ R/2πZ.(5.3)

More generally, let G = Rn/2πΓ0×K1×· · ·×Km be equipped with a rational metric g as in Definition 2.1.

Let Λ0 be the dual lattice of Γ0, and Λj be the weight lattice forKj , j = 1, · · · ,m. Let PN = ⊗mj=0ϕj(N−2∆j)

be a Littlewood-Paley projection of the product type as described in Section 3.3. Define the Schrodinger

kernel KN on G by

PNeit∆f = f ∗KN (t, ·) = KN (t, ·) ∗ f.(5.4)

Then

KN =

m∏j=0

KN,j ,(5.5)

where the KN,j ’s are respectively the Schrodinger kernels on each component of G

KN,0 =∑λ0∈Λ0

ϕ0(−|λ0|2

β0N2)e−itβ

−10 |λ0|2ei〈λ0,H0〉,

KN,j =∑λj∈Λ+

j

ϕj(−|λj + ρj |2 + |ρj |2

βjN2)eitβ

−1j (−|λj+ρj |2+|ρj |2)dλjχλj ,

j = 1, · · · ,m. Here the ρj ’s are defined in terms of (4.12). We also write

KN =∑λ∈G

ϕ(λ,N)e−itkλdλχλ,

STRICHARTZ ESTIMATES FOR THE SCHRODINGER FLOW ON COMPACT LIE GROUPS 13

where

λ = (λ0, . . . , λm) ∈ G = Λ0 × Λ+1 × · · · × Λ+

m,

−kλ = −β−10 |λ0|2 +

m∑j=1

β−1j (−|λj + ρj |2 + |ρj |2),(5.6)

ϕ(λ,N) = ϕ0(−|λ0|2

β0N2) ·

n∏j=1

ϕj(−|λj + ρj |2 + |ρj |2

βjN2),(5.7)

dλ =

m∏j=1

dλj , χλ = ei〈λ0,H0〉m∏j=1

χλj .

Tracking all the definitions, we get the following lemma.

Lemma 5.2. Let d, r be respectively the dimension and rank of G.

(i) |{λ ∈ G | kλ . N2}| . Nr.

(ii) dλ . Nd−r

2 , uniformly for all λ ∈ G such that kλ . N2.

Now we interpret the Strichartz estimates on G as Fourier restriction estimates.

Lemma 5.3. For a compact simply connected semisimple Lie group G and its weight lattice Λ, there exists

D ∈ N such that 〈λ1, λ2〉 ∈ D−1Z for all λ1, λ2 ∈ Λ.

Proof. Let Φ be the set of roots for G. Then by Lemma 4.3.5 in [Var84], 〈α, β〉 are rational numbers

for all α, β ∈ Φ. Let S = {α1, · · · , αr} ⊂ Φ be a system of simple roots. Since the set of fundamental

weights {w1, · · · , wn} forms a dual basis to { 2α1

〈α1,α1〉 , · · · ,2αr〈αr,αr〉} with respect to the Cartan-Killing form

〈 , 〉, and 〈αi, αj〉 are rational numbers for all i, j = 1, · · · , r, we have that the wj ’s can be expressed as

linear combinations of the αj ’s with rational coefficients. This implies that 〈wi, wj〉 are rational numbers

for all i, j = 1, · · · , r. Since there are only finitely many such numbers as 〈wi, wj〉, there exists D ∈ Nso that 〈wi, wj〉 ∈ D−1Z for all i, j = 1, · · · , r. Thus 〈λ1, λ2〉 ∈ D−1Z for all λ1, λ2 ∈ Λ, since Λ =

Zw1 + · · ·+ Zwn. �

For G = Rn/2πΓ0 × K1 × · · · × Km, by the previous lemma, there exists for each j = 1, · · · ,m some

Dj ∈ N such that 〈λ, µ〉 ∈ D−1j Z for all λ, µ ∈ Λ+

j , which implies by (4.12) that

−|λj + ρj |2 + |ρj |2 = −|λj |2 − 〈λj , 2ρj〉 ∈ D−1j Z

for all λj ∈ Λj . Also recall that we require that there exists some D ∈ N such that 〈u, v〉 ∈ D−1Z for all

u, v ∈ Γ0. This implies that there also exists some D0 ∈ N such that 〈λ, µ〉 ∈ D−10 Z for all λ, µ ∈ Λ0. By

Definition 2.1 of a rational metric, there exists some D∗ > 0 such that

β−10 , · · · , β−1

m ∈ D−1∗ N.

Define

T = 2πD∗ ·m∏j=0

Dj .(5.8)

Then (5.6) implies that Tkλ ∈ 2πZ, which then implies that the Schrodinger kernel as in (5.5) is periodic

in t with a period of T . Thus we may view the time variable t as living on the circle T = R/TZ. Now the

formal dual to the operator

T : L2(G)→ Lp(T×G), f 7→ PNeit∆(5.9)

14 Y. ZHANG

is computed to be

T∗ : Lp′(T×G)→ L2(G), F 7→

∫T

PNe−is∆F (s, ·) ds

T,(5.10)

and thus

TT∗ : Lp′(T×G)→ Lp(T×G), F 7→

∫T

P2Ne

i(t−s)∆F (s, ·) dsT

= KN ∗ F,(5.11)

where

KN =∑λ∈G

ϕ2(λ,N)e−itkλdλχλ = KN ∗KN .

Note that the cutoff function ϕ2(λ,N) still defines a Littlewood-Paley projection of the product type and KN

is the associated Schrodinger kernel. Now the argument of TT∗ says that the boundedness of the operators

(5.9), (5.10) and (5.11) are all equivalent, thus the Strichartz estimate in (3.1) is equivalent to the space-time

Strichartz estimate

(5.12) ‖KN ∗ F‖Lp(T×G) . Nd− 2(d+2)

p ‖F‖Lp′ (T×G).

We have the space-time Fourier transform on T×G as follows. For (n, λ) ∈ 2πT Z× G, we have

KN (n, λ) =

{ϕ(λ,N) · Iddλ×dλ , if n = −kλ,0, otherwise.

(5.13)

Similarly, for f ∈ L2(G), we have

(PNeit∆f(x))∧(n, λ) =

{ϕ(λ,N) · f(λ), if n = −kλ,0, otherwise.

(5.14)

For m(t) =∑n∈ 2π

T Z m(n)eitn, we compute

(mKN )∧(n, λ) = m(n+ kλ)ϕ(λ,N)Iddλ×dλ .(5.15)

6. The Stein-Tomas Argument

Throughout this section, S1 stands for the standard circle of unit length, and ‖ · ‖ stands for the distance

from 0 on S1. Define

Ma,q := {t ∈ S1 | ‖t− a

q‖ < 1

qN}

where

a ∈ Z≥0, q ∈ N, a < q, (a, q) = 1, q < N.

We call such Ma,q’s as major arcs, which are reminiscent of the Hardy-Littlewood circle method. We will

prove the following key dispersive estimate.

Theorem 6.1. Let KN be the Schrodinger kernel (5.5) and T be the period (5.8). Then

|KN (t, x)| . Nd

(√q(1 +N‖ t

2πD −aq ‖1/2))r

for t2πD ∈Ma,q, uniformly in x ∈ G.

Noting the product structure (5.5) of KN , the above theorem reduces to the cases on irreducible compo-

nents of G.

STRICHARTZ ESTIMATES FOR THE SCHRODINGER FLOW ON COMPACT LIE GROUPS 15

Theorem 6.2. (i) Given G = Td = Rd/2πΓ such that there exists D ∈ R for which 〈λ, µ〉 ∈ D−1Z for all

λ, µ ∈ Γ. Then the Schrodinger kernel

KN (t,H) =∑λ∈Λ

ϕ(|λ|2

N2)e−it|λ|

2+i〈λ,H〉

satisfies

|KN (t,H)| . (N

√q(1 +N‖ t

2πD −aq ‖

12 )

)d

for t2πD ∈Ma,q, uniformly in H ∈ Tn.

(ii) Let G be a compact simply connected semisimple Lie group. Let Λ be the weight lattice for which

〈λ, µ〉 ∈ D−1Z for all λ, µ ∈ Λ for some D ∈ R. Let KN be the Schrodinger kernel as defined in (5.2). Then

|KN (t, x)| . Nd

(√q(1 +N‖ t

2πD −aq ‖1/2))r

(6.1)

for t2πD ∈Ma,q, uniformly in x ∈ G.

We will prove this theorem in the next section. Now we show how this theorem implies Strichartz

estimates.

Theorem 6.3. Let G = Tn ×K1 × · · · ×Km be equipped with a rational metric g and T be a period of the

Schrodinger flow as in (5.8). Let d, r be the dimension and rank of G respectively. Let f ∈ L2(G), λ > 0

and define

mλ = µ{(t, x) ∈ T×G | |PNeit∆f(x)| > λ|}

where µ = dt · dµG, with dt being the standard measure on T = R/TZ and dµG being the Haar measure on

G. Let

p0 =2(r + 2)

r.

Then the following statements hold true.

(I)

mλ .ε Ndp02 −(d+2)+ελ−p0‖f‖p0

L2(G), for all λ & Nd2−

r4 , ε > 0.

(II)

mλ . Ndp2 −(d+2)λ−p‖f‖pL2(G), for all λ & N

d2−

r4 , p > p0.

(III)

‖PNeit∆f‖Lp(T×G) . N

d2−

d+2p ‖f‖L2(G)(6.2)

holds for all p ≥ 2 + 8r .

(IV) Assume it holds that

‖PNeit∆f‖Lp(T×G) .ε N

d2−

d+2p +ε‖f‖L2(G)(6.3)

for some p > p0, then (6.2) holds for all q > p.

The proof strategy of this theorem is a Stein-Tomas type argument, similar to the proofs of Propositions

3.82, 3.110, 3.113 in [Bou93]. The new ingredient is the non-abelian Fourier transform. We detail the proof

in the following.

16 Y. ZHANG

Let ω ∈ C∞c (R) such that ω ≥ 0, ω(x) = 1 for all |x| ≤ 1 and ω(x) = 0 for all |x| ≥ 2. Let N be a dyadic

natural number. Define

ω 1N2

:= ω(N2·),

ω 1NM

:= ω(NM ·)− ω(2NM ·),

where

1 ≤M < N, M dyadic.

Let

N1 =N

210, 1 ≤ Q < N1, Q dyadic.

Then ∑Q≤M≤N

ω 1NM

= 1, on

[− 1

NQ,

1

NQ

],(6.4)

∑Q≤M≤N

ω 1NM

= 0, outside

[− 2

NQ,

2

NQ

].(6.5)

Write

1 =∑

1≤Q≤N1

∑Q≤M≤N

∑

(a,q)=1,Q≤q<2Q

δa/q

∗ ω 1NM

(t

T) + ρ(t).(6.6)

Note the major arc disjointness property(a1

q1+

[− 2

NQ1,

2

NQ1

])∩(a2

q2+

[− 2

NQ2,

2

NQ2

])= ∅

for (ai, qi) = 1, Qi ≤ qi < 2Qi, i = 1, 2, Q1 ≤ Q2 ≤ N1. This in particular implies

0 ≤ ρ(t) ≤ 1, for all t ∈ R/TZ,(6.7)

∑

(a,q)=1,Q≤q<2Q

δa/q

∗ ω 1NM

(·T

)

∧

(0) =1

T

∫ T

0

∑(a,q)=1,Q≤q<2Q

δa/q

∗ ω 1NM

(t

T) dt ≤ 2Q2

NM,(6.8)

which implies

1 ≥ |ρ(0)| ≥ 1−∑

1≤Q≤N1

∑Q≤M≤N

∣∣∣∣∣∣∣∣ ∑

(a,q)=1,Q≤q<2Q

δa/q

∗ ω 1NM

(·T

)

∧

(0)

∣∣∣∣∣∣∣∣ ≥ 1− 8N1

N≥ 1

2.(6.9)

By Dirichlet’s lemma on rational approximations, for any tT ∈ S1, there exists a, q with a ∈ Z≥0, q ∈ N,

(a, q) = 1, q ≤ N , such that | tT −aq | <

1qN . If ρ( tT ) 6= 0, then (6.4) implies q > N1 = N

210 . This implies by

(6.1) and (6.7) that

‖ρ(t)KN (t, x)‖L∞(T×G) . Nd− r2 .(6.10)

STRICHARTZ ESTIMATES FOR THE SCHRODINGER FLOW ON COMPACT LIE GROUPS 17

Now define coefficients αQ,M such that ∑

(a,q)=1,Q≤q<2Q

δa/q

∗ ω 1NM

(·T

)

∧

(0) = αQ,M ρ(0),(6.11)

then (6.8) and (6.9) imply

αQ,M .Q2

NM.(6.12)

Write

KN (t, x) =∑Q≤N1

∑Q≤M≤N

KN (t, x)

∑(a,q)=1,Q≤q<2Q

δa/q

∗ ω 1NM

(·T

)

− αQ,Mρ (t)

+

1 +∑Q,M

αQ,M

KN (t, x)ρ(t),(6.13)

and define

ΛQ,M (t, x) := KN (t, x)

∑(a,q)=1,Q≤q<2Q

δa/q

∗ ω 1NM

(·T

)

− αQ,Mρ (t).(6.14)

Then from (6.1), (6.10), (6.12), we have

‖ΛQ,M‖L∞(T×G) . Nd− r2 (

M

Q)r/2.(6.15)

Next, we estimate ΛQ,M . From (5.15), for

n ∈ 2π

TZ ∼= T, λ ∈ G,

we have

ΛQ,M (n, λ) = λQ,M (n, λ) · Iddλ×dλ(6.16)

where

λQ,M (n, λ) = ϕ(λ,N)

∑(a,q)=1,Q≤q<2Q

δa/q

∧ · ω 1NM

(T ·)− αQ,M ρ

(n+ kλ).(6.17)

Note that (6.11) immediately implies

λQ,M (n, λ) = 0, for n+ kλ = 0.(6.18)

Let d(m,Q) denote the number of divisors of m less than Q, using Lemma 3.33 in [Bou93],

|(∑

(a,q)=1,Q≤q<2Q

δa/q)∧(Tn)| .ε d(

Tn

2π,Q)Q1+ε, n 6= 0, ε > 0,(6.19)

we get

|λQ,M (n, λ)| .ε ϕ(λ,N)Q1+ε

NMd(T (n+ kλ)

2π,Q) +

Q2

NM|ρ(n+ kλ)|.(6.20)

18 Y. ZHANG

Using

d(m,Q) .ε mε,

(6.19) and (6.6), we have

|ρ(n)| ≤∑

1≤Q≤N1

∑Q≤M≤N

d(Tn2π , Q)Q1+ε

NM.Nε

N, for n 6= 0, |n| . N2,(6.21)

thus

|λQ,M (n, λ)| .ε ϕ(λ,N)Q

NM

[Qεd(

T (n+ kλ)

2π,Q) +

Q

N1−ε

].ε ϕ(λ,N)

QNε

NM, for |n| . N2.(6.22)

Proposition 6.4. (i) Assume that f ∈ L1(T×G). Then

‖f ∗ ΛQ,M‖L∞(T×G) . Nd− r2 (

M

Q)r/2‖f‖L1(T×G).(6.23)

(ii) Assume that f ∈ L2(T×G). Assume also

(6.24) f(n, λ) = 0, for |n| & N2.

Then

‖f ∗ ΛQ,M‖L2(T×G) .εQNε

NM‖f‖L2(T×G),(6.25)

‖f ∗ ΛQ,M‖L2(T×G) .τ,BQ1+2τL

NM‖f‖L2(T×G) +M−1L−B/2Nd/2‖f‖L1(T×G).(6.26)

for all

L > 1, 0 < τ < 1, B >6

τ, N > (LQ)B .(6.27)

Proof. Using (6.15), we have

‖f ∗ ΛQ,M‖L∞(T×G) ≤ ‖f‖L1(T×G)‖ΛQ,M‖L∞(T×G) . Nd− r2 (

M

Q)r/2‖f‖L1(T×G).

This proves (i). (6.25) is a consequence of (4.4), (6.16), and (6.22). To prove (6.26), we use (4.1), (4.3) and

(6.16) to get

‖f ∗ ΛQ,M‖L2(T×G) =

∑n,λ

dλ‖f(n, λ)‖2HS · |λQ,M (n, λ)|21/2

,

which combined with (6.18), (6.20), and (6.21) yields

‖f ∗ ΛQ,M‖L2(T×G) .εQ1+ε

NM

∑n,λ

ϕ(λ,N)2dλ‖f(n, λ)‖2HSd(T (n+ kλ)

2π,Q)2

1/2

+Q2

MN2−ε ‖f‖L2(T×G).(6.28)

STRICHARTZ ESTIMATES FOR THE SCHRODINGER FLOW ON COMPACT LIE GROUPS 19

Using Lemma 3.47 in [Bou93] and Lemma 5.2, we have∣∣∣∣{(n, λ) | |n|, kλ . N2, d(T (n+ kλ)

2π,Q) > D}

∣∣∣∣.τ,B (D−BQτN2 +QB) · max

|m|.N2|{(n, λ) | n+ kλ = m}|

.τ,B (D−BQτN2 +QB) · |{λ ∈ G | kλ . N2}|

.τ,B (D−BQτN2 +QB) ·Nr.(6.29)

Now (4.5) gives

‖f(n, λ)‖2HS . dλ‖f‖2L1(T×G),

and Lemma 5.2 gives

|ϕ(λ,N)d2λ| . Nd−r,

which together with (6.29) imply

‖f ∗ ΛQ,M‖L2(T×G) .τ,B(Q1+εD

NM+

Q2

MN2−ε )‖f‖L2(T×G)

+Q1+ε

NM·Q · (D−B/2QτN +QB/2)Nd/2‖f‖L1(T×G).(6.30)

This implies (6.26) assuming the conditions in (6.27). �

Now interpolating (6.23) and (6.25), we get

‖f ∗ ΛQ,M‖Lp(T×G) .ε Nd− r2−

2d−r+2p +εM

r2−

r+2p Q−

r2 + r+2

p ‖f‖Lp′ (T×G).(6.31)

Interpolating (6.23) and (6.26) for

p >2(r + 2)

r+ 10τ, which implies σ =

r

2− r + 2 + 4τ

p> 0,(6.32)

we get

‖f ∗ ΛQ,M‖Lp(T×G) .τ,BNd− r2−

2d−r+2p M

r2−

r+2p Q−σL

2p ‖f‖Lp′ (T×G)

+Q−2r (1− 2

p )Mr2−

r+2p L−

Bp Nd− r2−

d−rp ‖f‖L1(T×G).(6.33)

Now we are ready to prove Theorem 6.3.

Proof of Theorem 6.3. Without loss of generality, we assume that ‖f‖L2(G) = 1. Then for F = PNeit∆f ,

(3.2) implies

‖F‖L2x. 1,(6.34)

‖F‖L∞x . Nd2 .(6.35)

Let

H = χ|F |>λ ·F

|F |.(6.36)

20 Y. ZHANG

Let ˜PN be a Littlewood-Paley projection of the product type such that ˜PN ◦ PN = PN . Let ˜KN be the

Schrodinger kernel associated to ˜PNeit∆. Then by (4.3), (5.13), and (5.14), we have

F ∗ ˜KN = F.

Let QN2 be the Littlewood-Paley projection operator on L2(T×G) defined by

(QN2H)∧ := ϕ(−kλ − n2

N4)H(n, λ)

for some bump function ϕ such that QN2 ◦PN = PN . Then by (4.2) and (5.14), we have

〈F,H〉L2t,x

= 〈QN2F,H〉L2t,x

= 〈F,QN2H〉L2t,x.

Then we can write

λmλ ≤ 〈F,H〉L2t,x

= 〈F ∗ ˜KN , QN2H〉L2t,x.

Using (4.1) and (4.3) again, we get

λmλ ≤ 〈F,QN2H ∗ ˜KN 〉L2t,x≤ ‖F‖L2

t,x‖QN2H ∗ ˜KN‖L2

t,x

. ‖QN2H ∗ ˜KN‖L2t,x

= 〈QN2H ∗ ˜KN , QN2H ∗ ˜KN 〉L2t,x

= 〈QN2H,QN2H ∗ ( ˜KN ∗ ˜KN )〉L2t,x.(6.37)

Let

H ′ = QN2H, KN = ˜KN ∗ ˜KN .

Note that H ′ by definition satisfies the assumption in (6.24) and we can apply Proposition 6.4. Also note

that KN is still a Schrodinger kernel associated to a Littlewood-Paley projection operator of the product

type. Finally note that the Bernstein type inequalities (3.2) and the definition (6.36) of H give

‖H ′‖Lpt,x . ‖H‖Lpt,x . m1p

λ .(6.38)

Write

Λ =∑

1≤Q≤N1

∑Q≤M≤N

ΛQ,M , KN = Λ + (KN − Λ),

where ΛQ,M is defined as in (6.14) except that KN is replaced by KN . We have by (6.37)

λ2m2λ . 〈H ′, H ′ ∗ Λ〉L2

t,x+ 〈H ′, H ′ ∗ (KN − Λ)〉L2

t,x

. ‖H ′‖Lp′t,x‖H ′ ∗ Λ‖Lpt,x + ‖H ′‖2L1

t,x‖KN − Λ‖L∞t,x .(6.39)

Using (6.31) for p = p0 := 2(r+2)r , then summing over Q,M , and noting (6.38), we have

‖H ′‖Lp′t,x‖H ′ ∗ Λ‖Lpt,x . N

d− 2d+4p0

+ε‖H ′‖2Lp′0t,x

. Nd− 2d+4p0

+εm2p′0λ .

From (6.10) and (6.12) we get

‖KN − Λ‖L∞t,x . Nd− r2 ,(6.40)

which implies

‖H ′‖2L1t,x‖KN − Λ‖L∞t,x . N

d− r2 ‖H ′‖2L1t,x. Nd− r2m2

λ.(6.41)

STRICHARTZ ESTIMATES FOR THE SCHRODINGER FLOW ON COMPACT LIE GROUPS 21

Then we have

λ2m2λ . N

d− 2d+4p0

+εm2p′0λ +Nd− r2m2

λ,

which implies for λ & Nd2−

r4

mλ .ε Np0( d2−

d+2p0

)+ελ−p0 .

Thus part (I) is proved. To prove part (II) for some fixed p, using part (I) and (6.35), it suffices to prove it

for λ & Nd2−ε. Summing (6.33) over Q,M in the range indicated by (6.27), we get

‖H ′ ∗ Λ1‖Lpt,x . LNd− 2d+4

p ‖H ′‖Lp′t,x

+ L−B/pNd− d+2p ‖H ′‖L1

t,x,(6.42)

where

Λ1 :=∑

Q<Q1,Q≤M≤N

ΛQ,M

and Q1 is the largest Q-value satisfying (6.27). For values Q ≥ Q1, use (6.31) to get

‖H ′ ∗ (Λ− Λ1)‖Lpt,x .ε Nd− 2d+4

p +εQ−( r2−

r+2p )

1 ‖H ′‖Lp′t,x.(6.43)

Using (6.39), (6.41), (6.42) and (6.43), we get

λ2m2λ . N

d− 2(d+2)p (L+

Nε

Qr2−

r+2p

1

)m2/p′

λ + L−B/pNd− d+2p m

1+ 1p′

λ +Nd− r2m2λ.

For λ & Nd2−

r4 , the last term of the above inequality can be dropped. Let Q1 = Nδ such that δ > 0 and

(LNδ)B < N(6.44)

such that (6.27) holds. Note that

L > 1 >Nε

Qr2−

r+2p

1

for p > p0 + 10τ and ε sufficiently small, thus

λ2m2λ . N

d− 2(d+2)p Lm

2/p′

λ + L−B/pNd− d+2p m

1+ 1p′

λ .

This implies

mλ . Np( d2−

d+2p )L

p2 λ−p +Np(d− d+2

p )L−Bλ−2p

. N−d−2(Nd/2

λ)pL

p2 +N−d−2(

Nd/2

λ)2pL−B .

Let

L = (Nd/2

λ)τ , B >

p

τ

and δ be sufficiently small so that (6.44) holds, then

mλ . N−d−2(

Nd/2

λ)p+

pτ2 .

Note that conditions for p, τ indicated in (6.32) implies that p + pτ2 can take any exponent > p0 = 2(r+2)

r .

This completes the proof of part (II).

22 Y. ZHANG

The proofs of parts (III) and (IV) are then identical to the proofs of Propositions 3.110 and 3.113 respec-

tively in [Bou93]. �

Proof of Theorem 2.2. Part (i) is a direct consequence of Theorem 6.3(III). Part (ii) is a direct consequence

of Theorem 6.3(IV) and the result from [BD15] that full Strichartz estimates hold on any torus with an

ε-loss. �

7. Dispersive Estimates on Major Arcs

In this section, we prove Theorem 6.2.

7.1. Weyl Type Sums on Rational Lattices.

Definition 7.1. Let L = Zw1 + · · · + Zwr be a lattice on an inner product space (V, 〈 , 〉). We say L is a

rational lattice provided that there exists some D ∈ R such that 〈wi, wj〉 ∈ D−1Z. We call the number D a

period of L.

By Lemma 5.3, any weight lattice Λ is a rational lattice with respect to the Cartan-Killing form. As a

sublattice of Λ, the root lattice Γ is also rational.

Let f be a function on Zr and define the difference operator Di by

(7.1) Dif(n1, · · · , nr) := f(n1, · · · , ni−1, ni + 1, ni+1, · · · , nr)− f(n1, · · · , nr)

for i = 1, · · · , r. The Leibniz rule for Di reads

Di(

n∏j=1

fj) =

n∑l=1

∑1≤k1<···<kl≤n

Difk1· · ·Difkl ·

∏j 6=k1,··· ,kl

1≤j≤n

fj .(7.2)

Note that there are 2n − 1 terms in the right side of the above formula.

Definition 7.2. Let L ∼= Zr be a lattice of rank r. Given A ∈ R, we say a function f on L is a pseudo-

polynomial of degree A provided for each n ∈ Z≥0,

|Di1 · · ·Dinf(n1, · · · , nr)| . NA−n(7.3)

holds uniformly in |ni| . N , i = 1, · · · , r, for all ij = 1, · · · , r, j = 1, · · · , n, and N ≥ 1.

A direct application of the Leibniz rule (7.2) gives the following lemma.

Lemma 7.3. Let L be a lattice and f, g two functions on L. Assume f, g are pseudo-polynomials of degrees

A,B respectively. Then f · g is a pseudo-polynomial of degree A+B.

Now we have the following estimate on Weyl type sums, which generalizes the classical Weyl inequality

in one dimension, as in Lemma 3.18 of [Bou93].

Lemma 7.4. Let L = Zw1 + · · ·+Zwr be a rational lattice in the inner product space (V, 〈 , 〉) with a period

D > 0. Let ϕ be a bump function on R and N ≥ 1, A ∈ R. Suppose f : L → C a pseudo-polynomial of

degree A. Let

F (t,H) =∑λ∈L

e−it|λ|2+i〈λ,H〉ϕ(

|λ|2

N2) · f(7.4)

STRICHARTZ ESTIMATES FOR THE SCHRODINGER FLOW ON COMPACT LIE GROUPS 23

for t ∈ R and H ∈ V . Then for t2πD ∈Ma,q, we have

|F (t,H)| . NA+r

(√q(1 +N‖ t

2πD −aq ‖1/2))r

(7.5)

uniformly in H ∈ V .

Note that Part (i) of Theorem 6.2 is a direct consequence of this lemma.

Proof. By the Weyl differencing trick, write

|F |2 =∑

λ1,λ2∈L

e−it(|λ1|2−|λ2|2)+i〈λ1−λ2,H〉ϕ(|λ1|2

N2)ϕ(|λ2|2

N2)f(λ1)f(λ2)

=∑

µ=λ1−λ2

e−it|µ|2+i〈µ,H〉

∑λ=λ2

e−i2t〈µ,λ〉ϕ(|µ+ λ|2

N2)ϕ(|λ|2

N2)f(µ+ λ)f(λ)

≤∑|µ|.N

|∑λ

e−i2t〈µ,λ〉ϕ(|µ+ λ|2

N2)ϕ(|λ|2

N2)f(µ+ λ)f(λ)|.

Now let L = Zw1 + · · ·+ Zwr. Write

λ =

r∑i=1

niwi,

and

g(λ) = ϕ(|µ+ λ|2

N2)ϕ(|λ|2

N2)f(µ+ λ)f(λ).

Note that as functions in λ ∈ L, both ϕ( |µ+λ|2N2 )| and ϕ( |λ|

2

N2 ) are pseudo-polynomials of degree 0, and

both f(µ + λ) and f(λ) are pseudo-polynomials of degree A, which implies by Lemma 7.3 that g(λ) is a

pseudo-polynomial of degree 2A. That is, g(λ) satisfies

|Di1 · · ·Ding(λ)| . N2A−n.(7.6)

uniformly for |λ| . N and N ≥ 1, for all i1, · · · , in ∈ {1, · · · , r}. Write∑λ∈L

e−i2t〈µ,λ〉g(λ) =∑

n1,··· ,nr∈Z(

r∏i=1

e−itni〈µ,2wi〉)g(λ).(7.7)

By summation by parts twice, we have∑n1∈Z

e−itn1〈µ,2w1〉g = (e−it〈µ,2w1〉

1− e−it〈µ,2w1〉)2∑n1∈Z

e−itn1〈µ,2w1〉D21g(n1, · · · , nr),(7.8)

then (7.7) becomes∑λ∈L

e−i2t〈µ,λ〉g = (e−it〈µ,2w1〉

1− e−it〈µ,2w1〉)2

∑n1,··· ,nr∈Z

(

r∏i=1

e−itni〈µ,2wi〉)D21g(n1, · · · , nr).

Then we can carry out the procedure of summation by parts twice with respect to other variables n2, · · · , nr.But we require that only when

|1− e−it〈µ,2wi〉| ≥ 1

N

24 Y. ZHANG

do we carry out the procedure to the variable ni. Using (7.6), we obtain

|∑λ

e−i2t〈µ,λ〉ϕ(|µ+ λ|2

N2)ϕ(|λ|2

N2)f(µ+ λ)f(λ)|

. N2A−rr∏i=1

1

(max{1− e−it〈µ,2wi〉, 1N })2

. N2A−rr∏i=1

1

(max{‖ 12π t〈µ, 2wi〉‖,

1N })2

.

Writing µ =∑rj=1mjwj , mj ∈ Z, we have

|F |2 . N2A−r∑

|mj |.N,j=1,··· ,r

r∏i=1

1

(max{‖ 12π t∑rj=1mj〈wj , 2wi〉‖, 1

N })2.

Let

ni =

r∑j=1

mj〈wj , 2wi〉 ·D, i = 1, · · · , r,(7.9)

where D > 0 is the period of L so that 〈wj , wi〉 ∈ D−1Z. Then ni ∈ Z. Note that the matrix (〈wj , 2wi〉D)i,jis non-degenerate, which implies that for each vector (n1, · · · , nr) ∈ Zr, there exists at most one vector

(m1, · · · ,mr) ∈ Zr so that (7.9) holds, thus

|F |2 . N2A−r∑|ni|.N,i=1,··· ,r

r∏i=1

1

(max{‖ t2πDni‖,

1N })2

. N2A−rr∏i=1

∑|ni|.N

1

(max{‖ t2πDni‖,

1N })2

.

Then by a standard estimate as in the proof of the classical Weyl inequality in one dimension, we have∑|ni|.N

1

(max{‖ t2πDni‖,

1N })2

.N3

(√q(1 +N‖ t

2πD −aq ‖1/2))2

,

which implies the desired result

|F |2 . N2A+2r

(√q(1 +N‖ t

2πD −aq ‖1/2))2r

.

�

Remark 7.5. Let λ0 be a constant vector in Rr and C a constant real number. Then we can slightly

generalize the form of the function F (t,H) in the above lemma into

F (t,H) =∑λ∈L

e−it|λ+λ0|2+i〈λ,H〉ϕ(|λ+ λ0|2 + C

N2) · f

such that the conclusion of the lemma still holds.

7.2. From a Chamber to the Whole Weight Lattice. To prove Part (ii) of Theorem 6.2, we first rewrite

the Schrodinger kernel as an exponential sum over the whole weight lattice Λ instead of just a chamber of

it, in order to apply Lemma 7.4.

STRICHARTZ ESTIMATES FOR THE SCHRODINGER FLOW ON COMPACT LIE GROUPS 25

Lemma 7.6. Recall that DP (H) =∑s∈W det s ei〈ρ,H〉 is the Weyl denominator. We have

KN (t, x) =eit|ρ|

2

(∏α∈P 〈α, ρ〉)DP (H)

∑λ∈Λ

e−it|λ|2+i〈λ,H〉ϕ(

|λ|2 − |ρ|2

N2)∏α∈P〈α, λ〉(7.10)

=eit|ρ|

2

(∏α∈P 〈α, ρ〉)|W |

∑λ∈Λ

e−it|λ|2

ϕ(|λ|2 − |ρ|2

N2)∏α∈P〈α, λ〉

∑s∈W det (s)ei〈s(λ),H〉∑s∈W det (s)ei〈s(ρ),H〉

(7.11)

Proof. To prove (7.11), first note that from Proposition 7.13 below,∏α∈P 〈α, ·〉 is an anti-invariant polyno-

mial, that is, ∏α∈P〈α, s(λ)〉 = det(s)

∏α∈P〈α, λ〉,(7.12)

for all λ ∈ ib∗. Recall that the Weyl group W acts on ib∗ isometrically, that is,

|s(λ)| = |λ|, for all s ∈W, λ ∈ ib∗.(7.13)

Also recalling the definition (4.12) of ρ and the definition (4.10) of the fundamental chamber C, we may

rewrite KN as in (5.2) into

KN (t, x) =eit|ρ|

2

(∏α∈P 〈α, ρ〉)DP

∑λ∈Λ∩C

e−it|λ|2

ϕ(|λ|2 − |ρ|2

N2)∏α∈P〈α, λ〉

∑s∈W

det(s)ei〈s(λ),H〉

Using the (7.12) and (7.13), we write

KN (t, x) =eit|ρ|

2

(∏α∈P 〈α, ρ〉)DP

∑s∈W

∑λ∈Λ∩C

e−it|λ|2

ϕ(|λ|2 − |ρ|2

N2)∏α∈P〈α, s(λ)〉ei〈s(λ),H〉

=eit|ρ|

2

(∏α∈P 〈α, ρ〉)DP

∑s∈W

∑λ∈Λ∩C

e−it|s(λ)|2ϕ(|s(λ)|2 − |ρ|2

N2)∏α∈P〈α, s(λ)〉ei〈s(λ),H〉

=eit|ρ|

2

(∏α∈P 〈α, ρ〉)DP

∑λ∈ts∈W s(Λ∩C)

e−it|λ|2

ϕ(|λ|2 − |ρ|2

N2)∏α∈P〈α, λ〉ei〈λ,H〉,(7.14)

which then implies by (4.11) that

KN (t, x) =eit|ρ|

2

(∏α∈P 〈α, ρ〉)DP

∑λ∈Λ

e−it|λ|2

ϕ(|λ|2 − |ρ|2

N2)∏α∈P〈α, λ〉ei〈λ,H〉.

This proves (7.10). To prove (7.11), write∑λ∈Λ

e−it|λ|2+i〈λ,H〉ϕ(

|λ|2 − |ρ|2

N2)∏α∈P〈α, λ〉 =

∑λ∈Λ

e−it|s(λ)|2+i〈s(λ),H〉ϕ(|s(λ)|2 − |ρ|2

N2)∏α∈P〈α, s(λ)〉,(7.15)

which implies using (7.12) and (7.13) that∑λ∈Λ

e−it|λ|2+i〈λ,H〉ϕ(

|λ|2 − |ρ|2

N2)∏α∈P〈α, λ〉 = det(s)

∑λ∈Λ

e−it|λ|2+i〈s(λ),H〉ϕ(

|λ|2 − |ρ|2

N2)∏α∈P〈α, λ〉,

which further implies∑λ∈Λ

e−it|λ|2+i〈λ,H〉ϕ(

|λ|2 − |ρ|2

N2)∏α∈P〈α, λ〉 =

1

|W |∑λ∈Λ

e−it|λ|2

ϕ(|λ|2 − |ρ|2

N2)∏α∈P〈α, λ〉

∑s∈W

det(s)ei〈s(λ),H〉.

This combined with (7.10) yields (7.11). �

26 Y. ZHANG

Example 7.7. Specializing (7.10) and (7.11) to the Schrodinger kernel (5.3) for G = SU(2), we get

KN (t, θ) =eit

eiθ − e−iθ∑m∈Z

e−itm2+imθϕ(

m2 − 1

N2)m(7.16)

=eit

2

∑m∈Z

e−itm2

ϕ(m2 − 1

N2)m · e

imθ − e−imθ

eiθ − e−iθ, θ ∈ R/2πZ.(7.17)

Corollary 7.8. (6.1) holds for the following two scenarios.

Scenario 1: x = 1G, where 1G is the identity element of G.

Scenario 2: ‖ 12π 〈α,H〉‖ &

1N for any x conjugate to expH. This is to say that the variable H is away from

all the cell walls {H | ‖ 12π 〈α,H〉‖ = 0 for some α ∈ P} by a distance of & 1

N .

Proof. Scenario 1: When x = 1G, the character equals χλ(1G) = dλ =∏α∈P 〈α, λ〉/

∏α∈P 〈α, ρ〉. Then by

(7.11), the Schrodinger kernel at x = 1G equals

(7.18) KN (t, 1G) =eit|ρ|

2

(∏α∈P 〈α, ρ〉)2|W |

∑λ∈Λ

e−it|λ|2

ϕ(|λ|2 − |ρ|2

N2)(∏α∈P〈α, λ〉)2.

Note that f(λ) =(∏

α∈P 〈α, λ〉)2

is a polynomial in the variable λ = n1w1 + · · ·+ nrwr ∈ Λ of degree 2|P |,which equals d− r by (4.9). Thus f is also a pseudo-polynomial of degree d− r. Then the desired estimate

is a direct consequence of Lemma 7.4.

Scenario 2: By Lemma 4.13.4 of Chapter 4 in [Var84], the Weyl denominator DP =∑s∈W (det s)ei〈s(ρ),H〉

can be rewritten as

DP = e−i〈ρ,H〉∏α∈P

(ei〈α,H〉 − 1).(7.19)

Note that

1 .|ei〈α,H〉 − 1|‖ 1

2π 〈α,H〉‖. 1.

Then by assumption the Weyl denominator satisfies

(7.20) |DP (H)| &∏α∈P‖ 1

2π〈α,H〉‖ & N−|P |.

Let

F =∑λ∈Λ

e−it|λ|2+i〈λ,H〉ϕ(

|λ|2 − |ρ|2

N2) · f,

where f =∏α∈P 〈α, λ〉. Note that f is a polynomial and thus also a pseudo-polynomial of degree |P | in λ.

Applying Lemma (7.4) to F we get

|KN (t, x)| = | eit|ρ|2

(∏α∈P 〈α, ρ〉)DP (H)

| · |F | . | 1

DP (H)| · |F | . N |P | · Nr+|P |

(√q(1 +N‖ t

2πD −aq ‖1/2))r

.

Recalling |P | = d−r2 , we establish (6.1) for Scenario 2. �

Example 7.9. We specialize the Schrodinger kernel (7.16) and (7.17) to the case of G = SU(2). Scenario

1 in the above corollary corresponds to when θ ∈ 2πZ and

KN (t, θ) =eit

2

∑m∈Z

e−itm2

ϕ(m2 − 1

N2)m2, |KN (t, θ)| .

∣∣∣∣∣∑m∈Z

e−itm2

ϕ(m2 − 1

N2)m2

∣∣∣∣∣ .(7.21)

STRICHARTZ ESTIMATES FOR THE SCHRODINGER FLOW ON COMPACT LIE GROUPS 27

Scenario 2 corresponds to when |eiθ − e−iθ| & 1N , equivalently, when θ is away from the cell walls {0, π} by

a distance & 1N . In this case,

|KN (t, θ)| .∣∣∣∣ 1

eiθ − e−iθ

∣∣∣∣ ·∣∣∣∣∣∑m∈Z

e−itm2+imθϕ(

m2 − 1

N2)m

∣∣∣∣∣ .(7.22)

Then we get the desired estimates for (7.21) and (7.22) using Lemma 7.4.

7.3. Pseudo-polynomial Behavior of Characters. We have established the key estimates (6.1) for when

the variable expH in the maximal torus is either the identity or away from all the cell walls by a distance of

& 1N . To establish (6.1) fully, we need to look at the scenarios when the variable expH is close to the some

of the cell walls within a distance of . 1N . In this section, we first deal with the scenario when the variable

expH is close to all the cell walls within a distance of . 1N . To achieve this end, we first prove the following

crucial lemma on the pseudo-polynomial behavior of characters.

Lemma 7.10. Let µ ∈ ib∗. For λ ∈ ib∗, define

χµ(λ,H) =

∑s∈W det (s)ei〈s(λ+µ),H〉∑s∈W det (s)ei〈s(ρ),H〉

.

Let L ∼= Zr be the weight lattice or the root lattice (or any sublattice of full rank of the weight lattice), and

viewing χµ(λ,H) as a function in λ ∈ L, we have

(7.23) |Di1 · · ·Dikχµ(λ,H)| . N

d−r2 −k

holds uniformly in |λ| . N , |H| . N−1, and N ≥ 1, for all k ∈ Z≥0. In other words, χµ(λ,H) is a

pseudo-polynomial of degree d−r2 in λ uniformly in |H| . N−1.

Using this lemma, applying Lemma 7.4 to the Schrodinger kernel KN in the form of (7.11), we immediately

get the following corollary.

Corollary 7.11. Inequality (6.1) holds uniformly when x ∈ G is conjugate to expH such that |H| . N−1.

In other words, when x is within . N−1 a distance from the identity 1G.

We now prove Lemma 7.10 for L ∼= Zw1 + · · ·+Zwr being the weight lattice (the case for the root lattice

or any other sublattice can be proved similarly). First note that as |H| . N−1 for N large enough, by (7.19),

we have ∣∣∣∣∏α∈P 〈α,H〉DP

∣∣∣∣ ≈ 1.

Thus it suffices to show (7.23) replacing χµ(λ,H) by

χµ1 (λ,H) =

∑s∈W det (s)ei〈s(λ+µ),H〉∏

α∈P 〈α,H〉.(7.24)

7.3.1. Approach 1: via BGG-Demazure Operators. The idea is to expand the numerator of χµ1 (λ,H) into a

power series of polynomials in H ∈ ib∗, which are anti-invariant with respect to the Weyl group W , and

then to estimate the quotients of these polynomial over the denominator∏α∈P 〈α,H〉. We will see that

these quotients are in fact polynomials in H ∈ ib∗, and can be more or less explicitly computed by the

BGG-Demazure operators. We now review the basic definitions and facts of the BGG-Demazure operators

and the related invariant theory. A good reference is Chapter IV in [Hil82].

From now on, we fix an inner product space (a, 〈 , 〉) and let Φ be an integral root system in the dual

space (a∗, 〈 , 〉). Let P (a) be the space of polynomial functions on a. The orthogonal group O(a) with

28 Y. ZHANG

respect to the inner product on a, in particular the Weyl group, acts on P (a) by

(sf)(H) := f(s−1H), s ∈ O(a), f ∈ P (a), H ∈ a.

Definition 7.12. For α ∈ a∗, let sα : a→ a denote the reflection about the hyperplane

{H ∈ a : α(H) = 0},

that is,

sα(H) := H − 2α(H)

〈α, α〉Hα

where H ∈ a. Here Hα corresponds to α through the identification a∼−→ a∗. Define the BGG-Demazure

operator ∆α : P (a)→ P (a) associated to α ∈ a∗ by

∆α(f) =f − sα(f)

α.

As an example, we compute ∆α(λm) for λ ∈ a∗.

∆α(λm) =λm − λ(· − 2 α

〈α,α〉Hα)m

α

=λm − (λ− 2 〈λ,α,〉〈α,α〉 α)m

α

=

m∑i=1

(−1)i−1

(m

i

)2i

〈α, α〉i〈λ, α〉iαi−1λm−i.(7.25)

This computation in particular implies that for any f ∈ P (a), the operator ∆α(f) lowers the degree of f by

at least 1.

Let P (a)W denote the subspace of P (a) that are invariant under the action of the Weyl group W , that is,

P (a)W := {f ∈ P (a) | sf = f for all s ∈W}.

We call P (a)W the space of invariant polynomials. We also define

P (a)Wdet := {f ∈ P (a) | sf = (det s)f for all s ∈W}.

We call P (a)Wdet the space of anti-invariant polynomials. We have the following proposition which states that

P (a)Wdet is a free P (a)W -module of rank 1.

Proposition 7.13 (Chapter II, Proposition 4.4 in [Hil82]). Define ddet ∈ P (a) by

ddet =∏α∈P

α.

Then ddet ∈ P (a)Wdet and

P (a)Wdet = ddet · P (a)W .

By the above proposition, given any anti-invariant polynomial f , we have f = d · g where g is invariant.

We call g the invariant part of f . The BGG-Demazure operators provide a procedure that computes the

invariant part of any anti-invariant polynomial. We describe this procedure as follows. The Weyl group W is

generated by the reflections sα1, · · · , sαr where S = {α1, · · · , αr} is the set of simple roots. Define the length

of s ∈W to be the smallest number k such that s can be written as s = sαi1 · · · sαik . The longest element s

in W is of length |P | = d−r2 , and such s is unique; see Section 1.8 in [Hum90]. Write s = sαi1 · · · sαiL . Set

δ = ∆αi1· · ·∆αiL

STRICHARTZ ESTIMATES FOR THE SCHRODINGER FLOW ON COMPACT LIE GROUPS 29

and note that it is well defined in the sense it does not depend on the particular choice of the decomposition

s = sαi1 · · · sαiL ; see Chapter IV, Proposition 1.7 in [Hil82].

Proposition 7.14 (Chapter IV, Proposition 1.6 in [Hil82]). We have

δf =|W |ddet

· f

for all f ∈ P (a)Wdet.

That is, the operator δ produces the invariant part of any anti-invariant polynomial (modulo a multiplica-

tive constant). As an example, we compute δ = ∆αi1· · ·∆αiL

on λm. Proceed inductively using (7.25), we

arrive at the following proposition.

Proposition 7.15. Let m ≥ L. Then

δ(λm) =∑

θ,a(α,β),b(γ),c(ζ),η∈Z

(−1)θ∏α≤β

〈αiα , αiβ 〉a(α,β)∏γ

〈λ, αiγ 〉b(γ)∏ζ

αc(ζ)iζ

λη

such that the following statements are true.

(1) In each term of the sum,∑γ b(γ) + η = m.

(2) In each term of the sum,∑ζ c(ζ) + η = m− L.

(3) In each term of the sum,∑γ b(γ)−

∑ζ c(ζ) = L.

(4) In each term of the sum, |a(α, β)| ≤ mL and b(γ), c(ζ), η = 0, 1, . . . ,m.

(5) There are in total less than 3mL terms in the sum.

Note that since each BGG-Demazure operator ∆αijin δ = ∆αi1

· · ·∆αiLlowers the degree of polynomials

by at least 1, δ lowers the degree by at least L. Thus

(7.26) δ(λm) = 0, for m < L.

Example 7.16. We specialize the discussion to the case M = SU(2). Recall that a∗ = Rw where w is the

fundamental weight, and Φ = {±α} with α = 2w. P (a) consists of polynomials in the variable λ ∈ R1

∼=7→

Rww

.

For λ ∈ R1

∼=7→

Rww

, and f ∈ P (a), we have

(δf)(λ) =f(λ)− f(−λ)

2λ,

δ(λm) =

{λm−1, m odd,

0, m even,

ddet(λ) = 2λ.(7.27)

We can now finish the proof of (7.23).

Proof of Lemma 7.10. Recall that it suffices to prove (7.23) replacing χµ(λ,H) by χµ1 (λ,H) in (7.24). Using

power series expansions, write∑s∈W

(det s)ei〈λ+µ,H〉 =∑s∈W

det s

∞∑m=0

1

m!(i〈s(λ+ µ), H〉)m

=

∞∑m=0

im

m!

∑s∈W

det s 〈s(λ+ µ), H〉m.(7.28)

30 Y. ZHANG

Note that

(7.29) fm(H) = fm(λ) = fm(λ,H) :=∑s∈W

det s 〈s(λ+ µ), H〉m

is an anti-invariant polynomial in H with respect to the Weyl group W , thus by Proposition 7.14,

fm(H) =ddet(H)

|W |· δfm(H) =

∏α∈P 〈α,H〉|W |

· δfm(H).

This implies that we can rewrite (7.24) as

χµ1(λ,H) =1

|W |

∞∑m=0

im

m!δfm(H).

Thus to prove (7.23), it suffices to prove that

∞∑m=0

1

m!|Di1 · · ·Dik (δfm(λ))| . NL−k,

for all k ∈ Z≥0, uniformly in |ni| . N , where λ = n1w1 + · · · + nrwr. Then by (7.29), it suffices to prove

that∞∑m=0

1

m!|Di1 · · ·Dik (δ [(s(λ+ µ))

m])| . NL−k, ∀s ∈W.

Without loss of generality, it suffices to show

(7.30)

∞∑m=0

1

m!|Di1 · · ·Dik (δ [(λ+ µ)m])| . NL−k.

Noting (7.26), it suffices to consider cases when m ≥ L. We apply Proposition 7.15 to write

δ((λ+ µ)m)(H)

=∑

θ,a(α,β),b(γ),c(ζ),η

(−1)θ∏α≤β

〈αiα , αiβ 〉a(α,β)∏γ

〈λ+ µ, αiγ 〉b(γ)∏ζ

〈αiζ , H〉c(ζ)〈λ+ µ,H〉η.(7.31)

First note that for λ = n1w1 + · · ·+ nrwr, |ni| . N , i = 1, · · · , r, we have

(7.32) 1 . |〈αi, αj〉| . 1, |〈λ+ µ, αi〉| . N,

and by the assumption |H| . N−1,

(7.33) |〈αi, H〉| . N−1, |〈λ+ µ,H〉| =

∣∣∣∣∣(

r∑i=1

ni〈wi, H〉

)+ 〈µ,H〉

∣∣∣∣∣ . 1.

These imply

|δ((λ+ µ)m)(H)| ≤∑

θ,a(α,β),b(γ),c(ζ),η

C∑α,β |a(α,β)|+

∑γ b(γ)+

∑ζ c(ζ)+ηN

∑γ c(γ)−

∑ζ c(ζ)(7.34)

for some constant C independent of m. Now we derive a similar estimate for Di (δ [(λ+ µ)m]) (H). By

(7.31),

Di (δ [(λ+ µ)m]) (H) =∑

θ,a(α,β),b(γ),c(ζ),η

(−1)θ∏α≤β

〈αiα , αiβ 〉a(α,β)∏ζ

〈αiζ , H〉c(ζ)

·Di

(∏γ

〈λ+ µ, αiγ 〉b(γ)〈λ+ µ,H〉η).(7.35)

STRICHARTZ ESTIMATES FOR THE SCHRODINGER FLOW ON COMPACT LIE GROUPS 31

For λ = n1w1 + · · ·+ nrwr, we compute

Di

(〈λ+ µ, αiγ 〉

)= 〈αi, αiγ 〉,

Di (〈λ+ µ,H〉) = 〈αi, H〉.

The above two formulas combined with (7.32), (7.33), and the Leibniz rule (7.2) for Di imply∣∣∣∣∣Di

(∏γ

〈λ+ µ, αiγ 〉b(γ)〈λ+ µ,H〉η)∣∣∣∣∣ ≤ C∑

γ b(γ)+ηN∑γ b(γ)−1.

This combined with (7.32), (7.33) and (7.35) implies

|Di (δ [(λ+ µ)m]) (H)| .∑

θ,a(α,β),b(γ),c(ζ),η

C∑α,β |a(α,β)|+

∑γ b(γ)+

∑ζ c(ζ)+ηN

∑γ b(γ)−

∑ζ c(ζ)−1.

Inductively, we have

|Di1 · · ·Dik (δ [(λ+ µ)m]) (H)| .∑

θ,a(α,β),b(γ),c(ζ),η

C∑α,β |a(α,β)|+

∑γ b(γ)+

∑ζ c(ζ)+ηN

∑γ b(γ)−

∑ζ c(ζ)−k,

for some constant C independent of m. This by Proposition 7.15 then implies

|Di1 · · ·Dik (δ [(λ+ µ)m]) (H)| ≤ 3mLCCmLNL−k ≤ CmNL−k

for some positive constant C independent of m. This estimate implies (7.30), noting that

(7.36)

∞∑m=0

Cm

m!. 1.

This finishes the proof. �

7.3.2. Approach 2: via Harish-Chandra’s Integral Formula. This very short approach expresses χµ1 (λ,H) as

an integral over the group G. We apply the Harish-Chandra’s integral formula (see [HC57]), which reads∑s∈W

det(s)e〈sλ,µ〉 =

∏α∈P 〈α, λ〉 ·

∏α∈P 〈α, µ〉∏

α∈P 〈α, ρ〉

∫G

e〈Adg(λ),µ〉 dg.

where λ, µ ∈ b∗C, and dg is the normalized Haar measure on G. Then we can rewrite χµ1 (λ,H) as

χµ1 (λ,H) =i|P |

∏α∈P 〈α, λ+ ρ〉∏α∈P 〈α, ρ〉

∫G

ei〈λ+ρ,Adg(H)〉 dg.

Note thati|P |

∏α∈P 〈α, λ+ ρ〉∏α∈P 〈α, ρ〉

is a polynomial in λ ∈ Λ of degree |P | = d−r2 . Also, as |H| . N−1, we have |Adg(H)| . N−1 uniformly in

g ∈ G, which implies that the integral

f(λ) =

∫G

ei〈λ+ρ,Adg(H)〉 dg

as a function in λ is a pseudo-polynomial of degree 0, uniformly in |H| . N−1. Then by the Leibniz rule,

χ′(λ,H) as a function of λ is a pseudo-polynomial of degree d−r2 , uniformly in |H| . N−1. This finishes the

proof of Lemma 7.10.

Remark 7.17. Note that Lemma 7.10 can be stated purely in terms of an integral root system without

mentioning the ambient compact Lie group, and it still holds true this way. It can be seen either by the

32 Y. ZHANG

approach via BGG-Demazure operators which is purely a root system theoretic argument, or by the fact that,

for any integral root system Φ, there associates to it a unique compact simply connected semisimple Lie

group equipped with this root system, thus the approach via Harish-Chandra’s integral formula still works,

even though the argument explicitly involves the group.

7.4. From the Weight Lattice to the Root Lattice. We say expH is a corner in the maximal torus

provided

‖ 1

2π〈α,H〉‖ = 0, for all α ∈ P.

In this section, we extend Corollary 7.11 to the scenarios when expH is within a distance of . N−1 from

some corner. That is, when

‖ 1

2π〈α,H〉‖ . N−1, for all α ∈ P.(7.37)

To this end, we rewrite the Schrodinger kernel KN (t, x) as a finite sum of exponential sums over the root

lattice:

KN (t, x) = C∑µ∈Λ/Γ

∑λ∈µ+Γ

e−it(|λ|2−|ρ|2)ϕ(

|λ|2 − |ρ|2

N2)

∏α∈P 〈α, λ〉∏α∈P 〈α, ρ〉

∑s∈W det (s)ei〈s(λ),H〉∑s∈W det (s)ei〈s(ρ),H〉

= C∑µ∈Λ/Γ

∑λ∈Γ

e−it(|λ+µ|2−|ρ|2)ϕ(|λ+ µ|2 − |ρ|2

N2)

∏α∈P 〈α, λ+ µ〉∏α∈P 〈α, ρ〉

∑s∈W det (s)ei〈s(λ+µ),H〉∑s∈W det (s)ei〈s(ρ),H〉

(7.38)

where C = eit|ρ|2

|W | .

Proposition 7.18. Let µ be an element in the weight lattice Λ and let

(7.39) KµN (t, x) =

∑λ∈Γ

e−it(|λ+µ|2−|ρ|2)ϕ(|λ+ µ|2 − |ρ|2

N2)

∏α∈P 〈α, λ+ µ〉∏α∈P 〈α, ρ〉

∑s∈W det (s)ei〈s(λ+µ),H〉∑s∈W det (s)ei〈s(ρ),H〉

where x is conjugate to expH. Then

|KµN (t, x)| . Nd

(√q(1 +N‖ t

2πD −aq ‖1/2))r

(7.40)

for t2πD ∈Ma,q, uniformly for ‖ 1

2π 〈α,H〉‖ . N−1 for all α ∈ P .

Using (7.38) and the finiteness of Λ/Γ, we have the following corollary.

Corollary 7.19. Inequality (6.1) holds for the case when ‖ 12π 〈α,H〉‖ . N

−1 for all α ∈ P .

To prove Proposition 7.18, we first prove a variant of Lemma 7.10.

Lemma 7.20. Let

χµ(λ,H) =

∑s∈W (det s)ei〈s(µ+λ),H〉

e−i〈ρ,H〉∏α∈P (ei〈α,H〉 − 1)

(7.41)

be defined as in Lemma 7.10. Assume in addition that µ ∈ Λ. Then χµ(λ,H) as a function in λ ∈ Γ is a

pseudo-polynomial of degree d−r2 , uniformly in H such that ‖ 1

2πα(H)‖ . N−1 for all α ∈ P .

Proof. For all H ∈ ib∗ such that ‖ 12π 〈α,H〉‖ . N−1 for all α ∈ P , by considering the dual basis of the

simple roots {α1, · · · , αr}, we can write

H = H1 +H2(7.42)

STRICHARTZ ESTIMATES FOR THE SCHRODINGER FLOW ON COMPACT LIE GROUPS 33

such that

| 1

2π〈αi, H1〉| = ‖

1

2π〈αi, H〉‖ . N−1, i = 1, · · · , r,(7.43)

and

(7.44) 〈αi, H2〉 ∈ 2πZ, i = 1, · · · , r.

This implies that expH2 is a corner and

|H1| . N−1.(7.45)

Then for λ ∈ Γ = Zα1 + · · ·+ Zαr,

χµ(λ,H) = χµ(λ,H1 +H2) =

∑s∈W det (s)ei〈s(λ+µ),H1〉ei〈s(µ),H2〉

e−i〈ρ,H1+H2〉∏α∈P

(ei〈α,H1〉 − 1

) .(7.46)

Note that, see Corollary 4.13.3 in [Var84], s(µ)−µ ∈ Γ for all µ ∈ Λ and s ∈W , which combined with (7.44)

implies

ei〈s(µ),H2〉 = ei〈µ,H2〉, for all µ ∈ Λ, s ∈W.

Then (7.46) becomes

χµ(λ,H) =ei〈µ,H2〉

e−i〈ρ,H2〉·∑s∈W det (s)ei〈s(λ+µ),H1〉

e−i〈ρ,H1〉∏α∈P

(ei〈α,H1〉 − 1

) = ei〈µ+ρ,H2〉 · χµ(λ,H1),(7.47)

which is a pseudo-polynomial in λ ∈ Γ of degree d−r2 uniformly in |H1| . N−1 by Lemma 7.10. �

Proof of Proposition 7.18. Since∏α∈P 〈α, λ+µ〉 is a polynomial, and thus also a pseudo-polynomial in λ of

degree |P | = d−r2 , and χµ(λ,H) is a pseudo-polynomial of degree d−r

2 uniformly in ‖ 12π 〈α,H〉‖ . N−1 for

all α ∈ P by the previous lemma,

f(λ) =

∏α∈P 〈α, λ+ µ〉∏α∈P 〈α, ρ〉

· χµ(λ,H)

is then a pseudo-polynomial of degree d− r uniformly in ‖ 12π 〈α,H〉‖ . N

−1 for all α ∈ P . Then the desired

result comes from a direct application of Lemma 7.4. �

Example 7.21. We specialize the discussion in this section to the case G = SU(2). Recall that Λ = Zw,

Γ = Zα with α = 2w, thus Λ/Γ ∼= {0, 1} · w. (7.38) specializes to

KN (t, θ) =eit

2

(K0N (t, θ) +K1

N (t, θ)),

where

K0N =

∑m=2k,k∈Z

e−itm2

ϕ(m2 − 1

N2)m · e

imθ − e−imθ

eiθ − e−iθ,

K1N =

∑m=2k+1,k∈Z

e−itm2

ϕ(m2 − 1

N2)m · e

imθ − e−imθ

eiθ − e−iθ,

34 Y. ZHANG

for θ ∈ R/2πZ. Condition (7.37) specializes to ‖ θπ‖ . N−1. Write θ = θ1 + θ2, where |θ1| . N−1, and

θ2 = 0, π. Then for m = 2k, k ∈ Z,

χm(θ) =1

e−iθ(ei2θ1 − 1)· (eimθ1 − e−imθ1)

=1

e−iθ(ei2θ1 − 1)·∞∑n=0

in

n!((mθ1)n − (−mθ1)n)

=θ1

e−iθ(ei2θ1 − 1)·∑n odd

in

n!(2θn−1

1 mn),(7.48)

and similarly for m = 2k + 1, k ∈ Z,

χm(θ) =eiθ2θ1

e−iθ(ei2θ1 − 1)·∑n odd

in

n!(2θn−1

1 mn).

Note that we are implicitly applying Proposition 7.14 so that

fn(θ1) := (mθ1)n − (−mθ1)n = θ1 · δfn =

{θ1 · 2θn−1

1 mn, n odd,

0, n even.

If |k| . N , then it is clear that

|DLχ2k| . N1−L, |DLχ2k+1| . N1−L, L ∈ Z≥0,

where D is the difference operator with respect to the variable k. These two inequalities will give the desired

estimates for K0N and K1

N respectively using the Weyl sum estimate Lemma 7.4 in 1 dimension.

7.5. Root Subsystems. To finish the proof of part (ii) of Theorem 6.2, considering Corollaries 7.8 and

7.19, it suffices to prove (6.1) in the scenarios when expH is away from all the corners by a distance of

& N−1 but stays close to some cell walls within a distance of . N−1. We will identify these other walls as

belonging to a root subsystem of the original root system Φ, and then we will decompose the character, the

weight lattice and thus the Schrodinger kernel according to this root subsystem.

7.5.1. Identifying Root Subsystems and Rewriting the Character. Given any H ∈ ib∗, let QH be the subset

of the set Φ of roots defined by

QH := {α ∈ Φ | ‖ 1

2π〈α,H〉‖ ≤ N−1}.

Thus

Φ \QH = {α ∈ Φ | ‖ 1

2π〈α,H〉‖ > N−1}.

Define

(7.49) ΦH := {α ∈ Φ | α lies in the Z-linear span of QH},

then ΦH ⊃ QH , and

(7.50) ‖ 1

2π〈α,H〉‖ . N−1, ∀α ∈ ΦH ,

with the implicit constant independent of H, and

(7.51) ‖ 1

2π〈α,H〉‖ > N−1, ∀α ∈ Φ \ ΦH .

Note that ΦH is Z-closed in Φ, that is, no element in Φ \ ΦH lies in the Z-linear span of ΦH .

Proposition 7.22. ΦH is an integral root system.

STRICHARTZ ESTIMATES FOR THE SCHRODINGER FLOW ON COMPACT LIE GROUPS 35

Proof. We check the requirements for an integral root system listed in (4.8). Parts (ii) and (iv) are automatic

from the fact that ΦH is a subset of Φ. Part (i) comes from the fact that ΦH is a Z-linear space. Part (iii)

follows from the fact that sαβ is a Z-linear combination of α and β, for all α, β ∈ ΦH , and the fact that ΦH

is a Z-linear space. �

Then we say that ΦH is a root subsystem of Φ.

Let WH be the Weyl group of ΦH . WH is generated by reflections sα for α ∈ ΦH and WH is a subgroup

of the Weyl group W of Φ. Let P be a positive system of roots of Φ and PH = P ∩ ΦH . Then PH is a

positive system of roots of ΦH . We rewrite the Weyl character as

χλ =

∑s∈W det s ei〈s(λ),H〉

e−i〈ρ,H〉∏α∈P (ei〈α,H〉 − 1)

=

1|WH |

∑sH∈WH

∑s∈W det(sHs) e

i〈(sHs)(λ),H〉

e−i〈ρ,H〉(∏

α∈P\PH (ei〈α,H〉 − 1))(∏

α∈PH (ei〈α,H〉 − 1))

=1

|WH |e−i〈ρ,H〉∏α∈P\PH (ei〈α,H〉 − 1)

∑s∈W

det s ·∑sH∈WH

det sH ei〈sH(s(λ)),H〉∏α∈PH (ei〈α,H〉 − 1)

= C(H)∑s∈W

det s ·∑sH∈WH

det sH ei〈sH(s(λ)),H〉∏α∈PH (ei〈α,H〉 − 1)

,(7.52)

where

C(H) :=1

|WH |e−i〈ρ,H〉∏α∈P\PH (ei〈α,H〉 − 1)

.(7.53)

Then by (7.51),

(7.54) |C(H)| . N |P\PH |.

Let VH be the R-linear span of ΦH in V = ib∗ and let H‖ be the orthogonal projection of H on VH . Let

H⊥ = H −H‖. Then H⊥ is orthogonal to VH and we have

χλ = C(H)∑s∈W

det s ·∑sH∈WH

det sH ei〈sH(s(λ)),H⊥+H‖〉∏α∈PH (ei〈α,H⊥+H‖〉 − 1)

= C(H)∑s∈W

det s ·∑sH∈WH

det sH ei〈s(λ),sH(H⊥)〉ei〈sH(s(λ)),H‖〉∏α∈PH (ei〈α,H‖〉 − 1)

.(7.55)

Note that since H⊥ is orthogonal to every root in ΦH , H⊥ is fixed by sα for any α ∈ ΦH , which in turn

implies that H⊥ is fixed by any sH ∈WH , that is, sH(H⊥) = H⊥. Then

χλ = C(H)∑s∈W

det s ·∑sH∈WH

det sH ei〈s(λ),H⊥〉ei〈sH(s(λ)),H‖〉∏α∈PH (ei〈α,H‖〉 − 1)

= C(H)∑s∈W

det s · ei〈s(λ),H⊥〉 ·∑sH∈WH

det sH ei〈sH(s(λ)),H‖〉∏α∈PH (ei〈α,H‖〉 − 1)

.(7.56)

Note that by the definition of H‖, we have

(7.57) ‖ 1

2π〈α,H‖〉‖ . N−1, ∀α ∈ ΦH .

36 Y. ZHANG

This means that expH‖ is a corner in the maximal torus of the compact semisimple Lie group associated to

the integral root system ΦH .

Using the above formula, we rewrite the Schrodinger kernel (7.11) as

KN (t, x) =C(H)eit|ρ|

2

(∏α∈P 〈α, ρ〉)|W |

∑s∈W

det s ·KN,s(t, x)(7.58)

where

KN,s(t, x) =∑λ∈Λ

ei〈s(λ),H⊥〉−it|λ|2ϕ(|λ|2 − |ρ|2

N2)

(∏α∈P〈α, λ〉

) ∑sH∈WH

det sH ei〈sH(s(λ)),H‖〉∏α∈PH (ei〈α,H‖〉 − 1)

.

Noting that for any s ∈ W , |s(λ)| = |λ|,∏α∈P 〈α, s(λ)〉 = det s

∏α∈P 〈α, λ〉 by Proposition 7.13, and

s(Λ) = Λ, we have

KN,s(t, x) = det s KN,1(t, x)

where 1 is the identity element in W . Then (7.58) becomes

KN (t, x) =C(H)eit|ρ|

2

(∏α∈P 〈α, ρ〉)

KN,1(t, x).(7.59)

Proposition 7.23. Recall that

(7.60) KN,1(t, x) =∑λ∈Λ

ei〈λ,H⊥〉−it|λ|2ϕ(

|λ|2 − |ρ|2

N2)

(∏α∈P〈α, λ〉

) ∑sH∈WH

det sH ei〈sH(λ),H‖〉∏α∈PH (ei〈α,H‖〉 − 1)

.

Then

|KN,1(t, x)| . Nd−|P\PH |(√q(1 +N | t

2πD −aq |1/2)

)r(7.61)

for t2πD ∈Ma,q, uniformly in x ∈ G.

Noting (7.54) and (7.59), the above proposition implies Part (ii) of Theorem 6.2.

Example 7.24. Figure 1 is an illustration of the decomposition of the maximal torus of SU(3) according to

the values of ‖ 12π 〈α,H〉‖, α ∈ Φ. Here P = {α1, α2, α3 = α1 + α2}. The three proper root subsystems of Φ

are {±αi}, i = 1, 2, 3. The association of ΦH to H is as follows.

H ∈ regions of color ⇔ ΦH = Φ,

H ∈ regions of color ⇔ ΦH = {±α1},

H ∈ regions of color ⇔ ΦH = {±α2},

H ∈ regions of color ⇔ ΦH = {±α3},

H ∈ regions of color ⇔ ΦH = ∅.

7.5.2. Decomposition of the Weight Lattice. To prove Proposition 7.23, we will make a decomposition of

the weight lattice Λ according to the root subsystem ΦH . First, we have the following lemma about root

subsystems. Let ProjU denote the orthogonal projection map from the ambient inner product space onto

the subspace U .

STRICHARTZ ESTIMATES FOR THE SCHRODINGER FLOW ON COMPACT LIE GROUPS 37

2π 2α1

〈α1,α1〉

2π 2α3

〈α3,α3〉

2π 2α2

〈α2,α2〉

HH‖

H⊥∼ N−1

Figure 1. Decomposition of the maximal torus of SU(3)according to the values of ‖ 1

2π 〈α,H〉‖, α ∈ ∆

Lemma 7.25. Let Φ be an integral root system in the space V with the associated weight lattice ΛΦ. Let Ψ be

a root subsystem of Φ. Then let ΓΨ and ΛΦ be the root lattice and weight lattice associated to Ψ respectively.

Let VΨ be the R-linear span of Ψ in V . Let ΥΨ be the image of the orthogonal projection of ΛΦ onto VΨ.

Then the following statements hold true.

(1) ΥΨ is a lattice and ΓΨ ⊂ ΥΨ ⊂ ΛΨ. In particular, the rank of ΥΨ equals the rank of ΓΨ as well as ΛΨ.

(2) Let the ranks of ΥΨ and ΛΦ be r and R respectively. Let {w1, · · · , wr} be a basis of ΥΨ. Pick any

{u1, · · · , ur} ⊂ ΛΦ such that ProjVΨ(ui) = wi, i = 1, · · · , r. Then we can extend {u1, · · · , ur} into a basis

38 Y. ZHANG

{u1, · · · , ur, ur+1, · · · , uR} of ΛΦ. Furthermore, we can pick {ur+1, · · · , uR} such that ProjVΨ(ui) = 0 for

i = r + 1, · · · , R.

Proof. (1) It’s clear that ΥΨ is a lattice. Let ΓΦ be the root lattice associated to Φ. Then ΓΨ ⊂ ΓΦ. Thus

ΓΨ = ProjVΨ(ΓΨ) ⊂ ProjVΨ

(ΓΦ) ⊂ ProjVΨ(ΛΦ) = ΥΨ.

On the other hand, for any µ ∈ ΛΦ, α ∈ ΓΨ, we have 〈ProjVΨ(µ), α〉 = 〈µ, α〉. This in particular implies

2〈ProjVΨ

(µ), α〉〈α, α〉

= 2〈µ, α〉〈α, α〉

∈ Z, for all µ ∈ ΛΦ, α ∈ ΓΨ.

This implies ProjVΨ(µ) ∈ ΛΨ for all µ ∈ ΛΦ, that is, ΥΨ = ProjVΨ

(ΛΦ) ⊂ ΛΨ.

(2) Let SΦ := Zu1 + · · · + Zur, then SΦ is a sublattice of ΛΦ of rank r. By the theory of modules over a

principal ideal domain, there exists a basis {u′1, · · · , u′R} of ΛΦ and positive integers d1|d2| · · · |dr such that

{d1u′1, · · · , dru′r} is a basis of SΦ. Then we must have d1 = d2 = · · · = dr = 1, since

Zd1ProjVΨ(u′1) + · · ·+ ZdrProjVΨ

(u′r) = ProjVΨ(SΦ)

= ProjVΨ(ΛΦ) ⊃ ZProjVΨ

(u′1) + · · ·+ ZProjVΨ(u′r)(7.62)

and that u′1, · · · , u′r are R-linear independent. Thus we have

SΦ = Zu1 + · · ·+ Zur = Zu′1 + · · ·+ Zu′r

and then {u1, · · · , ur, u′r+1, · · · , u′R} is also a basis of ΛΦ. Furthermore, by adding a Z-linear combination of

u1, · · · , ur to each of u′r+1, · · · , u′R, we can assume that ProjVΨ(u′i) = 0, for i = r + 1, · · · , R.

�

We apply the above lemma to the root subsystem ΦH of Φ. Let V = ib∗, VH be the R-linear span of ΦH

in V , ΓH be the root lattice for ΦH , and let

(7.63) ΥH := ProjVH (Λ).

Then by the above lemma, we have

(7.64) ΥH ⊃ ΓH .

Let rH be the rank of ΦH as well as of ΓH and ΥH , and let {w1, · · · , wrH} ⊂ ΥH such that

ΥH = Zw1 + · · ·+ ZwrH .

Pick {u1, · · · , urH} ⊂ Λ such that

ProjVH (ui) = wi, i = 1, · · · , rH .

Then by the above lemma, we can extend {u1, · · · , urH} into a basis {u1, · · · , ur} of Λ, such that

(7.65) ProjVH (ui) = 0, i = rH + 1, · · · , r,

with

Λ = Zu1 + · · ·+ Zur.

Set

Υ′H = Zu1 + · · ·+ ZurH ⊂ Λ,

STRICHARTZ ESTIMATES FOR THE SCHRODINGER FLOW ON COMPACT LIE GROUPS 39

then

ProjVH : Υ′H∼−→ ΥH .

Recalling (7.64), let Γ′H be the sublattice of Υ′H corresponding to ΓH ⊂ ΥH under this isomorphism. More

precisely, let {α1, · · · , αrH} be a simple system of roots for ΓH , then

ProjVH : Γ′H = Zα′1 + · · ·+ Zα′rH∼−→ ΓH = Zα1 + · · ·+ ZαrH , α′i 7→ αi, i = 1, · · · , rH ,(7.66)

and we have

(7.67) Υ′H/Γ′H∼= ΥH/ΓH , |Υ′H/Γ′H | = |ΥH/ΓH | <∞.

Decomposing the weight lattice as

Λ =⊔

µ∈Υ′H/Γ′H

(µ+ Γ′H + ZurH+1 + · · ·+ Zur) ,

we have

KN,1(t, x) =∑

µ∈Υ′H/Γ′H ,

λ′1=n1α′1+···+nrHα

′rH,

λ2=nrH+1urH+1+···+nrur

ei〈µ+λ′1+λ2,H⊥〉−it|µ+λ′1+λ2|2ϕ(

|µ+ λ′1 + λ2|2 − |ρ|2

N2)

·

(∏α∈P〈α, µ+ λ′1 + λ2〉

) ∑sH∈WH

det sH ei〈sH(µ+λ′1+λ2),H‖〉∏α∈PH (ei〈α,H‖〉 − 1)

.(7.68)

Note that (7.65) implies for λ2 = nrH+1urH+1 + · · ·+ nrur that

〈sH(λ2), H‖〉 = 〈λ2, sH(H‖)〉 = 0,

and (7.66) implies for λ′1 = n1α′1 + · · ·+ nrHα

′rH that

〈sH(λ′1), H‖〉 = 〈λ′1, sH(H‖)〉 = 〈λ1, sH(H‖)〉 = 〈sH(λ1), H‖〉

where λ1 = n1α1 + · · ·+ nrHαrH ∈ VH . Similarly, also note that

〈sH(µ), H‖〉 = 〈sH(µ‖), H‖〉, where µ‖ := ProjVH (µ).

Thus we write

KN,1(t, x) =∑

µ∈Υ′H/Γ′H

∑λ′1=n1α

′1+···+nrHα

′rH,

λ1=n1α1+···+nrHαrH ,λ2=nrH+1urH+1+···+nrur

ei〈µ+λ′1+λ2,H⊥〉−it|µ+λ′1+λ2|2ϕ(

|µ+ λ′1 + λ2|2 − |ρ|2

N2)

·

(∏α∈P〈α, µ+ λ′1 + λ2〉

) ∑sH∈WH

det sH ei〈sH(µ‖+λ1),H‖〉∏α∈PH (ei〈α,H‖〉 − 1)

.(7.69)

Remark 7.26. We have that in the above formula

χµ‖(λ1, H

‖) :=

∑sH∈WH

det sH ei〈sH(µ‖+λ1),H‖〉∏α∈PH (ei〈α,H‖〉 − 1)

is a character of the form (7.41). Also note that µ‖ ∈ ProjVH (Λ) lies in the weight lattice ΛH of ΦH by

Lemma 7.25.

Noting (7.67), Proposition 7.23 reduces to the following.

40 Y. ZHANG

Proposition 7.27. For µ ∈ Υ′H/Γ′H , let

KµN,1(t, x) : =

∑λ′1=n1α

′1+···+nrHα

′rH,

λ1=n1α1+···+nrHαrH ,λ2=nrH+1urH+1+···+nrur,

n1,··· ,nr∈Z

ei〈µ+λ′1+λ2,H⊥〉−it|µ+λ′1+λ2|2ϕ(

|µ+ λ′1 + λ2|2 − |ρ|2

N2)

·

(∏α∈P〈α, µ+ λ′1 + λ2〉

) ∑sH∈WH

det sH ei〈sH(µ‖+λ1),H‖〉∏α∈PH (ei〈α,H‖〉 − 1)

.(7.70)

Then

|KµN,1(t, x)| . Nd−|P\PH |(√

q(1 +N | t2πD −

aq |1/2)

)r(7.71)

for t2πD ∈Ma,q, uniformly in x ∈ G.

Proof. We apply Lemma 7.4 to the lattice Zα′1 + · · ·+ Zα′rH + ZurH+1 + · · ·+ Zur. Write

χµ‖(λ1, H

‖) =

∑sH∈WH

det sH ei〈sH(µ‖+λ1),H‖〉∏α∈PH (ei〈α,H‖〉 − 1)

Then it suffices to show that∣∣∣∣∣Di1 · · ·Dik

(∏α∈P〈α, µ+ λ′1 + λ2〉χµ

‖(λ1, H

‖)

)∣∣∣∣∣ . Nd−|P\PH |−r−k,

for 1 ≤ i1, · · · , ik ≤ r,

λ′1 = n1α′1 + · · ·+ nrHα

′rH ,

λ1 = n1α1 + · · ·+ nrHαrH ,

λ2 = nrH+1urH+1 + · · ·+ nrur,

uniformly in |ni| . N , i = 1, · · · , r. Since∏α∈P 〈α, µ+λ′1+λ2〉 is a polynomial and thus a pseudo-polynomial

of degree |P |, it suffices to show that∣∣∣Di1 · · ·Dik(χµ‖(λ1, H

‖))∣∣∣ . Nd−|P\PH |−r−|P |−k = N |PH |−k.(7.72)

Since χ(λ1) does not involve the variables nrH+1, · · · , nr, it suffices to prove (7.72) for 1 ≤ i1, · · · , ik ≤ rH

uniformly in |λ1| . N . But this follows by applying Lemma 7.20 to the root system ΦH , noting Remark

7.26. �

7.6. Lp Estimates. We prove in this section Lp(G) estimates of the Schrodinger kernel for p <∞. Though

we do not apply them to the proof of the main theorem, they encapsulate the essential ingredients in the

proof of the L∞(G)-estimate and are of independent interest.

Proposition 7.28. Let KN (t, x) be the Schrodinger kernel as in Proposition 6.1. Then for any p > 3, we

have

‖KN (t, ·)‖Lp(G) .Nd− dp

(√q(1 +N‖ t

2πD −aq ‖1/2))r

(7.73)

for t2πD ∈Ma,q.

STRICHARTZ ESTIMATES FOR THE SCHRODINGER FLOW ON COMPACT LIE GROUPS 41

Proof. As a linear combination of characters, the Schrodinger kernel KN (t, ·) is a central function. Then we

can apply to it the Weyl integration formula (4.16)

(7.74) ‖KN (t, ·)‖pLp(G) =1

|W |

∫B

|KN (t, b)|p|DP (b)|2 db

where B is the maximal torus with normalized Haar measure db. Recall that we can parametrize B = exp b

by H ∈ ib∗ ∼= b, and write

B ∼= ib∗/(2πZα∨1 + · · ·+ 2πZα∨r ) = [0, 2π)α∨1 + · · ·+ [0, 2π)α∨r ,(7.75)

where {α∨i = 2αi〈αi,αi〉 | i = 1, · · · , r} is the set of simple coroots associated to a system of simple roots

{αi | i = 1, · · · , r}.We have shown in Section 7.5 that each H ∈ ib∗ is associated to a root subsystem ΦH such that (7.50)

and (7.51) hold. Note that there are finitely many root subsystems of a given root system, thus B is covered

by finitely many subsets R of the form

R = {H ∈ B | ‖ 1

2π〈α,H〉‖ . N−1,∀α ∈ Ψ; ‖ 1

2π〈α,H〉‖ > N−1,∀α ∈ Φ \Ψ}(7.76)

where Ψ is a root subsystem of Φ. Thus to prove (7.73), using (7.74), it suffices to show∫R

|KN (t, expH)|p|Dp(expH)|2 dH .

(Nd

(√q(1 +N‖ t

2πD −aq ‖1/2))r

)pN−d.(7.77)

By (7.54), (7.59) and (7.61), we have

KN (t, expH) .1∏

α∈P\Q(ei〈α,H〉 − 1)· Nd−|P\Q|(√

q(1 +N | t2πD −

aq |1/2)

)rwhere P,Q are respectively the sets of positive roots of Φ and Ψ with P ⊃ Q. Recalling DP (expH) =∏α∈P (ei〈α,H〉 − 1), (7.77) is then reduced to

∫R

∣∣∣∣∣ 1∏α∈P\Q(ei〈α,H〉 − 1)

∣∣∣∣∣p−2

∣∣∣∣∣∣∏α∈Q

(ei〈α,H〉 − 1)

∣∣∣∣∣∣2

dH . Np|P\Q|−d.

Using

|ei〈α,H〉 − 1| . ‖ 1

2π〈α,H〉‖ . |ei〈α,H〉 − 1|,

it suffices to show ∫R

∣∣∣∣∣ 1∏α∈P\Q ‖

12π 〈α,H〉‖

∣∣∣∣∣p−2

∣∣∣∣∣∣∏α∈Q‖ 1

2π〈α,H〉‖

∣∣∣∣∣∣2

dH . Np|P\Q|−d.(7.78)

For each H ∈ B, we write

H = H ′ +H0

such that

‖ 1

2π〈α,H〉‖ = | 1

2π〈α,H ′〉|, 〈α,H0〉 ∈ 2πZ, ∀α ∈ P.

42 Y. ZHANG

We write

R ⊂⋃

H0∈B,〈α,H0〉∈2πZ,∀α∈P

R′ +H0(7.79)

where

R′ = {H ∈ B | | 1

2π〈α,H〉| . N−1,∀α ∈ Q; | 1

2π〈α,H〉| > N−1,∀α ∈ P \Q}.(7.80)

Note that 〈α, α∨i 〉 ∈ Z for all α ∈ P and i = 1, · · · , r due to the integrality of the root system, using (7.75),

we have that there are only finitely many H0 ∈ B such that 〈α,H0〉 ∈ 2πZ for all α ∈ P . Thus using (7.79),

(7.78) is further reduced to

∫R′

∣∣∣∣∣ 1∏α∈P\Q |

12π 〈α,H〉|

∣∣∣∣∣p−2

∣∣∣∣∣∣∏α∈Q| 1

2π〈α,H〉|

∣∣∣∣∣∣2

dH . Np|P\Q|−d.(7.81)

Now we reparametrize B ∼= [0, 2π)α∨1 + · · ·+ [0, 2π)α∨r by

H =

r∑i=1

tiwi, (t1, · · · , tr) ∈ D

where {wi, i = 1, · · · , r} are the fundamental weights such that 〈αi, wj〉 = δij|αi|2

2 , i, j = 1, · · · , r, and D is

a bounded domain in Rr. Then the normalized Haar measure dH equals

dH = Cdt1 · · · dtr

for some constant C. Let s ≤ r such that

{α1, · · · , αs} ⊂ P \Q,

{αs+1, · · · , αr} ⊂ Q.

Using (7.80), we estimate

∫R′

∣∣∣∣∣ 1∏α∈P\Q |

12π 〈α,H〉|

∣∣∣∣∣p−2

∣∣∣∣∣∣∏α∈Q| 1

2π〈α,H〉|

∣∣∣∣∣∣2

dH

.∫R′

1

|t1 · · · ts|p−2N (p−2)(|P\Q|−s)N−2|Q| dt1 · · · dtr(7.82)

. N (p−2)(|P\Q|−s)N−2|Q|∫|t1|,··· ,|ts|&N−1,

|ts+1|,··· ,|tr|.N−1

1

|t1 · · · ts|p−2dt1 · · · dtr.

If p > 3, the above is bounded by

. N (p−2)(|P\Q|−s)N−2|Q|Ns(p−3)−(r−s) = Np|P\Q|−d,

noting that 2|P \Q|+ 2|Q|+ r = 2|P |+ r = d. �

Remark 7.29. The requirement p > 3 is by no means optimal. The estimate in (7.82) may be improved to

lower the exponent p. We conjecture that (7.73) holds for all p > pr such that limr→∞ pr = 2, where r is the

rank of G.

STRICHARTZ ESTIMATES FOR THE SCHRODINGER FLOW ON COMPACT LIE GROUPS 43

Acknowledgments

I would like to thank my Ph.D. thesis advisors Prof. Monica Visan and Prof. Rowan Killip for their

constant guidance and support. I especially thank them for giving me all the freedom to choose problems

that suit my own interest. I would like to thank Prof. Raphael Rouquier and Prof. Terence Tao for sharing

their expertise on the BGG-Demazure operators and the Weyl type sums respectively. I would also like to

thank Jiayin Guo for pointing out several false conjectures of mine on root systems. My thanks also goes

to Prof. Guozhen Lu who encouraged me to publish the paper. I am thankful to the editors and referees

for their carefully reading the paper and their helpful suggestions on revising it. Last but not least, I am

thankful to Mengmeng for her company and encouragement.

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Department of Mathematics, University of Connecticut, Storrs, CT 06269

Email address: yunfeng.zhang@uconn.edu