Wave and Klein-Gordon equations on certain locally ... · 1/22 The Legacy of Joseph Fourier after 250 years TSIMF, Sanya, 17-21, December, 2018 Wave and Klein-Gordon equations on

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The Legacy of Joseph Fourier after 250 yearsTSIMF, Sanya, 17-21, December, 2018

Wave and Klein-Gordon equationson certain locally symmetric spaces

Hong-Wei ZHANG (Université d’Orléans)

Supervised by:Jean-Philippe Anker (Université d’Orléans)Nicolas Burq (Université Paris-Saclay)

Hong-Wei ZHANG Sanya, 19, December, 2018

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Contents

1 Background : A class of locally symmetric spaces2 Main result : Strichartz type estimates3 Recall : Wave kernel on symmetric spaces4 Key : Dispersion estimate on locally symmetri spaces5 Conclusion

InspirationA. Fotiadis, N. Mandouvalos, M. Marias.Schrödinger equations on locally symmetric spacesMath. Ann. 371 :1351–1374, 2018


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Background

G/K Riemannian symmetric space of the noncompact typeG semisimple Lie group, connected, noncompact, with finitecenterK maximal compact subgroup of G

Classification for RankG/K = 1 :

G/K dimensionHn(R) SO(n, 1)/SO(n) n

Hn(C) SU(n, 1)/SU(n) 2nHn(H) Sp(n, 1)/Sp(n) 4nH2(O) F−20

4 /Spin(9) 16

Γ discrete torsion-free subgroup of G M = Γ\G/K locally symmetric space


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Assumptions

Half sum of positive roots

ρ =12

∑α∈Σ+

mαα,

Critical exponent of the Poincaré series

δ(Γ) = inf {s > 0 | P(s; x , y) < +∞}

where P(s; x , y) is the Poincaré series

P(s; x , y) =∑γ∈Γ

e−sd(x ,γy), s > 0, x , y ∈ G/K

Assumption

(H1) : δ(Γ) < ρ λ0 = ρ2 > 0 [Corlette, Leuzinger] Vol(M) = +∞


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Assumptions

Assumption

(H2) : Γ convex cocompact

Conv(ΛΓ) convex hull of the limit set ΛΓ

Γ\Conv(ΛΓ) compact Γ convex cocompact


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Assumptions

M = Γ\G/K

NoncompactInfinite volumeCusp-free


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PDE problems

F (u) power-like nonlinearity

Nonlinear Klein-Gordon equation{∂2t u(t, x)−∆xu(t, x) + cu(t, x) = F (u(t, x)),

u(0, x) = f (x), ∂t |t=0u(t, x) = g(x),

where c > −ρ2.

Nonlinear wave equation{∂2t u(t, x)−∆xu(t, x) = F (u(t, x)),

u(0, x) = f (x), ∂t |t=0u(t, x) = g(x).


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PDE problems

Inhomogeneous linear wave equation

(LW )

{∂2t u(t, x)−∆xu(t, x) = F (t, x),

u(0, x) = f (x), ∂t |t=0u(t, x) = g(x).

Solution

u(t, x) = (cos tDx) f (x) +sin tDx

Dxg(x)︸︷︷︸

homogeneous

+

∫ t

0

sin(t − s)Dx

DxF (s, x)ds︸︷︷︸

inhomogeneous

où D = (−∆)1/2.


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Strichartz type estimate

Denote by

‖u‖Lp(I ;H−σ,q(M)) :=

(∫I‖(−∆)−

σ2 u‖pLq(M) dt

) 1p

.

Strichartz type estimate on M (Z. 2018)

Let (p, q) and (p, q) be two admissible couples, and let

σ ≥ n + 12

(12− 1

q

)and σ ≥ n + 1

2

(12− 1

q

).

Then all solutions u to the Cauchy problem (LW) satisfy thefollowing Strichartz type estimate :

‖∇R×Mu‖Lp(I ;H−σ,q(M)) . ‖f ‖H1(M) + ‖g‖L2(M)

+ ‖F‖Lp′ (I ;Hσ,q′ (M)) .


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Strichartz type estimate

Example. Admissible couples for n ≥ 4

A couple (p, q) is called admissible if(

1p ,

1q

)belongs to the triangle{(

1p,1q

)∈(

0,12

)×(

0,12

) ∣∣∣ 1p≥ n − 1

2

(12− 1

q

)}⋃{(

0,12

),

(12,12− 1

n − 1

)}.

1p

1q

12

12 −

1n−1

0 12Rn

1p

1q

12

012

12 −

1n−1


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Kernel estimates on the symmetric space G/K

D−σx e itDx f (x) = f ∗ ωσt (x) =∫G/K ω

σt (d(x , y))f (y)dy

Inverse spherical Fourier transform formula

ωσt (r) = cst.

∫ +∞

0

dλ

|c(λ)|2{λ2 + ρ2}−σ2 e it

√λ2+ρ2

ϕλ(r)

Split up∫ +∞0

dλ|c(λ)|2 =

∫ 20

dλ|c(λ)|2χ0(λ) +

∫ +∞1

dλ|c(λ)|2χ∞(λ)

and ωσt = ωσ,0t + ωσ,∞t accordinglyωσ,∞t has a logarithmic singularity on the sphere r = t whenσ = n+1

2 , we consider the analytic family of operators

W σ,∞t :=

eσ2

Γ(n+12 − σ

)χ∞(Dx)D−σx e itDx ,

in the vertical strip 0 ≤ Reσ ≤ n+12 , and denote by ωσ,∞t the

kernel associated.Hong-Wei ZHANG Sanya, 19, December, 2018

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Kernel estimates on the symmetric space G/K

[Anker-Pierfelice 2014]

Estimate of ωσ,0t

|ωσ,0t (r)| .

{ϕ0(r), ∀t ∈ R, ∀r ≥ 0|t|−

32 (1 + r)ϕ0(r), ∀|t| ≥ 1, ∀ 0 ≤ r ≤ |t|2

Estimate of ωσ,∞t for Reσ = n+12

|ωσ,∞t (r)| .

{|t|−

n−12 e−ρr , ∀0 < |t| < 1, if n ≥ 3

|t|−N(1 + r)Nϕ0(r), ∀|t| ≥ 1, ∀N ∈ N

Ground spherical function

ϕ0(r) � (1 + r)e−ρr


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Wave kernel on M

Consider the series

ωσt (x , y) :=∑γ∈Γ

ωσt (y−1γx), ∀x , y ∈ G

where ωσt is the half-wave kernel on G/K .

Cartan decomposition (G = K (exp a+)K )

y−1γx = kγ(expHγ)k ′γ with Hγ ∈ a+ and kγ , k′γ ∈ K

bi-K -invariance of ωσt

|ωσt (x , y)| .∑γ∈Γ

|ωσ,0t (expHγ)|+∑γ∈Γ

|ωσ,∞t (expHγ)|

(H1) : δ(Γ) < ρ convergence of the series.


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Wave operator on M

Wave operator on G/K :

W σt f (x) = f ∗ ωσt (x) =

∫G/K

ωσt (y−1x)f (y)dy , f ∈ C∞c (G/K )

Wave operator on M = Γ\G/K :

W σt f (x) =

∫G/K

ωσt (y−1x)f (y)dy , f ∈ C∞c (Γ\G/K )

By summing over Γ, the wave operator on M is given by

W σt f (x) =

∫Γ\G/K

∑γ∈Γ

f (γy)ωσt (y−1γx)

dy

=

∫Mωσt (x , y)f (y)dy


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Key result

Dispersion estimate on M (Z. 2018)

For n ≥ 3, 2 < q < +∞ and σ ≥ (n + 1)(

12 −

1q

),∥∥∥W σ

t

∥∥∥Lq′ (M)→Lq(M)

.

|t|−(n−1)(

12−

1q

)if 0 < |t| < 1,

|t|−32 if |t| ≥ 1.

At the endpoint q = 2, t 7→ e itD is a one-parameter group ofunitary operators on L2(M)

In dimension n = 2, there is an additional logarithmic factor inthe small time bound, which becomes

|t|−(

12−

1q

)(1− log |t|)1− 2

q


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Sketch : L1 → L∞ estimate

Lemma. Uniform upper boundary of the Poincaré series (Z. 2018)

If Γ is convex cocompact, then there exists a constant C > 0 suchthat for all x , y ∈ G/K ,

P(s; x , y) =∑γ∈Γ

e−sd(x ,γy) ≤ CP(s; 0, 0),

where 0 = eK denotes the origin of G/K .

Denote by F a compact fundamental domain containing 0 forthe action of Γ on Conv(ΛΓ) This lemma holds for all x , y ∈ Conv(ΛΓ)

Denote by π⊥ the orthogonal projection from G/K toConv(ΛΓ), then for all x , y ∈ G/K , γ ∈ Γ,

π⊥(γx) = γπ⊥(x) and d(π⊥(x), π⊥(y)) ≤ d(x , y)

P(s; x , y) . P(s;π⊥(x), π⊥(y))


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Sketch : L2 → L2 estimate

Lemma. L2 Kunze-Stein phenomenon on M (Z. 2018)

Let ψ be a reasonable bi-K -invariant functions on G , e.g., in theSchwartz class. Then

‖. ∗ ψ‖L2(M)→L2(M) ≤∫G|ψ(x)|ϕ0(x)dx .

RemarkSimilar result holds on higher rank locally symmetric spaces underthe assumption (H1) : δ(Γ) < ρ.

Spherical functions in the noncompact case

ϕλ(x) =

∫Ke(iλ+ρ)A(kg)dk, λ ∈ a∗C,

where A(kg) is the unique a-component in the Iwasawadecomposition (G = N (exp a)K ) of kg .


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Direct integral decomposition

L2(Γ\G ) ∼=∫ ⊕GHπdν(π)

and

L2(Γ\G/K ) = L2(M) ∼=∫ ⊕GK

(Hπ)Kdν(π)

accordingly, whereGK is the spherical subdual of G , the unitary dual of G(Hπ)K = Ceπ is one-dimensional for every π ∈ GK

In the rank one case, GK consists ofthe unitary spherical principal series π±λ (λ ∈ R/± 1)the trivial representation π±iρ = 1the complementary series π±iλ (λ ∈ I )


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Remark

(H1) : δ(Γ) < ρ λ0 = ρ2

−∆ acts on (Hπ)K by multiplication by λ2 + ρ2

only tempered representations are involved |ϕλ| ≤ ϕ0

Right convolution by ψ ∈ S(K\G/K ) acts on (Hπλ)K bymultiplication by

Hψ(λ) =

∫Gψ(x)ϕλ(x)dx ,

‖. ∗ ψ‖L2(M)→L2(M) ≤ supλ∈R

∣∣∣∣∫Gψ(x)ϕλ(x)dx

∣∣∣∣ ≤ ∫G|ψ(x)|ϕ0(x)dx


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Comparaison with symmetric spaces

Admissible range for n ≥ 4

1p

1q

12

12 −

1n−1

0 12

Figure.1. On M = Γ\G/K

1p

1q

12

12 −

1n−1

0 12

Figure.2. On G/K

Lack of a stronger dispersive property∥∥∥W σt

∥∥∥Lq′ (M)→Lq(M)

. |t|−(n−1) max(

12−

1q, 12−

1q

), ∀ 0 < |t| < 1


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Questions

Handled the remaining range δ(Γ) ∈ [ρ, 2ρ] ?

Substitute for convex cocompactness in higher rank ?

Sharp kernel estimates on higher rank symmetric spaces ?


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Wave and Klein-Gordon equations on certain locally ... · 1/22 The Legacy of Joseph Fourier after 250 years TSIMF, Sanya, 17-21, December, 2018 Wave and Klein-Gordon equations on