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• ❛❜♦r ❋r❛♠❡ ❉❡❝♦♠♣♦s✐t✐♦♥s ♦❢ ❊✈♦❧✉t✐♦♥ ❖♣❡r❛t♦rs ❛♥❞

❆♣♣❧✐❝❛t✐♦♥s

▼✐❝❤❡❧❡ ❇❡rr❛

❯♥✐✈❡rs✐tà ❞❡❣❧✐ ❙t✉❞✐ ❞✐ ❚♦r✐♥♦ ❉✐♣❛rt✐♠❡♥t♦ ❞✐ ▼❛t❡♠❛t✐❝❛ ✏●✐✉s❡♣♣❡ P❡❛♥♦✑

❳❳❳■■■ ❈♦♥✈❡❣♥♦ ◆❛③✐♦♥❛❧❡ ❞✐ ❆♥❛❧✐s✐ ❆r♠♦♥✐❝❛ ❆❧❜❛✱ ✶✼✲✷✵ ●✐✉❣♥♦ ✷✵✶✸

❖✉t❧✐♥❡

• ●❡❧❢❛♥❞✲❙❤✐❧♦✈ ❙♣❛❝❡s
• ●❛❜♦r ❋r❛♠❡s
• ●❛❜♦r ❋r❛♠❡ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs
• ❍❡❛t ❊q✉❛t✐♦♥
• ●❡♥❡r❛❧✐③❡❞ ❍❡❛t ❊q✉❛t✐♦♥
• ❍❛r♠♦♥✐❝ ❘❡♣✉❧s♦r

✶ ✴ ✶✽

• ❡❧❢❛♥❞✲❙❤✐❧♦✇ ❙♣❛❝❡s

s, r ≥ 0✳ f ∈ Ss

r(Rd) ✐❢ f ∈ S(Rd) ❛♥❞ t❤❡r❡ ❡①✐st A, B > 0 s✉❝❤ t❤❛t

sup

x∈Rd |xα∂βf(x)| A|α|B|β|(α!)r(β!)s,

α, β, ∈ Zd

+.

❊q✉✐✈❛❧❡♥t❧② f ∈ Ss

r(Rd) ✐✛ t❤❡r❡ ❡①✐st h, k > 0 s✉❝❤ t❤❛t

feh|x|1/r∞ < ∞ ❛♥❞ ˆ fek|ω|1/s∞ < ∞. ✭✶✮ ❚❤❡ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ✐♥t❡r❝❤❛♥❣❡s t❤❡ ✐♥❞❡①❡s✿f ∈ Ss

r(Rd) ⇔ ˆ

f ∈ Sr

s(Rd).

■❢ 0 ≤ s1 ≤ s2 ❛♥❞ 0 ≤ r1 ≤ r2 t❤❡♥ Ss1

r1 (Rd) ⊆ Ss2 r2 (Rd).

S1/2

1/2(Rd) ✐s t❤❡ s♠❛❧❧❡st ♥♦♥ tr✐✈✐❛❧ ●❡❧❢❛♥❞✲❙❤✐❧♦✇ s♣❛❝❡✳

✷ ✴ ✶✽

• ❛❜♦r ❋r❛♠❡s

g ∈ S(Rd)\{0}✱ Λ := αZd × βZd, α, β > 0 ✳ G(g, α, β) :=

• gm,n = MnTmg, (m, n) ∈ Λ
• ,

✇❤❡r❡ Mng(x) = e2πinxg(x) ❛♥❞ Tmg(x) = g(x − m)✳ G(g, α, β) ✐s ❛ ❢r❛♠❡ ❢♦r L2(Rd) ✐✛ t❤❡r❡ ❡①✐st 0 < A ≤ B < +∞✿ Af2

2 ≤

• (m,n) ∈ Λ

|f, gm,n|2 ≤ Bf2

2,

∀f ∈ L2(Rd).

✸ ✴ ✶✽

■❢ G(g, α, β) ✐s ❛ ❢r❛♠❡✱ t❤❡♥ f =

• (m,n)∈Λ

f, gm,nγm,n =

• (m,n)∈Λ

f, γm,ngm,n, ∀ f ∈ L2(Rd), G(γ, α, β) ❞✉❛❧ ❢r❛♠❡✱ γ ❞✉❛❧ ✇✐♥❞♦✇✳

❚❤❡♦r❡♠ ✭❙❡✐♣✲❲❛❧❧sté♥ ✶✾✾✷✱●rö❝❤❡♥✐❣✱ ▲②✉❜❛rs❦✐✐ ✷✵✵✽✮

• ▲❡t g = e−π|x|2, x ∈ Rd✳ ❚❤❡♥ G(g, α, β) ✐s ❛ ❢r❛♠❡ ✐✛ αβ < 1✳
• ▲❡t G(g, α, β) ❜❡ ❛s ❛❜♦✈❡✳ ❚❤❡♥ t❤❡ ❞✉❛❧ ✇✐♥❞♦✇ γ s❛t✐s❢②✿

|γ(x)| + | γ(x)| ≤ Ce−π|x|2, ∀ x ∈ Rd. ◆♦t✐❝❡ t❤❛t γ ∈ S1/2

1/2✳ ❲❡ ✉s❡ ●❛✉ss✐❛♥ ●❛❜♦r ❢r❛♠❡s ✐♥ ♦✉r ♥✉♠❡r✐❝❛❧

❛♣♣❧✐❝❛t✐♦♥s ✇✐t❤ ❧❛tt✐❝❡ ♣❛r❛♠❡t❡rs α = 1, β = 1

2.

✹ ✴ ✶✽

• ❛❜♦r ❋r❛♠❡ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs

T ∈ L(L2(Rd))✱ G(g, α, β) ●❛❜♦r ❢r❛♠❡ ✇✐t❤ ❞✉❛❧ ✇✐♥❞♦✇ γ ❛♥❞ f ∈ L2(Rd)✳

• Tf = T

 

• (m,n)∈Λ

f, γm,ngm,n   =

• (m,n)∈Λ

f, γm,nTgm,n

• Tf =
• (m′,n′)∈Λ

Tf, gm′,n′γm′,n′ Tf =

• (m,n),(m′,n′)∈Λ

Tgm,n, gm′,n′

• Tm,n,m′,n′

f, γm,n

• c(f)m,n

γm′,n′, Tm,n,m′,n′ ✐s t❤❡ ●❛❜♦r ▼❛tr✐①

✺ ✴ ✶✽

❯s❡✿

• Tf = σ(D)f = F −1

σ ˆ f

• ❋♦✉r✐❡r ▼✉❧t✐♣❧✐❡r✳
• Tf = µ(A)f

▼❡t❛♣❧❡❝t✐❝ ❖♣❡r❛t♦r✳

❚❤❡♦r❡♠ ✭❈♦r❞❡r♦✱ ◆✐❝♦❧❛✱ ❘♦❞✐♥♦✱ ✷✵✶✷✮

▲❡t s ≥ 1

2, g ∈ Ss s(Rd) ❛♥❞ σ ∈ C∞(Rd)✳❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✿

• ❚❤❡ s②♠❜♦❧ σ s❛t✐s❢②❡s

|∂ασ(z)| C|α|(α!)s, ∀ z ∈ R2d, ∀ α ∈ Z2d

+

✭✷✮

• ❚❤❡r❡ ❡①✐sts ε > 0 s✉❝❤ t❤❛t

|σπ(λ)g, π(µ)g)| e−ε|λ−µ|

1 s ,

∀ λ, µ ∈ αZd × βZd ✭✸✮

✇❤❡r❡ π(λ)g = MnTng✱ ✇✐t❤ λ = (m, n) ❛♥❞ µ = (m′, n′)✳

✻ ✴ ✶✽

❍❡❛t ❊q✉❛t✐♦♥

❋♦r α > 0 t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠    ∂tu − α∆u = 0 u(0, x) = u0(x), t ∈ R, x ∈ Rd, ❤❛s s♦❧✉t✐♦♥ u(t, x) = σα(t, D)u0 =

• Rd e2πix·ωσα(t, ω)

u0(ω)dω, ✭✹✮ ✇✐t❤ σα(t, ω) = e−4π2αt|ω|2✳

❚❤❡♦r❡♠

▲❡t G(g, 1, 1

2) ●❛✉ss✐❛♥ ●❛❜♦r ❢r❛♠❡✱ t❤❡ ●❛❜♦r ▼❛tr✐① ❛ss♦❝✐❛t❡❞ t♦ ✭✹✮ ✐s

|σα(t, D)gm,n, gm′,n′| = (2 + 4παt)− d

2 e−π[n2+n′2+ 1 2+4παt((m−m′)2+(n+n′)2)]

✭✺✮

(m, n) ∈ Zd × 1

2 Zd.

✼ ✴ ✶✽

❈♦❡✣❝✐❡♥ts✬ ❉❡❝❛②✲ ❍❡❛t ❊q✉❛t✐♦♥

❈♦❡✣❝✐❡♥ts✬ ❉❡❝❛② ✲ ❍❡❛t ❊q✉❛t✐♦♥ ❢♦r

• σ(t, D)g0,0, gm′,n′
• ✱ ❞✐♠❡♥s✐♦♥ d = 2✳

✽ ✴ ✶✽

❈♦♥t♦✉r ♣❧♦t

❈♦♥t♦✉r ♣❧♦t ♦❢

• σ(t, D)g0,0, gm′,n′
• ✳

✾ ✴ ✶✽

• ❡♥❡r❛❧✐③❡❞ ❍❡❛t ❊q✉❛t✐♦♥

❋♦r k ∈ Z+✱ t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠    ∂tu(−∆)ku = 0, u(0, x) = u0(x), t ∈ R, x ∈ R, ❤❛s s♦❧✉t✐♦♥ u(t, x) = σk(t, D)u0 =

• Rd e2πix·ωσk(t, ω)

u0(ω)dω, ✭✻✮ ✇✐t❤ σk(t, ω) = e−t(2πω)2k.

❚❤❡♦r❡♠

▲❡t G(g, α, β) ❜❡ ❛ ●❛✉ss✐❛♥✳ ❚❤❡♥ |σ(t, D)gm,n, gm′,n′| ≤ Ct,ke−˜

εt,k2− 1

s |(m,n)−(m′,n′)| 1 s ,

✇✐t❤ s =

2k 2k−1✱ ˜

εk,t = 2k−1

4k 1 4πkt

• 1

2k−1 2− k 2k−1 ❛♥❞ Ct,k = |4πkt| k−1 2k−1 ✳

❈♦♥s✐st❡♥t ✇✐t❤ ✭✸✮✳

✶✵ ✴ ✶✽

❖❜s❡r✈❡ |σk(t, D)gm,n, gm′,n′| ≤ e−π(n2−n′2)

• F
• e−t(2πω)2k

θm,n,m′,n′

• ∗
• F
• e−2πω2

θm,n,m′,n′

• ,

✇❤❡r❡ θm,n,m′,n′ := (m − m′) + i(n + n′)✳ ❲❡ ❤❛✈❡ t♦ ✜♥❞ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ❢♦r

• F
• e−t(2πω)2k
• .

■t ✐s ♥♦t ♣♦ss✐❜❧❡ t♦ ❝♦♠♣✉t❡ t❤❡ ●❛❜♦r ▼❛tr✐① ❡①♣❧✐❝✐t❧② s✐♥❝❡ t❤❡ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ♦❢ σ(t, ω) = e−t(2πω)2k ❝❛♥♥♦t ❜❡ ❝❛❧❝✉❧❛t❡❞ ❞✐r❡❝t❧②✳

✶✶ ✴ ✶✽

❆s②♠♣t♦t✐❝ ■♥t❡❣r❛t✐♦♥

❯s✐♥❣ t❤❡ ❆s②♠♣t♦t✐❝ ✐♥t❡❣r❛t✐♦♥✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ✜♥❞ ❛♥ ✉♣♣❡r ❜♦✉♥❞✦

❚❤❡♦r❡♠

▲❡t f(x) = e−αx2k✱ ✇✐t❤ α > 0 ❛♥❞ k ≥ 1✳ ❚❤❡♥ ˆ f(ω) s❛t✐s✜❡s✿ | ˆ f(ω)| ≤ Ck,αe−εk,αω

2k 2k−1 ,

✇❤❡r❡ Ck,α =

|2kα|

k−1 2k−1

(2π)

2k(k−1) 2k−1

❛♥❞ εk,α = (2π)

2k 2k−1 2k−1

2k 1 2kα

• 1

2k−1 ✳

◆♦t✐❝❡ t❤❛t σk(x, t) = e−t(2πx)2k ❢✉❧✜❧❧s ❝♦♥❞✐t✐♦♥ ✭✶✮ ✇❤✐t s = 2k−1

2k , r = 1 2k.

❚❤✉s σ ∈ Ss

r✳ ■♥ ♣❛rt✐❝✉❧❛r σ ❢✉❧❧✜❧❧s ✭✷✮✳

✶✷ ✴ ✶✽

❈♦❡✣❝✐❡♥ts✬ ❉❡❝❛②✲ ●❡♥❡r❛❧✐③❡❞ ❍❡❛t ❊q✉❛t✐♦♥

❈♦❡✣❝✐❡♥ts✬ ❉❡❝❛② ✲ ●❡♥❡r❛❧✐③❡❞ ❍❡❛t ❊q✉❛t✐♦♥ ❢♦r

• σ2(t, D)g0,0, gm′,n′
• ✱

❞✐♠❡♥s✐♦♥ d = 1✳

✶✸ ✴ ✶✽

❍❛r♠♦♥✐❝ ❘❡♣✉❧s♦r

❚❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠      i∂tu −

1 4π ∆u + π|x|2 = 0

u(0, x) = u0(x), x ∈ Rd, ❤❛s s♦❧✉t✐♦♥✿ u(t, x) = Tu0(x) = 1 cosh(t)

• Rd e

πi tanh(t)x2+ 2πix·ω

cosh(t) −πi tanh(t)·ω2

• u0(ω)dω.

❚❤❡♦r❡♠

❚❤❡ ●❛❜♦r ▼❛tr✐① ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ♦♣❡r❛t♦r T ✐s ❣✐✈❡♥ ❜②✿ Tgm,n, gm′,n′ = Ce − π

2 [m2+n2+n′2+m′2+2 tanh(t)(mn−m′n′)−2(mm′−nn′)],

✇❤❡r❡ C =

eiψ (2 cosh(t))

d 2 ❛♥❞ |eiψ| = 1. ✶✹ ✴ ✶✽

❈♦❡✣❝✐❡♥ts✬ ❉❡❝❛②✲ ❍❛r♠♦♥✐❝ ❘❡♣✉❧s♦r

❈♦❡✣❝✐❡♥ts✬ ❉❡❝❛② ✲ ❍❛r♠♦♥✐❝ ❘❡♣✉❧s♦r

• Tg0,0, gm′,n′
• ✱ ❞✐♠❡♥s✐♦♥ d = 2✳

✶✺ ✴ ✶✽

❈♦♥t♦✉r ♣❧♦ts

❈♦♥t♦✉r ♣❧♦t ♦❢

• T(t, D)g0,0, gm′,n′
• ❛♥❞
• T(t, D)g1, 1

2 , gm′,n′

• ✳

✶✻ ✴ ✶✽

❲❤❛t✬s ♥❡①t

Tf =

• (m,n),(m′,n′)∈Λ

Tgm,n, gm′,n′

• Tm,n,m′,n′

f, γm,n

• c(f)m,n

γm′,n′

• ❈♦♥str✉❝t ❛❧❣♦r✐t❤♠s t♦ r❡♣r❡s❡♥t ❣❡♥❡r❛❧ s♦❧✉t✐♦♥s ♦❢ ❙❝❤rö❞✐♥❣❡r✲t②♣❡

❈❛✉❝❤② Pr♦❜❧❡♠s✳

• ❆♣♣❧② ●❛❜♦r ❞❡❝♦♠♣♦s✐t✐♦♥ t♦ ❛♥❛❧②③❡ ❋♦✉r✐❡r ■♥t❡❣r❛❧ ❖♣❡r❛t♦rs✳

✶✼ ✴ ✶✽

❘❡❢❡r❡♥❝❡s

❊✳ ❈♦r❞❡r♦✱ ❋✳ ◆✐❝♦❧❛✱ ❛♥❞ ▲✳ ❘♦❞✐♥♦✱ ●❛❜♦r r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ❡✈♦❧✉t✐♦♥ ♦♣❡r❛t♦rs✱ ❆r❳✐✈ ❡✲♣r✐♥ts ✭✷✵✶✷✮✳ ▼✳ ❙✳ P✳ ❊❛st❤❛♠✱ ❆s②♠♣t♦t✐❝ ❢♦r♠✉❧❛❡ ♦❢ ▲✐♦✉✈✐❧❧❡✲●r❡❡♥ t②♣❡ ❢♦r ❤✐❣❤❡r✲♦r❞❡r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ❏✳ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝✳ ✭✷✮ ✷✽ ✭✶✾✽✸✮✱ ♥♦✳ ✸✱ ✺✵✼✕✺✶✽✳ ▼❘ ✼✷✹✼✷✶ ✭✽✻❛✿✸✹✵✽✼✮ ❑❛r❧❤❡✐♥③ ●rö❝❤❡♥✐❣ ❛♥❞ ❨✉r✐✐ ▲②✉❜❛rs❦✐✐✱ ●❛❜♦r ❢r❛♠❡s ✇✐t❤ ❍❡r♠✐t❡ ❢✉♥❝t✐♦♥s✱ ❈✳ ❘✳ ▼❛t❤✳ ❆❝❛❞✳ ❙❝✐✳ P❛r✐s ✸✹✹ ✭✷✵✵✼✮✱ ♥♦✳ ✸✱ ✶✺✼✕✶✻✷✳ ▼❘ ✷✷✾✷✷✽✵ ✭✷✵✵✼❦✿✹✷✵✾✹✮ ■✳ ▼✳ ●❡❧❢❛♥❞ ❛♥❞ ●✳ ❊✳ ❙❤✐❧♦✈✱ ●❡♥❡r❛❧✐③❡❞ ❢✉♥❝t✐♦♥s✳ ❱♦❧✳ ✷✲✸✱ ❆❝❛❞❡♠✐❝ Pr❡ss✱ ◆❡✇ ❨♦r❦✱ ✶✾✻✼✳ ❉♦♥ ❇ ❍✐♥t♦♥✱ ❆s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ s♦❧✉t✐♦♥s ♦❢ ✭r② ✭♠✮✮✭❦✮±q②❂ ✵✱ ❏♦✉r♥❛❧ ♦❢ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✹ ✭✶✾✻✽✮✱ ✺✾✵✕✺✾✻✳ ❑r✐st✐❛♥ ❙❡✐♣ ❛♥❞ ❘♦❜❡rt ❲❛❧❧sté♥✱ ❉❡♥s✐t② t❤❡♦r❡♠s ❢♦r s❛♠♣❧✐♥❣ ❛♥❞ ✐♥t❡r♣♦❧❛t✐♦♥ ✐♥ t❤❡ ❇❛r❣♠❛♥♥✲❋♦❝❦ s♣❛❝❡✳ ■■✱ ❏✳ ❘❡✐♥❡ ❆♥❣❡✇✳ ▼❛t❤✳ ✹✷✾ ✭✶✾✾✷✮✱ ✶✵✼✕✶✶✸✳ ▼❘ ✶✶✼✸✶✶✽ ✭✾✸❣✿✹✻✵✷✻❜✮

✶✽ ✴ ✶✽