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❙♠♦♦t❤❡rs ❢♦r ❡✣❝✐❡♥t ♠✉❧t✐❣r✐❞ ♠❡t❤♦❞s ✐♥ ■●❆

❈❧❡♠❡♥s ❍♦❢r❡✐t❤❡r✱ ❙t❡❢❛♥ ❚❛❦❛❝s✱ ❲❛❧t❡r ❩✉❧❡❤♥❡r ❉❉✷✸✱ ❏✉❧② ✷✵✶✺ s✉♣♣♦rt❡❞ ❜②

❚❤❡ ✇♦r❦ ✇❛s ❢✉♥❞❡❞ ❜② t❤❡ ❆✉str✐❛♥ ❙❝✐❡♥❝❡ ❋✉♥❞ ✭❋❲❋✮✿ ◆❋◆ ❙✶✶✼ ✭✜rst ❛♥❞ t❤✐r❞ ❛✉t❤♦r✮ ❛♥❞ ❏✸✸✻✷✲◆✷✺ ✭s❡❝♦♥❞ ❛✉t❤♦r✮✳

❖✉t❧✐♥❡

Pr❡❧✐♠✐♥❛r✐❡s ▼♦❞❡❧ ♣r♦❜❧❡♠ ■●❆ ❞✐s❝r❡t✐③❛t✐♦♥ ▼✉❧t✐❣r✐❞ ❛❧❣♦r✐t❤♠ ▼✉❧t✐❣r✐❞ ❢r❛♠❡✇♦r❦ ❇❛s✐s✲✐♥❞❡♣❡♥❞❡♥t s♠♦♦t❤❡r ❙♠♦♦t❤❡r ❢♦r ♦♥❡ ❞✐♠❡♥s✐♦♥ ❙♠♦♦t❤❡r ❢♦r t✇♦ ❞✐♠❡♥s✐♦♥s ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❛♥❞ ❖✉t❧♦♦❦

❖✉t❧✐♥❡

Pr❡❧✐♠✐♥❛r✐❡s ▼♦❞❡❧ ♣r♦❜❧❡♠ ■●❆ ❞✐s❝r❡t✐③❛t✐♦♥ ▼✉❧t✐❣r✐❞ ❛❧❣♦r✐t❤♠ ▼✉❧t✐❣r✐❞ ❢r❛♠❡✇♦r❦ ❇❛s✐s✲✐♥❞❡♣❡♥❞❡♥t s♠♦♦t❤❡r ❙♠♦♦t❤❡r ❢♦r ♦♥❡ ❞✐♠❡♥s✐♦♥ ❙♠♦♦t❤❡r ❢♦r t✇♦ ❞✐♠❡♥s✐♦♥s ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❛♥❞ ❖✉t❧♦♦❦

P♦✐ss♦♥ ♠♦❞❡❧ ♣r♦❜❧❡♠

❉♦♠❛✐♥ Ω ∈ R❞

• ✐✈❡♥ ❢✉♥❝t✐♦♥ ❢ ∈ ▲✷(Ω)

❋✐♥❞ ✉ ∈ ❍✶(Ω) s✉❝❤ t❤❛t −∆✉ = ❢ ✐♥ Ω ∂✉ ∂♥ = ✵ ♦♥ ∂Ω

❱❛r✐❛t✐♦♥❛❧ ❢♦r♠✉❧❛t✐♦♥

❉♦♠❛✐♥ Ω ∈ R❞

• ✐✈❡♥ ❢✉♥❝t✐♦♥ ❢ ∈ ▲✷(Ω)

❋✐♥❞ ✉ ∈ ❍✶(Ω) s✉❝❤ t❤❛t (✉, ˜ ✉)❍✶(Ω) = (❢ , ˜ ✉)▲✷(Ω) ❢♦r ❛❧❧ ˜ ✉ ∈ ❍✶(Ω)✳

❖✉t❧✐♥❡

Pr❡❧✐♠✐♥❛r✐❡s ▼♦❞❡❧ ♣r♦❜❧❡♠ ■●❆ ❞✐s❝r❡t✐③❛t✐♦♥ ▼✉❧t✐❣r✐❞ ❛❧❣♦r✐t❤♠ ▼✉❧t✐❣r✐❞ ❢r❛♠❡✇♦r❦ ❇❛s✐s✲✐♥❞❡♣❡♥❞❡♥t s♠♦♦t❤❡r ❙♠♦♦t❤❡r ❢♦r ♦♥❡ ❞✐♠❡♥s✐♦♥ ❙♠♦♦t❤❡r ❢♦r t✇♦ ❞✐♠❡♥s✐♦♥s ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❛♥❞ ❖✉t❧♦♦❦

❙♣❧✐♥❡ s♣❛❝❡s ✐♥ ♦♥❡ ❞✐♠❡♥s✐♦♥

◮ ▲❡t Ω = (✵, ✶)

✐s t❤❡ ✉♥✐❢♦r♠ s✉❜❞✐✈✐s✐♦♥ ♦❢ ✵ ✶ ✐♥ ♥ ♥✵✷ s✉❜✐♥t❡r✈❛❧s ❚✐ ♦❢ ❧❡♥❣t❤ ❤ ❤✵✷ ❢♦r ✵ ✶ ✷ ❙♣❧✐♥❡ s♣❛❝❡ ❙♣ ❦ ✐s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ s♣❧✐♥❡ ❢✉♥❝t✐♦♥s ✐♥ ❈ ❦ ✱ ✇❤✐❝❤ ❛r❡ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ♣ ♦♥ ❡❛❝❤ s✉❜✐♥t❡r✈❛❧ ✐♥ ✳ ▼❛①✐♠✉♠ s♠♦♦t❤♥❡ss✿ ❙♣ ❙♣ ♣

✶

❙t❛♥❞❛r❞ ❇✲s♣❧✐♥❡ ❜❛s✐s✿

✶ ♣

①

♠ ♣

①

❙♣❧✐♥❡ s♣❛❝❡s ✐♥ ♦♥❡ ❞✐♠❡♥s✐♦♥

◮ ▲❡t Ω = (✵, ✶) ◮ Mℓ ✐s t❤❡ ✉♥✐❢♦r♠ s✉❜❞✐✈✐s✐♦♥ ♦❢ Ω = (✵, ✶) ✐♥ ♥ℓ = ♥✵✷ℓ

s✉❜✐♥t❡r✈❛❧s ❚✐ ♦❢ ❧❡♥❣t❤ ❤ℓ := ❤✵✷−ℓ ❢♦r ℓ = ✵, ✶, ✷, . . . ❙♣❧✐♥❡ s♣❛❝❡ ❙♣ ❦ ✐s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ s♣❧✐♥❡ ❢✉♥❝t✐♦♥s ✐♥ ❈ ❦ ✱ ✇❤✐❝❤ ❛r❡ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ♣ ♦♥ ❡❛❝❤ s✉❜✐♥t❡r✈❛❧ ✐♥ ✳ ▼❛①✐♠✉♠ s♠♦♦t❤♥❡ss✿ ❙♣ ❙♣ ♣

✶

❙t❛♥❞❛r❞ ❇✲s♣❧✐♥❡ ❜❛s✐s✿

✶ ♣

①

♠ ♣

①

❙♣❧✐♥❡ s♣❛❝❡s ✐♥ ♦♥❡ ❞✐♠❡♥s✐♦♥

◮ ▲❡t Ω = (✵, ✶) ◮ Mℓ ✐s t❤❡ ✉♥✐❢♦r♠ s✉❜❞✐✈✐s✐♦♥ ♦❢ Ω = (✵, ✶) ✐♥ ♥ℓ = ♥✵✷ℓ

s✉❜✐♥t❡r✈❛❧s ❚✐ ♦❢ ❧❡♥❣t❤ ❤ℓ := ❤✵✷−ℓ ❢♦r ℓ = ✵, ✶, ✷, . . .

◮ ❙♣❧✐♥❡ s♣❛❝❡ ❙♣,❦,ℓ(Ω) ✐s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ s♣❧✐♥❡ ❢✉♥❝t✐♦♥s ✐♥

❈ ❦(Ω)✱ ✇❤✐❝❤ ❛r❡ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ♣ ♦♥ ❡❛❝❤ s✉❜✐♥t❡r✈❛❧ ✐♥ Mℓ✳ ▼❛①✐♠✉♠ s♠♦♦t❤♥❡ss✿ ❙♣ ❙♣ ♣

✶

❙t❛♥❞❛r❞ ❇✲s♣❧✐♥❡ ❜❛s✐s✿

✶ ♣

①

♠ ♣

①

❙♣❧✐♥❡ s♣❛❝❡s ✐♥ ♦♥❡ ❞✐♠❡♥s✐♦♥

◮ ▲❡t Ω = (✵, ✶) ◮ Mℓ ✐s t❤❡ ✉♥✐❢♦r♠ s✉❜❞✐✈✐s✐♦♥ ♦❢ Ω = (✵, ✶) ✐♥ ♥ℓ = ♥✵✷ℓ

s✉❜✐♥t❡r✈❛❧s ❚✐ ♦❢ ❧❡♥❣t❤ ❤ℓ := ❤✵✷−ℓ ❢♦r ℓ = ✵, ✶, ✷, . . .

◮ ❙♣❧✐♥❡ s♣❛❝❡ ❙♣,❦,ℓ(Ω) ✐s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ s♣❧✐♥❡ ❢✉♥❝t✐♦♥s ✐♥

❈ ❦(Ω)✱ ✇❤✐❝❤ ❛r❡ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ♣ ♦♥ ❡❛❝❤ s✉❜✐♥t❡r✈❛❧ ✐♥ Mℓ✳

◮ ▼❛①✐♠✉♠ s♠♦♦t❤♥❡ss✿ ❙♣,ℓ(Ω) := ❙♣,♣−✶,ℓ(Ω)

❙t❛♥❞❛r❞ ❇✲s♣❧✐♥❡ ❜❛s✐s✿

✶ ♣

①

♠ ♣

①

❙♣❧✐♥❡ s♣❛❝❡s ✐♥ ♦♥❡ ❞✐♠❡♥s✐♦♥

◮ ▲❡t Ω = (✵, ✶) ◮ Mℓ ✐s t❤❡ ✉♥✐❢♦r♠ s✉❜❞✐✈✐s✐♦♥ ♦❢ Ω = (✵, ✶) ✐♥ ♥ℓ = ♥✵✷ℓ

s✉❜✐♥t❡r✈❛❧s ❚✐ ♦❢ ❧❡♥❣t❤ ❤ℓ := ❤✵✷−ℓ ❢♦r ℓ = ✵, ✶, ✷, . . .

◮ ❙♣❧✐♥❡ s♣❛❝❡ ❙♣,❦,ℓ(Ω) ✐s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ s♣❧✐♥❡ ❢✉♥❝t✐♦♥s ✐♥

❈ ❦(Ω)✱ ✇❤✐❝❤ ❛r❡ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ♣ ♦♥ ❡❛❝❤ s✉❜✐♥t❡r✈❛❧ ✐♥ Mℓ✳

◮ ▼❛①✐♠✉♠ s♠♦♦t❤♥❡ss✿ ❙♣,ℓ(Ω) := ❙♣,♣−✶,ℓ(Ω) ◮ ❙t❛♥❞❛r❞ ❇✲s♣❧✐♥❡ ❜❛s✐s✿ φ(✶) ♣,ℓ(①), . . . , φ(♠ℓ) ♣,ℓ (①)

❙♣❧✐♥❡ s♣❛❝❡s ✐♥ t✇♦ ❛♥❞ ♠♦r❡ ❞✐♠❡♥s✐♦♥s

◮ ▲❡t Ω = (✵, ✶)❞

❚❡♥s♦r ♣r♦❞✉❝t ❇✲s♣❧✐♥❡s✿

✐ ♠ ❥ ♣

① ②

✐ ♣

①

❥ ♣

② ❋♦r ✵ ✶ ❞ ✇✐t❤ ❞ ✶✿ ❙♣ ❞❡♥♦t❡s t❡♥s♦r ♣r♦❞✉❝t s♣❧✐♥❡ s♣❛❝❡ ▼♦r❡ ❣❡♥❡r❛❧ ❞♦♠❛✐♥s✿ ❣❡♦♠❡tr② ♠❛♣♣✐♥❣ ❋♦r r❡❣✉❧❛r ❣❡♦♠❡tr② ♠❛♣♣✐♥❣s✿ ♠✉❧t✐❣r✐❞ ❢♦r ♣❛r❛♠❡t❡r ❞♦♠❛✐♥ ❝❛♥ ❜❡ ✉s❡❞ ❛s ♣r❡❝♦♥❞✐t✐♦♥❡r

❙♣❧✐♥❡ s♣❛❝❡s ✐♥ t✇♦ ❛♥❞ ♠♦r❡ ❞✐♠❡♥s✐♦♥s

◮ ▲❡t Ω = (✵, ✶)❞ ◮ ❚❡♥s♦r ♣r♦❞✉❝t ❇✲s♣❧✐♥❡s✿ ϕ(✐+♠ℓ❥) ♣,ℓ

(①, ②) = φ(✐)

♣,ℓ(①)φ(❥) ♣,ℓ(②)

❋♦r ✵ ✶ ❞ ✇✐t❤ ❞ ✶✿ ❙♣ ❞❡♥♦t❡s t❡♥s♦r ♣r♦❞✉❝t s♣❧✐♥❡ s♣❛❝❡ ▼♦r❡ ❣❡♥❡r❛❧ ❞♦♠❛✐♥s✿ ❣❡♦♠❡tr② ♠❛♣♣✐♥❣ ❋♦r r❡❣✉❧❛r ❣❡♦♠❡tr② ♠❛♣♣✐♥❣s✿ ♠✉❧t✐❣r✐❞ ❢♦r ♣❛r❛♠❡t❡r ❞♦♠❛✐♥ ❝❛♥ ❜❡ ✉s❡❞ ❛s ♣r❡❝♦♥❞✐t✐♦♥❡r

❙♣❧✐♥❡ s♣❛❝❡s ✐♥ t✇♦ ❛♥❞ ♠♦r❡ ❞✐♠❡♥s✐♦♥s

◮ ▲❡t Ω = (✵, ✶)❞ ◮ ❚❡♥s♦r ♣r♦❞✉❝t ❇✲s♣❧✐♥❡s✿ ϕ(✐+♠ℓ❥) ♣,ℓ

(①, ②) = φ(✐)

♣,ℓ(①)φ(❥) ♣,ℓ(②) ◮ ❋♦r Ω = (✵, ✶)❞ ✇✐t❤ ❞ > ✶✿ ❙♣,ℓ(Ω) ❞❡♥♦t❡s t❡♥s♦r ♣r♦❞✉❝t

s♣❧✐♥❡ s♣❛❝❡ ▼♦r❡ ❣❡♥❡r❛❧ ❞♦♠❛✐♥s✿ ❣❡♦♠❡tr② ♠❛♣♣✐♥❣ ❋♦r r❡❣✉❧❛r ❣❡♦♠❡tr② ♠❛♣♣✐♥❣s✿ ♠✉❧t✐❣r✐❞ ❢♦r ♣❛r❛♠❡t❡r ❞♦♠❛✐♥ ❝❛♥ ❜❡ ✉s❡❞ ❛s ♣r❡❝♦♥❞✐t✐♦♥❡r

❙♣❧✐♥❡ s♣❛❝❡s ✐♥ t✇♦ ❛♥❞ ♠♦r❡ ❞✐♠❡♥s✐♦♥s

◮ ▲❡t Ω = (✵, ✶)❞ ◮ ❚❡♥s♦r ♣r♦❞✉❝t ❇✲s♣❧✐♥❡s✿ ϕ(✐+♠ℓ❥) ♣,ℓ

(①, ②) = φ(✐)

♣,ℓ(①)φ(❥) ♣,ℓ(②) ◮ ❋♦r Ω = (✵, ✶)❞ ✇✐t❤ ❞ > ✶✿ ❙♣,ℓ(Ω) ❞❡♥♦t❡s t❡♥s♦r ♣r♦❞✉❝t

s♣❧✐♥❡ s♣❛❝❡

◮ ▼♦r❡ ❣❡♥❡r❛❧ ❞♦♠❛✐♥s✿ ❣❡♦♠❡tr② ♠❛♣♣✐♥❣

❋♦r r❡❣✉❧❛r ❣❡♦♠❡tr② ♠❛♣♣✐♥❣s✿ ♠✉❧t✐❣r✐❞ ❢♦r ♣❛r❛♠❡t❡r ❞♦♠❛✐♥ ❝❛♥ ❜❡ ✉s❡❞ ❛s ♣r❡❝♦♥❞✐t✐♦♥❡r

❙♣❧✐♥❡ s♣❛❝❡s ✐♥ t✇♦ ❛♥❞ ♠♦r❡ ❞✐♠❡♥s✐♦♥s

◮ ▲❡t Ω = (✵, ✶)❞ ◮ ❚❡♥s♦r ♣r♦❞✉❝t ❇✲s♣❧✐♥❡s✿ ϕ(✐+♠ℓ❥) ♣,ℓ

(①, ②) = φ(✐)

♣,ℓ(①)φ(❥) ♣,ℓ(②) ◮ ❋♦r Ω = (✵, ✶)❞ ✇✐t❤ ❞ > ✶✿ ❙♣,ℓ(Ω) ❞❡♥♦t❡s t❡♥s♦r ♣r♦❞✉❝t

s♣❧✐♥❡ s♣❛❝❡

◮ ▼♦r❡ ❣❡♥❡r❛❧ ❞♦♠❛✐♥s✿ ❣❡♦♠❡tr② ♠❛♣♣✐♥❣ ◮ ❋♦r r❡❣✉❧❛r ❣❡♦♠❡tr② ♠❛♣♣✐♥❣s✿ ♠✉❧t✐❣r✐❞ ❢♦r ♣❛r❛♠❡t❡r

❞♦♠❛✐♥ ❝❛♥ ❜❡ ✉s❡❞ ❛s ♣r❡❝♦♥❞✐t✐♦♥❡r

❉✐s❝r❡t✐③❛t✐♦♥

❱❛r✐❛t✐♦♥❛❧ ❢♦r♠✉❧❛t✐♦♥✿ ❋✐♥❞ ✉ℓ ∈ ❙♣,ℓ(Ω) s✉❝❤ t❤❛t (✉ℓ, ˜ ✉ℓ)❍✶(Ω) = (❢ , ˜ ✉ℓ)▲✷(Ω) ❢♦r ❛❧❧ ˜ ✉ℓ ∈ ❙♣,ℓ(Ω) ▼❛tr✐①✲✈❡❝t♦r ♥♦t❛t✐♦♥✿ ❑ ✉ ❢

❉✐s❝r❡t✐③❛t✐♦♥

❱❛r✐❛t✐♦♥❛❧ ❢♦r♠✉❧❛t✐♦♥✿ ❋✐♥❞ ✉ℓ ∈ ❙♣,ℓ(Ω) s✉❝❤ t❤❛t (✉ℓ, ˜ ✉ℓ)❍✶(Ω) = (❢ , ˜ ✉ℓ)▲✷(Ω) ❢♦r ❛❧❧ ˜ ✉ℓ ∈ ❙♣,ℓ(Ω) ▼❛tr✐①✲✈❡❝t♦r ♥♦t❛t✐♦♥✿ ❑ℓ✉ℓ = ❢ ℓ

❖✉t❧✐♥❡

Pr❡❧✐♠✐♥❛r✐❡s ▼♦❞❡❧ ♣r♦❜❧❡♠ ■●❆ ❞✐s❝r❡t✐③❛t✐♦♥ ▼✉❧t✐❣r✐❞ ❛❧❣♦r✐t❤♠ ▼✉❧t✐❣r✐❞ ❢r❛♠❡✇♦r❦ ❇❛s✐s✲✐♥❞❡♣❡♥❞❡♥t s♠♦♦t❤❡r ❙♠♦♦t❤❡r ❢♦r ♦♥❡ ❞✐♠❡♥s✐♦♥ ❙♠♦♦t❤❡r ❢♦r t✇♦ ❞✐♠❡♥s✐♦♥s ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❛♥❞ ❖✉t❧♦♦❦

▼✉❧t✐❣r✐❞ ♠❡t❤♦❞

❖♥❡ st❡♣ ♦❢ t❤❡ ♠✉❧t✐❣r✐❞ ♠❡t❤♦❞ ♦♥ ❣r✐❞ ❧❡✈❡❧ ❦ ❛♣♣❧✐❡❞ t♦ ✐t❡r❛t❡ ✉(✵,✵)

ℓ

= ✉(✵)

ℓ

❛♥❞ r✐❣❤t✲❤❛♥❞✲s✐❞❡ ❢ ℓ t♦ ♦❜t❛✐♥ ①(✶)

ℓ

✐s ❣✐✈❡♥ ❜②✿

◮ ❆♣♣❧② ν s♠♦♦t❤✐♥❣ st❡♣s

✉(✵,♠)

ℓ

= ✉(✵,♠−✶)

ℓ

+ τ ❑ −✶

ℓ

(❢ ℓ − ❑ℓ✉(✵,♠−✶)

ℓ

) ❢♦r ♠ = ✶, . . . , ν✳ ❆♣♣❧② ❝♦❛rs❡✲❣r✐❞ ❝♦rr❡❝t✐♦♥

❈♦♠♣✉t❡ ❞❡❢❡❝t ❛♥❞ r❡str✐❝t t♦ ❝♦❛rs❡r ❣r✐❞ ❙♦❧✈❡ ♣r♦❜❧❡♠ ♦♥ ❝♦❛rs❡r ❣r✐❞ Pr♦❧♦♥❣❛t❡ ❛♥❞ ❛❞❞ r❡s✉❧t

■❢ r❡❛❧✐③❡❞ ❡①❛❝t❧② ✭t✇♦✲❣r✐❞ ♠❡t❤♦❞✮✿ ✉ ✶ ✉ ✵ ■

✶❑ ✶ ✶■ ✶ ❢

❑❦✉ ✵ ❙t❛♥❞❛r❞ ❛r❣✉♠❡♥ts✿ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t✇♦✲❣r✐❞ ♠❡t❤♦❞ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♠✉❧t✐❣r✐❞ ♠❡t❤♦❞ ✭❲✲❝②❝❧❡✮

▼✉❧t✐❣r✐❞ ♠❡t❤♦❞

❖♥❡ st❡♣ ♦❢ t❤❡ ♠✉❧t✐❣r✐❞ ♠❡t❤♦❞ ♦♥ ❣r✐❞ ❧❡✈❡❧ ❦ ❛♣♣❧✐❡❞ t♦ ✐t❡r❛t❡ ✉(✵,✵)

ℓ

= ✉(✵)

ℓ

❛♥❞ r✐❣❤t✲❤❛♥❞✲s✐❞❡ ❢ ℓ t♦ ♦❜t❛✐♥ ①(✶)

ℓ

✐s ❣✐✈❡♥ ❜②✿

◮ ❆♣♣❧② ν s♠♦♦t❤✐♥❣ st❡♣s

✉(✵,♠)

ℓ

= ✉(✵,♠−✶)

ℓ

+ τ ❑ −✶

ℓ

(❢ ℓ − ❑ℓ✉(✵,♠−✶)

ℓ

) ❢♦r ♠ = ✶, . . . , ν✳

◮ ❆♣♣❧② ❝♦❛rs❡✲❣r✐❞ ❝♦rr❡❝t✐♦♥

◮ ❈♦♠♣✉t❡ ❞❡❢❡❝t ❛♥❞ r❡str✐❝t t♦ ❝♦❛rs❡r ❣r✐❞ ◮ ❙♦❧✈❡ ♣r♦❜❧❡♠ ♦♥ ❝♦❛rs❡r ❣r✐❞ ◮ Pr♦❧♦♥❣❛t❡ ❛♥❞ ❛❞❞ r❡s✉❧t

■❢ r❡❛❧✐③❡❞ ❡①❛❝t❧② ✭t✇♦✲❣r✐❞ ♠❡t❤♦❞✮✿ ✉ ✶ ✉ ✵ ■

✶❑ ✶ ✶■ ✶ ❢

❑❦✉ ✵ ❙t❛♥❞❛r❞ ❛r❣✉♠❡♥ts✿ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t✇♦✲❣r✐❞ ♠❡t❤♦❞ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♠✉❧t✐❣r✐❞ ♠❡t❤♦❞ ✭❲✲❝②❝❧❡✮

▼✉❧t✐❣r✐❞ ♠❡t❤♦❞

❖♥❡ st❡♣ ♦❢ t❤❡ ♠✉❧t✐❣r✐❞ ♠❡t❤♦❞ ♦♥ ❣r✐❞ ❧❡✈❡❧ ❦ ❛♣♣❧✐❡❞ t♦ ✐t❡r❛t❡ ✉(✵,✵)

ℓ

= ✉(✵)

ℓ

❛♥❞ r✐❣❤t✲❤❛♥❞✲s✐❞❡ ❢ ℓ t♦ ♦❜t❛✐♥ ①(✶)

ℓ

✐s ❣✐✈❡♥ ❜②✿

◮ ❆♣♣❧② ν s♠♦♦t❤✐♥❣ st❡♣s

✉(✵,♠)

ℓ

= ✉(✵,♠−✶)

ℓ

+ τ ❑ −✶

ℓ

(❢ ℓ − ❑ℓ✉(✵,♠−✶)

ℓ

) ❢♦r ♠ = ✶, . . . , ν✳

◮ ❆♣♣❧② ❝♦❛rs❡✲❣r✐❞ ❝♦rr❡❝t✐♦♥

◮ ❈♦♠♣✉t❡ ❞❡❢❡❝t ❛♥❞ r❡str✐❝t t♦ ❝♦❛rs❡r ❣r✐❞ ◮ ❙♦❧✈❡ ♣r♦❜❧❡♠ ♦♥ ❝♦❛rs❡r ❣r✐❞ ◮ Pr♦❧♦♥❣❛t❡ ❛♥❞ ❛❞❞ r❡s✉❧t

■❢ r❡❛❧✐③❡❞ ❡①❛❝t❧② ✭t✇♦✲❣r✐❞ ♠❡t❤♦❞✮✿ ✉(✶)

ℓ

= ✉(✵,ν)

ℓ

+ ■ ℓ

ℓ−✶❑ −✶ ℓ−✶■ ℓ−✶ ℓ

(❢ ℓ − ❑❦✉(✵,ν)

ℓ

)

◮ ❙t❛♥❞❛r❞ ❛r❣✉♠❡♥ts✿ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t✇♦✲❣r✐❞ ♠❡t❤♦❞ ⇒

❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♠✉❧t✐❣r✐❞ ♠❡t❤♦❞ ✭❲✲❝②❝❧❡✮

■♥t❡r❣r✐❞ tr❛♥s❢❡r

◮ ◆❡st❡❞ s♣❛❝❡s✿ ❙♣,ℓ−✶(Ω) ⊂ ❙♣,ℓ(Ω)

❚❤❡ ♣r♦❧♦♥❣❛t✐♦♥ ■

✶ ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣

❑♥♦t ✐♥s❡rt✐♦♥ ❛❧❣♦r✐t❤♠

❚❤❡ r❡str✐❝t✐♦♥ ✐s ✐ts tr❛♥s♣♦s❡✿ ■

✶

■

✶ ❚

■♥t❡r❣r✐❞ tr❛♥s❢❡r

◮ ◆❡st❡❞ s♣❛❝❡s✿ ❙♣,ℓ−✶(Ω) ⊂ ❙♣,ℓ(Ω) ◮ ❚❤❡ ♣r♦❧♦♥❣❛t✐♦♥ ■ ℓ ℓ−✶ ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣

◮ ❑♥♦t ✐♥s❡rt✐♦♥ ❛❧❣♦r✐t❤♠

❚❤❡ r❡str✐❝t✐♦♥ ✐s ✐ts tr❛♥s♣♦s❡✿ ■

✶

■

✶ ❚

■♥t❡r❣r✐❞ tr❛♥s❢❡r

◮ ◆❡st❡❞ s♣❛❝❡s✿ ❙♣,ℓ−✶(Ω) ⊂ ❙♣,ℓ(Ω) ◮ ❚❤❡ ♣r♦❧♦♥❣❛t✐♦♥ ■ ℓ ℓ−✶ ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣

◮ ❑♥♦t ✐♥s❡rt✐♦♥ ❛❧❣♦r✐t❤♠

◮ ❚❤❡ r❡str✐❝t✐♦♥ ✐s ✐ts tr❛♥s♣♦s❡✿ ■ ℓ−✶ ℓ

= (■ ℓ

ℓ−✶)❚

❍❛❝❦❜✉s❝❤✲❧✐❦❡ ❝♦♥✈❡r❣❡♥❝❡ ❛♥❛❧②s✐s

❈♦♥✈❡r❣❡♥❝❡ ♦❢ t✇♦✲❣r✐❞ ♠❡t❤♦❞ ✇✐t❤ r❛t❡ q

❈❆❈❙

✱ ✐✳❡✳✱ ✉∗

ℓ − ✉(✶) ℓ ▲ℓ ≤ q✉∗ ℓ − ✉(✵) ℓ ▲ℓ,

✐♥ ♠❛tr✐① ♥♦t❛t✐♦♥ ❚ ❙

▲

q✱ ✐s ❣✉❛r❛♥t❡❡❞ ❜② ▲✶ ✷❚ ❑

✶▲✶ ✷

❈❆ ✭❛♣♣r♦①✐♠❛t✐♦♥ ♣r♦♣❡rt②✮ ▲

✶ ✷❑ ❙ ▲ ✶ ✷

❈❙

✶

✭s♠♦♦t❤✐♥❣ ♣r♦♣❡rt②✮

❍❛❝❦❜✉s❝❤✲❧✐❦❡ ❝♦♥✈❡r❣❡♥❝❡ ❛♥❛❧②s✐s

❈♦♥✈❡r❣❡♥❝❡ ♦❢ t✇♦✲❣r✐❞ ♠❡t❤♦❞ ✇✐t❤ r❛t❡ q

❈❆❈❙

✱ ✐✳❡✳✱ ✉∗

ℓ − ✉(✶) ℓ ▲ℓ ≤ q✉∗ ℓ − ✉(✵) ℓ ▲ℓ,

✐♥ ♠❛tr✐① ♥♦t❛t✐♦♥ ❚ℓ❙ν

ℓ ▲ℓ ≤ q✱

✐s ❣✉❛r❛♥t❡❡❞ ❜② ▲✶ ✷❚ ❑

✶▲✶ ✷

❈❆ ✭❛♣♣r♦①✐♠❛t✐♦♥ ♣r♦♣❡rt②✮ ▲

✶ ✷❑ ❙ ▲ ✶ ✷

❈❙

✶

✭s♠♦♦t❤✐♥❣ ♣r♦♣❡rt②✮

❍❛❝❦❜✉s❝❤✲❧✐❦❡ ❝♦♥✈❡r❣❡♥❝❡ ❛♥❛❧②s✐s

❈♦♥✈❡r❣❡♥❝❡ ♦❢ t✇♦✲❣r✐❞ ♠❡t❤♦❞ ✇✐t❤ r❛t❡ q = ❈❆❈❙

ν

✱ ✐✳❡✳✱ ✉∗

ℓ − ✉(✶) ℓ ▲ℓ ≤ q✉∗ ℓ − ✉(✵) ℓ ▲ℓ,

✐♥ ♠❛tr✐① ♥♦t❛t✐♦♥ ❚ℓ❙ν

ℓ ▲ℓ ≤ q✱

✐s ❣✉❛r❛♥t❡❡❞ ❜② ▲✶/✷

ℓ

❚ℓ❑ −✶

ℓ

▲✶/✷

ℓ

≤ ❈❆ ✭❛♣♣r♦①✐♠❛t✐♦♥ ♣r♦♣❡rt②✮ ▲−✶/✷

ℓ

❑ℓ❙ν

ℓ ▲−✶/✷ ℓ

≤ ❈❙ν−✶ ✭s♠♦♦t❤✐♥❣ ♣r♦♣❡rt②✮

❖✉t❧✐♥❡

Pr❡❧✐♠✐♥❛r✐❡s ▼♦❞❡❧ ♣r♦❜❧❡♠ ■●❆ ❞✐s❝r❡t✐③❛t✐♦♥ ▼✉❧t✐❣r✐❞ ❛❧❣♦r✐t❤♠ ▼✉❧t✐❣r✐❞ ❢r❛♠❡✇♦r❦ ❇❛s✐s✲✐♥❞❡♣❡♥❞❡♥t s♠♦♦t❤❡r ❙♠♦♦t❤❡r ❢♦r ♦♥❡ ❞✐♠❡♥s✐♦♥ ❙♠♦♦t❤❡r ❢♦r t✇♦ ❞✐♠❡♥s✐♦♥s ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❛♥❞ ❖✉t❧♦♦❦

❇❛s✐s✲✭✐♥✮❞❡♣❡♥❞❡♥t s♠♦♦t❤❡r

◮ ❘✐❝❤❛r❞s♦♥ s♠♦♦t❤❡r✿

✉(✵,♠)

ℓ

= ✉(✵,♠−✶)

ℓ

+ τ ❑

✶

(❢ ℓ − ❑ℓ✉(✵,♠−✶)

ℓ

), ✐✳❡✳✱ ❑ℓ := ■

▲❡♠♠❛

❆ss✉♠❡ t❤❛t ❑ ▲ ❛♥❞ t❤❛t ✐s ❝❤♦s❡♥ s✉❝❤ t❤❛t ✵ ✶ ▲

✶ ✷❑ ▲ ✶ ✷

❚❤❡♥ t❤❡ s♠♦♦t❤✐♥❣ ♣r♦♣❡rt② ✐s s❛t✐s✜❡❞ ❢♦r ❈❙

✶✳

Pr♦♦❢✿ ❙t❛♥❞❛r❞ ❡✐❣❡♥✈❛❧✉❡ ❛♥❛❧②s✐s✳ ❈♦♥✈❡r❣❡♥❝❡ r❛t❡✿ q

❈❆❈❙ ❈❆

✶

❇❛s✐s✲✐♥❞❡♣❡♥❞❡♥t s♠♦♦t❤❡r

◮ ❘✐❝❤❛r❞s♦♥ s♠♦♦t❤❡r ✭✐♥ ❝♦♥t✐♥✉♦✉s ✉♥❞❡rst❛♥❞✐♥❣✮✿

✉(✵,♠)

ℓ

= ✉(✵,♠−✶)

ℓ

+ τ ❑ −✶

ℓ

(❢ ℓ − ❑ℓ✉(✵,♠−✶)

ℓ

), ✇❤❡r❡ ❑ℓ := ▲ℓ ✐s t❤❡ ❘✐❡s③ ✐s♦♠♦r♣❤✐s♠

▲❡♠♠❛

❆ss✉♠❡ t❤❛t ❑ ▲ ❛♥❞ t❤❛t ✐s ❝❤♦s❡♥ s✉❝❤ t❤❛t ✵ ✶ ▲

✶ ✷❑ ▲ ✶ ✷

❚❤❡♥ t❤❡ s♠♦♦t❤✐♥❣ ♣r♦♣❡rt② ✐s s❛t✐s✜❡❞ ❢♦r ❈❙

✶✳

Pr♦♦❢✿ ❙t❛♥❞❛r❞ ❡✐❣❡♥✈❛❧✉❡ ❛♥❛❧②s✐s✳ ❈♦♥✈❡r❣❡♥❝❡ r❛t❡✿ q

❈❆❈❙ ❈❆

✶

❇❛s✐s✲✐♥❞❡♣❡♥❞❡♥t s♠♦♦t❤❡r

◮ ❘✐❝❤❛r❞s♦♥ s♠♦♦t❤❡r ✭✐♥ ❝♦♥t✐♥✉♦✉s ✉♥❞❡rst❛♥❞✐♥❣✮✿

✉(✵,♠)

ℓ

= ✉(✵,♠−✶)

ℓ

+ τ ❑ −✶

ℓ

(❢ ℓ − ❑ℓ✉(✵,♠−✶)

ℓ

), ✇❤❡r❡ ❑ℓ := ▲ℓ ✐s t❤❡ ❘✐❡s③ ✐s♦♠♦r♣❤✐s♠

▲❡♠♠❛

❆ss✉♠❡ t❤❛t ❑ℓ := ▲ℓ ❛♥❞ t❤❛t τ ✐s ❝❤♦s❡♥ s✉❝❤ t❤❛t ✵ < τ ≤ ✶ ▲−✶/✷

ℓ

❑ℓ▲−✶/✷

ℓ

• .

❚❤❡♥ t❤❡ s♠♦♦t❤✐♥❣ ♣r♦♣❡rt② ✐s s❛t✐s✜❡❞ ❢♦r ❈❙ = τ −✶✳ Pr♦♦❢✿ ❙t❛♥❞❛r❞ ❡✐❣❡♥✈❛❧✉❡ ❛♥❛❧②s✐s✳ ❈♦♥✈❡r❣❡♥❝❡ r❛t❡✿ q

❈❆❈❙ ❈❆

✶

❇❛s✐s✲✐♥❞❡♣❡♥❞❡♥t s♠♦♦t❤❡r

◮ ❘✐❝❤❛r❞s♦♥ s♠♦♦t❤❡r ✭✐♥ ❝♦♥t✐♥✉♦✉s ✉♥❞❡rst❛♥❞✐♥❣✮✿

✉(✵,♠)

ℓ

= ✉(✵,♠−✶)

ℓ

+ τ ❑ −✶

ℓ

(❢ ℓ − ❑ℓ✉(✵,♠−✶)

ℓ

), ✇❤❡r❡ ❑ℓ := ▲ℓ ✐s t❤❡ ❘✐❡s③ ✐s♦♠♦r♣❤✐s♠

▲❡♠♠❛

❆ss✉♠❡ t❤❛t ❑ℓ := ▲ℓ ❛♥❞ t❤❛t τ ✐s ❝❤♦s❡♥ s✉❝❤ t❤❛t ✵ < τ ≤ ✶ ▲−✶/✷

ℓ

❑ℓ▲−✶/✷

ℓ

• .

❚❤❡♥ t❤❡ s♠♦♦t❤✐♥❣ ♣r♦♣❡rt② ✐s s❛t✐s✜❡❞ ❢♦r ❈❙ = τ −✶✳ Pr♦♦❢✿ ❙t❛♥❞❛r❞ ❡✐❣❡♥✈❛❧✉❡ ❛♥❛❧②s✐s✳ ❈♦♥✈❡r❣❡♥❝❡ r❛t❡✿ q = ❈❆❈❙

ν

= ❈❆τ −✶

ν

❖✉t❧✐♥❡

Pr❡❧✐♠✐♥❛r✐❡s ▼♦❞❡❧ ♣r♦❜❧❡♠ ■●❆ ❞✐s❝r❡t✐③❛t✐♦♥ ▼✉❧t✐❣r✐❞ ❛❧❣♦r✐t❤♠ ▼✉❧t✐❣r✐❞ ❢r❛♠❡✇♦r❦ ❇❛s✐s✲✐♥❞❡♣❡♥❞❡♥t s♠♦♦t❤❡r ❙♠♦♦t❤❡r ❢♦r ♦♥❡ ❞✐♠❡♥s✐♦♥ ❙♠♦♦t❤❡r ❢♦r t✇♦ ❞✐♠❡♥s✐♦♥s ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❛♥❞ ❖✉t❧♦♦❦

❇❛s✐s✲✐♥❞❡♣❡♥❞❡♥t s♠♦♦t❤❡r ❢♦r ♦♥❡ ❞✐♠❡♥s✐♦♥

◮ ❈❧❛ss✐❝❛❧ ❛♥❛❧②s✐s✿ ■♥ ▲✷(Ω)

▲ ❤

✷▼ ✐s t❤❡ ♣r♦♣❡r❧② s❝❛❧❡❞ ♠❛ss ♠❛tr✐①

Pr♦♦❢s ❢♦r st❛♥❞❛r❞ s♠♦♦t❤❡rs ✭❘✐❝❤❛r❞s♦♥✱ ❏❛❝♦❜✐✱

• ❛✉ss✲❙❡✐❞❡❧✮ ✉s❡ t❤❛t t❤❡ ♠❛ss ♠❛tr✐① ✐s s♣❡❝tr❛❧❧②

❡q✉✐✈❛❧❡♥t t♦ ✐ts ❞✐❛❣♦♥❛❧✳

❚❤✐s ✐s tr✉❡ ❢♦r ❇✲s♣❧✐♥❡s✱ ❜✉t ❞❡t❡r✐♦r❛t❡ ❢♦r ✐♥❝r❡❛s❡❞ ♣✳

❯s❡ ❑ ▲ ❤

✷▼ ✭♠❛ss✲❘✐❝❤❛r❞s♦♥ s♠♦♦t❤❡r✮

❇❛s✐s✲✐♥❞❡♣❡♥❞❡♥t s♠♦♦t❤❡r ❢♦r ♦♥❡ ❞✐♠❡♥s✐♦♥

◮ ❈❧❛ss✐❝❛❧ ❛♥❛❧②s✐s✿ ■♥ ▲✷(Ω) ◮ ▲ℓ = ❤−✷ ℓ ▼ℓ ✐s t❤❡ ♣r♦♣❡r❧② s❝❛❧❡❞ ♠❛ss ♠❛tr✐①

Pr♦♦❢s ❢♦r st❛♥❞❛r❞ s♠♦♦t❤❡rs ✭❘✐❝❤❛r❞s♦♥✱ ❏❛❝♦❜✐✱

• ❛✉ss✲❙❡✐❞❡❧✮ ✉s❡ t❤❛t t❤❡ ♠❛ss ♠❛tr✐① ✐s s♣❡❝tr❛❧❧②

❡q✉✐✈❛❧❡♥t t♦ ✐ts ❞✐❛❣♦♥❛❧✳

❚❤✐s ✐s tr✉❡ ❢♦r ❇✲s♣❧✐♥❡s✱ ❜✉t ❞❡t❡r✐♦r❛t❡ ❢♦r ✐♥❝r❡❛s❡❞ ♣✳

❯s❡ ❑ ▲ ❤

✷▼ ✭♠❛ss✲❘✐❝❤❛r❞s♦♥ s♠♦♦t❤❡r✮

❇❛s✐s✲✐♥❞❡♣❡♥❞❡♥t s♠♦♦t❤❡r ❢♦r ♦♥❡ ❞✐♠❡♥s✐♦♥

◮ ❈❧❛ss✐❝❛❧ ❛♥❛❧②s✐s✿ ■♥ ▲✷(Ω) ◮ ▲ℓ = ❤−✷ ℓ ▼ℓ ✐s t❤❡ ♣r♦♣❡r❧② s❝❛❧❡❞ ♠❛ss ♠❛tr✐① ◮ Pr♦♦❢s ❢♦r st❛♥❞❛r❞ s♠♦♦t❤❡rs ✭❘✐❝❤❛r❞s♦♥✱ ❏❛❝♦❜✐✱

• ❛✉ss✲❙❡✐❞❡❧✮ ✉s❡ t❤❛t t❤❡ ♠❛ss ♠❛tr✐① ✐s s♣❡❝tr❛❧❧②

❡q✉✐✈❛❧❡♥t t♦ ✐ts ❞✐❛❣♦♥❛❧✳

❚❤✐s ✐s tr✉❡ ❢♦r ❇✲s♣❧✐♥❡s✱ ❜✉t ❞❡t❡r✐♦r❛t❡ ❢♦r ✐♥❝r❡❛s❡❞ ♣✳

❯s❡ ❑ ▲ ❤

✷▼ ✭♠❛ss✲❘✐❝❤❛r❞s♦♥ s♠♦♦t❤❡r✮

❇❛s✐s✲✐♥❞❡♣❡♥❞❡♥t s♠♦♦t❤❡r ❢♦r ♦♥❡ ❞✐♠❡♥s✐♦♥

◮ ❈❧❛ss✐❝❛❧ ❛♥❛❧②s✐s✿ ■♥ ▲✷(Ω) ◮ ▲ℓ = ❤−✷ ℓ ▼ℓ ✐s t❤❡ ♣r♦♣❡r❧② s❝❛❧❡❞ ♠❛ss ♠❛tr✐① ◮ Pr♦♦❢s ❢♦r st❛♥❞❛r❞ s♠♦♦t❤❡rs ✭❘✐❝❤❛r❞s♦♥✱ ❏❛❝♦❜✐✱

• ❛✉ss✲❙❡✐❞❡❧✮ ✉s❡ t❤❛t t❤❡ ♠❛ss ♠❛tr✐① ✐s s♣❡❝tr❛❧❧②

❡q✉✐✈❛❧❡♥t t♦ ✐ts ❞✐❛❣♦♥❛❧✳

◮ ❚❤✐s ✐s tr✉❡ ❢♦r ❇✲s♣❧✐♥❡s✱ ❜✉t ❞❡t❡r✐♦r❛t❡ ❢♦r ✐♥❝r❡❛s❡❞ ♣✳

❯s❡ ❑ ▲ ❤

✷▼ ✭♠❛ss✲❘✐❝❤❛r❞s♦♥ s♠♦♦t❤❡r✮

❇❛s✐s✲✐♥❞❡♣❡♥❞❡♥t s♠♦♦t❤❡r ❢♦r ♦♥❡ ❞✐♠❡♥s✐♦♥

◮ ❈❧❛ss✐❝❛❧ ❛♥❛❧②s✐s✿ ■♥ ▲✷(Ω) ◮ ▲ℓ = ❤−✷ ℓ ▼ℓ ✐s t❤❡ ♣r♦♣❡r❧② s❝❛❧❡❞ ♠❛ss ♠❛tr✐① ◮ Pr♦♦❢s ❢♦r st❛♥❞❛r❞ s♠♦♦t❤❡rs ✭❘✐❝❤❛r❞s♦♥✱ ❏❛❝♦❜✐✱

• ❛✉ss✲❙❡✐❞❡❧✮ ✉s❡ t❤❛t t❤❡ ♠❛ss ♠❛tr✐① ✐s s♣❡❝tr❛❧❧②

❡q✉✐✈❛❧❡♥t t♦ ✐ts ❞✐❛❣♦♥❛❧✳

◮ ❚❤✐s ✐s tr✉❡ ❢♦r ❇✲s♣❧✐♥❡s✱ ❜✉t ❞❡t❡r✐♦r❛t❡ ❢♦r ✐♥❝r❡❛s❡❞ ♣✳

◮ ❯s❡

❑ℓ := ▲ℓ := ❤−✷

ℓ ▼ℓ ✭♠❛ss✲❘✐❝❤❛r❞s♦♥ s♠♦♦t❤❡r✮

▼❛ss✲❘✐❝❤❛r❞s♦♥ s♠♦♦t❤❡r

◮ ❯s❡

❑ℓ := ▲ℓ := ❤−✷

ℓ ▼ℓ

❈♦♥❞✐t✐♦♥ ❢♦r ❝❤♦✐❝❡ ♦❢ r❡❛❞s ❛s✿ s✉♣

✉

♠

✵

✉

❑

❤

✶ ✉ ▼

s✉♣

✉ ❙♣ ✵

✉

❍✶

❤

✶ ✉ ▲✷ ✶ ✷

❛♥❞ ❈❙

✶

◆❡❡❞✿ r♦❜✉st ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t② ❢♦r s♣❧✐♥❡ s♣❛❝❡ ▲♦❝❛❧ ❋♦✉r✐❡r ❛♥❛❧②s✐s s✉❣❣❡sts✿ ❙✉❝❤ ❛ r♦❜✉st ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t② ❤♦❧❞s ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❬❍♦❢r❡✐t❤❡r✱ ❩✉❧❡❤♥❡r ✷✵✶✹❪✿ ❚❤✐s ❛♣♣r♦❛❝❤ ❞♦❡s ♥♦t ✇♦r❦ ✇❡❧❧ ❈❤♦♦s❡ ✉♣ ①

✶ ♣

① ♠❛① ✵ ❤ ①

♣ ❛♥❞ ♦❜t❛✐♥✿

❙✉❝❤ ❛ r♦❜✉st ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t② ❞♦❡s ♥♦t ❤♦❧❞ s✉♣ ♣ ♣

✷

♣✷

▼❛ss✲❘✐❝❤❛r❞s♦♥ s♠♦♦t❤❡r

◮ ❯s❡

❑ℓ := ▲ℓ := ❤−✷

ℓ ▼ℓ ◮ ❈♦♥❞✐t✐♦♥ ❢♦r ❝❤♦✐❝❡ ♦❢ τ r❡❛❞s ❛s✿

s✉♣

✉ℓ∈R♠ℓ\{✵}

✉ℓ❑ℓ ❤−✶

ℓ ✉ℓ▼ℓ

= s✉♣

✉ℓ∈❙♣,ℓ(Ω)\{✵}

|✉ℓ|❍✶(Ω) ❤−✶

ℓ ✉ℓ▲✷(Ω)

≤ τ −✶/✷. ❛♥❞ ❈❙ = τ −✶ ◆❡❡❞✿ r♦❜✉st ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t② ❢♦r s♣❧✐♥❡ s♣❛❝❡ ▲♦❝❛❧ ❋♦✉r✐❡r ❛♥❛❧②s✐s s✉❣❣❡sts✿ ❙✉❝❤ ❛ r♦❜✉st ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t② ❤♦❧❞s ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❬❍♦❢r❡✐t❤❡r✱ ❩✉❧❡❤♥❡r ✷✵✶✹❪✿ ❚❤✐s ❛♣♣r♦❛❝❤ ❞♦❡s ♥♦t ✇♦r❦ ✇❡❧❧ ❈❤♦♦s❡ ✉♣ ①

✶ ♣

① ♠❛① ✵ ❤ ①

♣ ❛♥❞ ♦❜t❛✐♥✿

❙✉❝❤ ❛ r♦❜✉st ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t② ❞♦❡s ♥♦t ❤♦❧❞ s✉♣ ♣ ♣

✷

♣✷

▼❛ss✲❘✐❝❤❛r❞s♦♥ s♠♦♦t❤❡r

◮ ❯s❡

❑ℓ := ▲ℓ := ❤−✷

ℓ ▼ℓ ◮ ❈♦♥❞✐t✐♦♥ ❢♦r ❝❤♦✐❝❡ ♦❢ τ r❡❛❞s ❛s✿

s✉♣

✉ℓ∈R♠ℓ\{✵}

✉ℓ❑ℓ ❤−✶

ℓ ✉ℓ▼ℓ

= s✉♣

✉ℓ∈❙♣,ℓ(Ω)\{✵}

|✉ℓ|❍✶(Ω) ❤−✶

ℓ ✉ℓ▲✷(Ω)

≤ τ −✶/✷. ❛♥❞ ❈❙ = τ −✶

◮ ◆❡❡❞✿ r♦❜✉st ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t② ❢♦r s♣❧✐♥❡ s♣❛❝❡

▲♦❝❛❧ ❋♦✉r✐❡r ❛♥❛❧②s✐s s✉❣❣❡sts✿ ❙✉❝❤ ❛ r♦❜✉st ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t② ❤♦❧❞s ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❬❍♦❢r❡✐t❤❡r✱ ❩✉❧❡❤♥❡r ✷✵✶✹❪✿ ❚❤✐s ❛♣♣r♦❛❝❤ ❞♦❡s ♥♦t ✇♦r❦ ✇❡❧❧ ❈❤♦♦s❡ ✉♣ ①

✶ ♣

① ♠❛① ✵ ❤ ①

♣ ❛♥❞ ♦❜t❛✐♥✿

❙✉❝❤ ❛ r♦❜✉st ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t② ❞♦❡s ♥♦t ❤♦❧❞ s✉♣ ♣ ♣

✷

♣✷

▼❛ss✲❘✐❝❤❛r❞s♦♥ s♠♦♦t❤❡r

◮ ❯s❡

❑ℓ := ▲ℓ := ❤−✷

ℓ ▼ℓ ◮ ❈♦♥❞✐t✐♦♥ ❢♦r ❝❤♦✐❝❡ ♦❢ τ r❡❛❞s ❛s✿

s✉♣

✉ℓ∈R♠ℓ\{✵}

✉ℓ❑ℓ ❤−✶

ℓ ✉ℓ▼ℓ

= s✉♣

✉ℓ∈❙♣,ℓ(Ω)\{✵}

|✉ℓ|❍✶(Ω) ❤−✶

ℓ ✉ℓ▲✷(Ω)

≤ τ −✶/✷. ❛♥❞ ❈❙ = τ −✶

◮ ◆❡❡❞✿ r♦❜✉st ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t② ❢♦r s♣❧✐♥❡ s♣❛❝❡ ◮ ▲♦❝❛❧ ❋♦✉r✐❡r ❛♥❛❧②s✐s s✉❣❣❡sts✿

❙✉❝❤ ❛ r♦❜✉st ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t② ❤♦❧❞s ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❬❍♦❢r❡✐t❤❡r✱ ❩✉❧❡❤♥❡r ✷✵✶✹❪✿ ❚❤✐s ❛♣♣r♦❛❝❤ ❞♦❡s ♥♦t ✇♦r❦ ✇❡❧❧ ❈❤♦♦s❡ ✉♣ ①

✶ ♣

① ♠❛① ✵ ❤ ①

♣ ❛♥❞ ♦❜t❛✐♥✿

❙✉❝❤ ❛ r♦❜✉st ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t② ❞♦❡s ♥♦t ❤♦❧❞ s✉♣ ♣ ♣

✷

♣✷

▼❛ss✲❘✐❝❤❛r❞s♦♥ s♠♦♦t❤❡r

◮ ❯s❡

❑ℓ := ▲ℓ := ❤−✷

ℓ ▼ℓ ◮ ❈♦♥❞✐t✐♦♥ ❢♦r ❝❤♦✐❝❡ ♦❢ τ r❡❛❞s ❛s✿

s✉♣

✉ℓ∈R♠ℓ\{✵}

✉ℓ❑ℓ ❤−✶

ℓ ✉ℓ▼ℓ

= s✉♣

✉ℓ∈❙♣,ℓ(Ω)\{✵}

|✉ℓ|❍✶(Ω) ❤−✶

ℓ ✉ℓ▲✷(Ω)

≤ τ −✶/✷. ❛♥❞ ❈❙ = τ −✶

◮ ◆❡❡❞✿ r♦❜✉st ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t② ❢♦r s♣❧✐♥❡ s♣❛❝❡ ◮ ▲♦❝❛❧ ❋♦✉r✐❡r ❛♥❛❧②s✐s s✉❣❣❡sts✿

❙✉❝❤ ❛ r♦❜✉st ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t② ❤♦❧❞s

◮ ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❬❍♦❢r❡✐t❤❡r✱ ❩✉❧❡❤♥❡r ✷✵✶✹❪✿

❚❤✐s ❛♣♣r♦❛❝❤ ❞♦❡s ♥♦t ✇♦r❦ ✇❡❧❧ ❈❤♦♦s❡ ✉♣ ①

✶ ♣

① ♠❛① ✵ ❤ ①

♣ ❛♥❞ ♦❜t❛✐♥✿

❙✉❝❤ ❛ r♦❜✉st ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t② ❞♦❡s ♥♦t ❤♦❧❞ s✉♣ ♣ ♣

✷

♣✷

▼❛ss✲❘✐❝❤❛r❞s♦♥ s♠♦♦t❤❡r

◮ ❯s❡

❑ℓ := ▲ℓ := ❤−✷

ℓ ▼ℓ ◮ ❈♦♥❞✐t✐♦♥ ❢♦r ❝❤♦✐❝❡ ♦❢ τ r❡❛❞s ❛s✿

s✉♣

✉ℓ∈R♠ℓ\{✵}

✉ℓ❑ℓ ❤−✶

ℓ ✉ℓ▼ℓ

= s✉♣

✉ℓ∈❙♣,ℓ(Ω)\{✵}

|✉ℓ|❍✶(Ω) ❤−✶

ℓ ✉ℓ▲✷(Ω)

≤ τ −✶/✷. ❛♥❞ ❈❙ = τ −✶

◮ ◆❡❡❞✿ r♦❜✉st ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t② ❢♦r s♣❧✐♥❡ s♣❛❝❡ ◮ ▲♦❝❛❧ ❋♦✉r✐❡r ❛♥❛❧②s✐s s✉❣❣❡sts✿

❙✉❝❤ ❛ r♦❜✉st ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t② ❤♦❧❞s

◮ ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❬❍♦❢r❡✐t❤❡r✱ ❩✉❧❡❤♥❡r ✷✵✶✹❪✿

❚❤✐s ❛♣♣r♦❛❝❤ ❞♦❡s ♥♦t ✇♦r❦ ✇❡❧❧

◮ ❈❤♦♦s❡ ✉♣,ℓ(①) := φ(✶) ♣,ℓ(①) = ♠❛①{✵, ❤ℓ − ①}♣ ❛♥❞ ♦❜t❛✐♥✿

❙✉❝❤ ❛ r♦❜✉st ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t② ❞♦❡s ♥♦t ❤♦❧❞ s✉♣ ♣ ♣

✷

♣✷

▼❛ss✲❘✐❝❤❛r❞s♦♥ s♠♦♦t❤❡r

◮ ❯s❡

❑ℓ := ▲ℓ := ❤−✷

ℓ ▼ℓ ◮ ❈♦♥❞✐t✐♦♥ ❢♦r ❝❤♦✐❝❡ ♦❢ τ r❡❛❞s ❛s✿

s✉♣

✉ℓ∈R♠ℓ\{✵}

✉ℓ❑ℓ ❤−✶

ℓ ✉ℓ▼ℓ

= s✉♣

✉ℓ∈❙♣,ℓ(Ω)\{✵}

|✉ℓ|❍✶(Ω) ❤−✶

ℓ ✉ℓ▲✷(Ω)

≤ τ −✶/✷. ❛♥❞ ❈❙ = τ −✶

◮ ◆❡❡❞✿ r♦❜✉st ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t② ❢♦r s♣❧✐♥❡ s♣❛❝❡ ◮ ▲♦❝❛❧ ❋♦✉r✐❡r ❛♥❛❧②s✐s s✉❣❣❡sts✿

❙✉❝❤ ❛ r♦❜✉st ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t② ❤♦❧❞s

◮ ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❬❍♦❢r❡✐t❤❡r✱ ❩✉❧❡❤♥❡r ✷✵✶✹❪✿

❚❤✐s ❛♣♣r♦❛❝❤ ❞♦❡s ♥♦t ✇♦r❦ ✇❡❧❧

◮ ❈❤♦♦s❡ ✉♣,ℓ(①) := φ(✶) ♣,ℓ(①) = ♠❛①{✵, ❤ℓ − ①}♣ ❛♥❞ ♦❜t❛✐♥✿

❙✉❝❤ ❛ r♦❜✉st ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t② ❞♦❡s ♥♦t ❤♦❧❞ ⇒ s✉♣ ... ≥ ♣ ⇒ τ ≤ ♣−✷ ⇒ ν ≥ ♣✷

■♥✈❡rs❡ ✐♥❡q✉❛❧✐t②

❚❤❡♦r❡♠ ✭❬❚✳✱ ❚❛❦❛❝s ✷✵✶✺❪✮

❋♦r ❛❧❧ ℓ ∈ N✵ ❛♥❞ ♣ ∈ N✱ |✉ℓ|❍✶ ≤ ✷ √ ✸❤−✶

ℓ ✉ℓ▲✷

✐s s❛t✐s✜❡❞ ❢♦r ❛❧❧ ✉ℓ ∈ ❙♣,ℓ(Ω)✱ ✇❤❡r❡ ❙♣,ℓ(✵, ✶) ✐s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ✉ℓ ∈ ❙♣,ℓ(✵, ✶) ✇❤♦s❡ ♦❞❞ ❞❡r✐✈❛t✐✈❡s ✈❛♥✐s❤ ❛t t❤❡ ❜♦✉♥❞❛r②✿ ∂✷❧+✶ ∂①✷❧+✶ ✉ℓ(✵) = ∂✷❧+✶ ∂①✷❧+✶ ✉ℓ(✶) = ✵ ❢♦r ❛❧❧ ❧ ∈ N✵ ✇✐t❤ ✷❧ + ✶ < ♣. ◆♦t❡✿ ❙♣ ❤❛s ♦♥❧② ♣ ✭❢♦r ♣ ❡✈❡♥✮ ♦r ♣ ✶ ✭❢♦r ♣ ♦❞❞✮ ❞✐♠❡♥s✐♦♥s✳

■♥✈❡rs❡ ✐♥❡q✉❛❧✐t②

❚❤❡♦r❡♠ ✭❬❚✳✱ ❚❛❦❛❝s ✷✵✶✺❪✮

❋♦r ❛❧❧ ℓ ∈ N✵ ❛♥❞ ♣ ∈ N✱ |✉ℓ|❍✶ ≤ ✷ √ ✸❤−✶

ℓ ✉ℓ▲✷

✐s s❛t✐s✜❡❞ ❢♦r ❛❧❧ ✉ℓ ∈ ❙♣,ℓ(Ω)✱ ✇❤❡r❡ ❙♣,ℓ(✵, ✶) ✐s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ✉ℓ ∈ ❙♣,ℓ(✵, ✶) ✇❤♦s❡ ♦❞❞ ❞❡r✐✈❛t✐✈❡s ✈❛♥✐s❤ ❛t t❤❡ ❜♦✉♥❞❛r②✿ ∂✷❧+✶ ∂①✷❧+✶ ✉ℓ(✵) = ∂✷❧+✶ ∂①✷❧+✶ ✉ℓ(✶) = ✵ ❢♦r ❛❧❧ ❧ ∈ N✵ ✇✐t❤ ✷❧ + ✶ < ♣. ◆♦t❡✿ ❙⊥

♣,ℓ(Ω) ❤❛s ♦♥❧② ♣ ✭❢♦r ♣ ❡✈❡♥✮ ♦r ♣ − ✶ ✭❢♦r ♣ ♦❞❞✮

❞✐♠❡♥s✐♦♥s✳

■❞❡❛✿ ❜♦✉♥❞❛r② ❝♦rr❡❝t✐♦♥

❚❤❡♦r❡♠ ✭❬❍♦❢r❡✐t❤❡r✱ ❚✳✱ ❩✉❧❡❤♥❡r ✷✵✶✺❪✮

❋♦r ❛❧❧ ℓ ∈ N✵ ❛♥❞ ♣ ∈ N✱ |✉ℓ|❍✶ ≤ √ ✷(✶ + ✹ √ ✻)❤−✶

ℓ (✉ℓ✷ ▲✷ + |HΓ,ℓ✉ℓ,Γ|✷ ❍✶)✶/✷

✐s s❛t✐s✜❡❞ ❢♦r ❛❧❧ ✉ℓ = ✉Γ,ℓ + ✉■,ℓ ∈ ❙♣,ℓ(Ω)✱ ✇❤❡r❡ ✉■,ℓ ∈ ❙(■)

♣,ℓ(Ω) ⊆

❙♣,ℓ(Ω)✱ ✉Γ,ℓ ∈ (❙(■)

♣,ℓ(Ω))⊥ ❛♥❞ HΓ,ℓ ✐s t❤❡

❞✐s❝r❡t❡ ❤❛r♠♦♥✐❝ ❡①t❡♥s✐♦♥✳ Pr♦♦❢✿ ♥❡❡❞ ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t② ♦♥ ❙♣,ℓ(Ω) ❛♥❞ r♦❜✉st ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ❡st✐♠❛t❡ ❛❧s♦ ♦♥ ❙♣,ℓ(Ω) ✭❝❢✳ ❬❚✳✱ ❚❛❦❛❝s ✷✵✶✺❪✮✳

❇♦✉♥❞❛r② ❝♦rr❡❝t✐♦♥

❘❡♦r❞❡r ✈❛r✐❛❜❧❡s s✉❝❤ t❤❛t✿ ❑ℓ = ❑ΓΓ,ℓ ❑ ❚

■Γ,ℓ

❑■Γ,ℓ ❑■■,ℓ

• ,

✉ℓ = ✉Γ,ℓ ✉■,ℓ

• ❈♦♠♣✉t❡ t❤❡ ❞✐s❝r❡t❡ ❍❛r♠♦♥✐❝ ❡①t❡♥s✐♦♥✿

✉

❍✶

✉ ❑

✶ ■■ ❑■

✉

❑

✉

❑ ❑ ❚

■

❑

✶ ■■

❑■

• ❡♥❡r❛❧✐③❡❞ ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t②✿

✉

❑

✷ ✶ ✹ ✻ ✉

▲

✇✐t❤ ❑ ▲ ❤

✷▼

❑ ❑ ❚

■

❑

✶ ■■ ❑■

✵ ✵ ✵ ❑ ❙♠♦♦t❤✐♥❣ ♣r♦♣❡rt② ❢♦r ✵ ✷

✶ ✶

✹ ✻

✷

❇♦✉♥❞❛r② ❝♦rr❡❝t✐♦♥

❘❡♦r❞❡r ✈❛r✐❛❜❧❡s s✉❝❤ t❤❛t✿ ❑ℓ = ❑ΓΓ,ℓ ❑ ❚

■Γ,ℓ

❑■Γ,ℓ ❑■■,ℓ

• ,

✉ℓ = ✉Γ,ℓ ✉■,ℓ

• ❈♦♠♣✉t❡ t❤❡ ❞✐s❝r❡t❡ ❍❛r♠♦♥✐❝ ❡①t❡♥s✐♦♥✿

|HΓ,ℓ✉Γ,ℓ|❍✶(Ω) =

• ✉Γ,ℓ

−❑ −✶

■■,ℓ❑■Γ,ℓ✉Γ,ℓ

• ❑ℓ

= ✉Γ,ℓ❑ΓΓ,ℓ−❑ ❚

■Γ,ℓ❑ −✶ ■■,ℓ❑■Γ,ℓ.

• ❡♥❡r❛❧✐③❡❞ ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t②✿

✉

❑

✷ ✶ ✹ ✻ ✉

▲

✇✐t❤ ❑ ▲ ❤

✷▼

❑ ❑ ❚

■

❑

✶ ■■ ❑■

✵ ✵ ✵ ❑ ❙♠♦♦t❤✐♥❣ ♣r♦♣❡rt② ❢♦r ✵ ✷

✶ ✶

✹ ✻

✷

❇♦✉♥❞❛r② ❝♦rr❡❝t✐♦♥

❘❡♦r❞❡r ✈❛r✐❛❜❧❡s s✉❝❤ t❤❛t✿ ❑ℓ = ❑ΓΓ,ℓ ❑ ❚

■Γ,ℓ

❑■Γ,ℓ ❑■■,ℓ

• ,

✉ℓ = ✉Γ,ℓ ✉■,ℓ

• ❈♦♠♣✉t❡ t❤❡ ❞✐s❝r❡t❡ ❍❛r♠♦♥✐❝ ❡①t❡♥s✐♦♥✿

|HΓ,ℓ✉Γ,ℓ|❍✶(Ω) =

• ✉Γ,ℓ

−❑ −✶

■■,ℓ❑■Γ,ℓ✉Γ,ℓ

• ❑ℓ

= ✉Γ,ℓ❑ΓΓ,ℓ−❑ ❚

■Γ,ℓ❑ −✶ ■■,ℓ❑■Γ,ℓ.

• ❡♥❡r❛❧✐③❡❞ ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t②✿

✉ℓ❑ℓ ≤ √ ✷(✶ + ✹ √ ✻)✉ℓ▲ℓ ✇✐t❤

• ❑ℓ := ▲ℓ := ❤−✷

ℓ ▼ℓ +

❑ΓΓ,ℓ − ❑ ❚

■Γ,ℓ❑ −✶ ■■,ℓ❑■Γ,ℓ

✵ ✵ ✵

• ❑ℓ :=

. ❙♠♦♦t❤✐♥❣ ♣r♦♣❡rt② ❢♦r ✵ ✷

✶ ✶

✹ ✻

✷

❇♦✉♥❞❛r② ❝♦rr❡❝t✐♦♥

❘❡♦r❞❡r ✈❛r✐❛❜❧❡s s✉❝❤ t❤❛t✿ ❑ℓ = ❑ΓΓ,ℓ ❑ ❚

■Γ,ℓ

❑■Γ,ℓ ❑■■,ℓ

• ,

✉ℓ = ✉Γ,ℓ ✉■,ℓ

• ❈♦♠♣✉t❡ t❤❡ ❞✐s❝r❡t❡ ❍❛r♠♦♥✐❝ ❡①t❡♥s✐♦♥✿

|HΓ,ℓ✉Γ,ℓ|❍✶(Ω) =

• ✉Γ,ℓ

−❑ −✶

■■,ℓ❑■Γ,ℓ✉Γ,ℓ

• ❑ℓ

= ✉Γ,ℓ❑ΓΓ,ℓ−❑ ❚

■Γ,ℓ❑ −✶ ■■,ℓ❑■Γ,ℓ.

• ❡♥❡r❛❧✐③❡❞ ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t②✿

▲−✶/✷

ℓ

❑ℓ▲−✶/✷

ℓ

≤ ✷(✶ + ✹ √ ✻)✷ ✇✐t❤

• ❑ℓ := ▲ℓ := ❤−✷

ℓ ▼ℓ +

❑ΓΓ,ℓ − ❑ ❚

■Γ,ℓ❑ −✶ ■■,ℓ❑■Γ,ℓ

✵ ✵ ✵

• ❑ℓ :=

. ❙♠♦♦t❤✐♥❣ ♣r♦♣❡rt② ❢♦r ✵ ✷

✶ ✶

✹ ✻

✷

❇♦✉♥❞❛r② ❝♦rr❡❝t✐♦♥

❘❡♦r❞❡r ✈❛r✐❛❜❧❡s s✉❝❤ t❤❛t✿ ❑ℓ = ❑ΓΓ,ℓ ❑ ❚

■Γ,ℓ

❑■Γ,ℓ ❑■■,ℓ

• ,

✉ℓ = ✉Γ,ℓ ✉■,ℓ

• ❈♦♠♣✉t❡ t❤❡ ❞✐s❝r❡t❡ ❍❛r♠♦♥✐❝ ❡①t❡♥s✐♦♥✿

|HΓ,ℓ✉Γ,ℓ|❍✶(Ω) =

• ✉Γ,ℓ

−❑ −✶

■■,ℓ❑■Γ,ℓ✉Γ,ℓ

• ❑ℓ

= ✉Γ,ℓ❑ΓΓ,ℓ−❑ ❚

■Γ,ℓ❑ −✶ ■■,ℓ❑■Γ,ℓ.

• ❡♥❡r❛❧✐③❡❞ ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t②✿

▲−✶/✷

ℓ

❑ℓ▲−✶/✷

ℓ

≤ ✷(✶ + ✹ √ ✻)✷ ✇✐t❤

• ❑ℓ := ▲ℓ := ❤−✷

ℓ ▼ℓ +

❑ΓΓ,ℓ − ❑ ❚

■Γ,ℓ❑ −✶ ■■,ℓ❑■Γ,ℓ

✵ ✵ ✵

• ❑ℓ :=

. ⇒ ❙♠♦♦t❤✐♥❣ ♣r♦♣❡rt② ❢♦r ✵ < τ ≤ ✷−✶(✶ + ✹ √ ✻)−✷

❘♦❜✉st ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ❡st✐♠❛t❡

❚❤❡♦r❡♠ ✭❬❚✳✱ ❚❛❦❛❝s ✷✵✶✺❪✮

❋♦r ❡❛❝❤ ✉ ∈ ❍✶(Ω)✱ ❡❛❝❤ ℓ ∈ N✵ ❛♥❞ ♣ ∈ N✿ (■ − Π♣,ℓ)✉▲✷(Ω) ≤ ✷ √ ✷ ❤ℓ|✉|❍✶(Ω) ✐s s❛t✐s✜❡❞✱ ✇❤❡r❡ Π♣,ℓ ✐s t❤❡ ❍✶✲♦rt❤♦❣♦♥❛❧ ♣r♦❥❡❝t✐♦♥ t♦ ❙♣,ℓ(Ω)✳ ❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♣r♦♣❡rt② ❢♦❧❧♦✇s ✉s✐♥❣ st❛♥❞❛r❞ ❛r❣✉♠❡♥ts ✭❢♦r t❤❡ ❤

✷▼ ♣❛rt✮

✉

❑

✉

❍✶

✉

❍✶

✉

❑ ✭❢♦r t❤❡ ❑ ♣❛rt✮

❘♦❜✉st ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ❡st✐♠❛t❡

❚❤❡♦r❡♠ ✭❬❚✳✱ ❚❛❦❛❝s ✷✵✶✺❪✮

❋♦r ❡❛❝❤ ✉ ∈ ❍✶(Ω)✱ ❡❛❝❤ ℓ ∈ N✵ ❛♥❞ ♣ ∈ N✿ (■ − Π♣,ℓ)✉▲✷(Ω) ≤ ✷ √ ✷ ❤ℓ|✉|❍✶(Ω) ✐s s❛t✐s✜❡❞✱ ✇❤❡r❡ Π♣,ℓ ✐s t❤❡ ❍✶✲♦rt❤♦❣♦♥❛❧ ♣r♦❥❡❝t✐♦♥ t♦ ❙♣,ℓ(Ω)✳ ❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♣r♦♣❡rt② ❢♦❧❧♦✇s ✉s✐♥❣

◮ st❛♥❞❛r❞ ❛r❣✉♠❡♥ts ✭❢♦r t❤❡ ❤−✷ ℓ ▼ℓ ♣❛rt✮ ◮ ✉ℓ ❑ℓ = |HΓ,ℓ✉Γ,ℓ|❍✶(Ω) ≤ |✉ℓ|❍✶(Ω) = ✉ℓ❑ℓ ✭❢♦r t❤❡

❑ℓ ♣❛rt✮

❈♦♥✈❡r❣❡♥❝❡ t❤❡♦r❡♠

❚❤❡♦r❡♠ ✭❬❍♦❢r❡✐t❤❡r✱ ❚✳✱ ❩✉❧❡❤♥❡r ✷✵✶✺❪✮

❚❤❡ t✇♦✲❣r✐❞ ♠❡t❤♦❞ ❝♦♥✈❡r❣❡s ✇✐t❤ r❛t❡ ❈❆❈❙

ν

✐❢ ν > ❈❆❈❙ s♠♦♦t❤✐♥❣ st❡♣s ❛r❡ ❛♣♣❧✐❡❞✳ ❚❤❡ ❝♦♥st❛♥ts ❈❆ ❛♥❞ ❈❙ ❞♦ ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ❣r✐❞ s✐③❡ ❤ℓ ❛♥❞ t❤❡ s♣❧✐♥❡ ❞❡❣r❡❡ ♣✳ ❚❤❡ ❡①t❡♥s✐♦♥ t♦ t❤❡ ❲✲❝②❝❧❡ ♠✉❧t✐❣r✐❞ ♠❡t❤♦❞ ✐s st❛♥❞❛r❞✳

❖✉t❧✐♥❡

Pr❡❧✐♠✐♥❛r✐❡s ▼♦❞❡❧ ♣r♦❜❧❡♠ ■●❆ ❞✐s❝r❡t✐③❛t✐♦♥ ▼✉❧t✐❣r✐❞ ❛❧❣♦r✐t❤♠ ▼✉❧t✐❣r✐❞ ❢r❛♠❡✇♦r❦ ❇❛s✐s✲✐♥❞❡♣❡♥❞❡♥t s♠♦♦t❤❡r ❙♠♦♦t❤❡r ❢♦r ♦♥❡ ❞✐♠❡♥s✐♦♥ ❙♠♦♦t❤❡r ❢♦r t✇♦ ❞✐♠❡♥s✐♦♥s ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❛♥❞ ❖✉t❧♦♦❦

▼❛tr✐❝❡s ❢r♦♠ t❡♥s♦r✲♣r♦❞✉❝t s♣❧✐♥❡s

◮ ❚❤❡ ♠❛ss ♠❛tr✐① ❤❛s ❛ t❡♥s♦r✲♣r♦❞✉❝t str✉❝t✉r❡✿

Mℓ = ▼ℓ ⊗ ▼ℓ ❚❤❡ st✐✛♥❡ss ♠❛tr✐① ✐s t❤❡ s✉♠ ♦❢ t✇♦ t❡♥s♦r✲♣r♦❞✉❝t ♠❛tr✐❝❡s✿ ❑ ▼ ▼ ❑ ❚❡♥s♦r✲♣r♦❞✉❝t str✉❝t✉r❡ ❝❛♥ ❜❡ ✉s❡❞ ❢♦r ✐♥✈❡rt✐♥❣ ✱ ❜✉t ♥♦t ❢♦r ✐♥✈❡rt✐♥❣

▼❛tr✐❝❡s ❢r♦♠ t❡♥s♦r✲♣r♦❞✉❝t s♣❧✐♥❡s

◮ ❚❤❡ ♠❛ss ♠❛tr✐① ❤❛s ❛ t❡♥s♦r✲♣r♦❞✉❝t str✉❝t✉r❡✿

Mℓ = ▼ℓ ⊗ ▼ℓ

◮ ❚❤❡ st✐✛♥❡ss ♠❛tr✐① ✐s t❤❡ s✉♠ ♦❢ t✇♦ t❡♥s♦r✲♣r♦❞✉❝t

♠❛tr✐❝❡s✿ Kℓ = ❑ℓ ⊗ ▼ℓ + ▼ℓ ⊗ ❑ℓ ❚❡♥s♦r✲♣r♦❞✉❝t str✉❝t✉r❡ ❝❛♥ ❜❡ ✉s❡❞ ❢♦r ✐♥✈❡rt✐♥❣ ✱ ❜✉t ♥♦t ❢♦r ✐♥✈❡rt✐♥❣

▼❛tr✐❝❡s ❢r♦♠ t❡♥s♦r✲♣r♦❞✉❝t s♣❧✐♥❡s

◮ ❚❤❡ ♠❛ss ♠❛tr✐① ❤❛s ❛ t❡♥s♦r✲♣r♦❞✉❝t str✉❝t✉r❡✿

Mℓ = ▼ℓ ⊗ ▼ℓ

◮ ❚❤❡ st✐✛♥❡ss ♠❛tr✐① ✐s t❤❡ s✉♠ ♦❢ t✇♦ t❡♥s♦r✲♣r♦❞✉❝t

♠❛tr✐❝❡s✿ Kℓ = ❑ℓ ⊗ ▼ℓ + ▼ℓ ⊗ ❑ℓ

◮ ❚❡♥s♦r✲♣r♦❞✉❝t str✉❝t✉r❡ ❝❛♥ ❜❡ ✉s❡❞ ❢♦r ✐♥✈❡rt✐♥❣ Mℓ✱ ❜✉t

♥♦t ❢♦r ✐♥✈❡rt✐♥❣ Kℓ

❙♠♦♦t❤❡r ❢♦r t✇♦ ❞✐♠❡♥s✐♦♥s

• Kℓ,♦r✐❣ =

❑ℓ ⊗ ▼ℓ + ▼ℓ ⊗ ❑ℓ = (❤−✷

ℓ ▼ℓ +

❑ℓ) ⊗ ▼ℓ + ▼ℓ ⊗ (❤−✷

ℓ ▼ℓ +

❑ℓ) = ✷❤−✷

ℓ ▼ℓ ⊗ ▼ℓ +

❑ℓ ⊗ ▼ℓ + ▼ℓ ⊗ ❑ℓ

• ❤✷

ℓ

❑ℓ ⊗ ❑ℓ

• t❡♥s♦r✲♣r♦❞✉❝t

− ❤✷

ℓ

❑ℓ ⊗ ❑ℓ

• ❧♦✇✲r❛♥❦ ❝♦rr❡❝t✐♦♥

=: Kℓ, ✇❤❡r❡ ❑ℓ = ❤−✶

ℓ ▼ℓ +

❑ℓ✳

❙❤❡r♠❛♥♥ ▼♦rr✐ss♦♥ ❲♦♦❞❜✉rr② ❢♦r♠✉❧❛

▲❡t ❆ ∈ R◆×◆✱ ❇ ∈ R♥×♥✱ P ∈ R♥×◆ ✇✐t❤ ❢✉❧❧ r❛♥❦✳ ❚❤❡♥✿ (❆ − P❇P❚)−✶ = ❆−✶ + ❆−✶P(❇−✶ − P❚❆−✶P)−✶P❚❆−✶ ❚❤❡ ✐♥✈❡rs✐♦♥ ♦❢ ❆ ❑ ❑ ❝❛♥ ❜❡ ❞♦♥❡ ✉s✐♥❣ t❡♥s♦r✲♣r♦❞✉❝t str✉❝t✉r❡ ❛♥❞ s♦❧✈❡rs ❢♦r ♦♥❡ ❞✐♠❡♥s✐♦♥ ❚❤❡ r❡st ❧✐✈❡s ♦♥❧② ✐♥ t❤❡ ❜♦✉♥❞❛r② ❧❛②❡r ❖♣t✐♠❛❧ ♦r❞❡r✿ ♠✉❧t✐❣r✐❞ s♦❧✈❡r ❤❛s t❤❡ s❛♠❡ ♦r❞❡r ♦❢ ❝♦♠♣❧❡①✐t② ❛s t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✇✐t❤ ❑

❙❤❡r♠❛♥♥ ▼♦rr✐ss♦♥ ❲♦♦❞❜✉rr② ❢♦r♠✉❧❛

▲❡t ❆ ∈ R◆×◆✱ ❇ ∈ R♥×♥✱ P ∈ R♥×◆ ✇✐t❤ ❢✉❧❧ r❛♥❦✳ ❚❤❡♥✿ (❆ − P❇P❚)−✶ = ❆−✶ + ❆−✶P(❇−✶ − P❚❆−✶P)−✶P❚❆−✶

◮ ❚❤❡ ✐♥✈❡rs✐♦♥ ♦❢ ❆ =

❑ℓ ⊗ ❑ℓ ❝❛♥ ❜❡ ❞♦♥❡ ✉s✐♥❣ t❡♥s♦r✲♣r♦❞✉❝t str✉❝t✉r❡ ❛♥❞ s♦❧✈❡rs ❢♦r ♦♥❡ ❞✐♠❡♥s✐♦♥ ❚❤❡ r❡st ❧✐✈❡s ♦♥❧② ✐♥ t❤❡ ❜♦✉♥❞❛r② ❧❛②❡r ❖♣t✐♠❛❧ ♦r❞❡r✿ ♠✉❧t✐❣r✐❞ s♦❧✈❡r ❤❛s t❤❡ s❛♠❡ ♦r❞❡r ♦❢ ❝♦♠♣❧❡①✐t② ❛s t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✇✐t❤ ❑

❙❤❡r♠❛♥♥ ▼♦rr✐ss♦♥ ❲♦♦❞❜✉rr② ❢♦r♠✉❧❛

▲❡t ❆ ∈ R◆×◆✱ ❇ ∈ R♥×♥✱ P ∈ R♥×◆ ✇✐t❤ ❢✉❧❧ r❛♥❦✳ ❚❤❡♥✿ (❆ − P❇P❚)−✶ = ❆−✶ + ❆−✶P(❇−✶ − P❚❆−✶P)−✶P❚❆−✶

◮ ❚❤❡ ✐♥✈❡rs✐♦♥ ♦❢ ❆ =

❑ℓ ⊗ ❑ℓ ❝❛♥ ❜❡ ❞♦♥❡ ✉s✐♥❣ t❡♥s♦r✲♣r♦❞✉❝t str✉❝t✉r❡ ❛♥❞ s♦❧✈❡rs ❢♦r ♦♥❡ ❞✐♠❡♥s✐♦♥

◮ ❚❤❡ r❡st ❧✐✈❡s ♦♥❧② ✐♥ t❤❡ ❜♦✉♥❞❛r② ❧❛②❡r

❖♣t✐♠❛❧ ♦r❞❡r✿ ♠✉❧t✐❣r✐❞ s♦❧✈❡r ❤❛s t❤❡ s❛♠❡ ♦r❞❡r ♦❢ ❝♦♠♣❧❡①✐t② ❛s t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✇✐t❤ ❑

❙❤❡r♠❛♥♥ ▼♦rr✐ss♦♥ ❲♦♦❞❜✉rr② ❢♦r♠✉❧❛

▲❡t ❆ ∈ R◆×◆✱ ❇ ∈ R♥×♥✱ P ∈ R♥×◆ ✇✐t❤ ❢✉❧❧ r❛♥❦✳ ❚❤❡♥✿ (❆ − P❇P❚)−✶ = ❆−✶ + ❆−✶P(❇−✶ − P❚❆−✶P)−✶P❚❆−✶

◮ ❚❤❡ ✐♥✈❡rs✐♦♥ ♦❢ ❆ =

❑ℓ ⊗ ❑ℓ ❝❛♥ ❜❡ ❞♦♥❡ ✉s✐♥❣ t❡♥s♦r✲♣r♦❞✉❝t str✉❝t✉r❡ ❛♥❞ s♦❧✈❡rs ❢♦r ♦♥❡ ❞✐♠❡♥s✐♦♥

◮ ❚❤❡ r❡st ❧✐✈❡s ♦♥❧② ✐♥ t❤❡ ❜♦✉♥❞❛r② ❧❛②❡r ◮ ❖♣t✐♠❛❧ ♦r❞❡r✿ ♠✉❧t✐❣r✐❞ s♦❧✈❡r ❤❛s t❤❡ s❛♠❡ ♦r❞❡r ♦❢

❝♦♠♣❧❡①✐t② ❛s t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✇✐t❤ ❑ℓ

❈♦♥✈❡r❣❡♥❝❡ t❤❡♦r❡♠

❚❤❡♦r❡♠ ✭❬❍♦❢r❡✐t❤❡r✱ ❚✳✱ ❩✉❧❡❤♥❡r ✷✵✶✺❪✮

❚❤❡ t✇♦✲❣r✐❞ ♠❡t❤♦❞ ❝♦♥✈❡r❣❡s ✇✐t❤ r❛t❡ ❈❆❈❙

ν

✐❢ ν > ❈❆❈❙ s♠♦♦t❤✐♥❣ st❡♣s ❛r❡ ❛♣♣❧✐❡❞✳ ❚❤❡ ❝♦♥st❛♥ts ❈❆ ❛♥❞ ❈❙ ❞♦ ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ❣r✐❞ s✐③❡ ❤ℓ ❛♥❞ t❤❡ s♣❧✐♥❡ ❞❡❣r❡❡ ♣✳ ❚❤❡ ❡①t❡♥s✐♦♥ t♦ t❤❡ ❲✲❝②❝❧❡ ♠✉❧t✐❣r✐❞ ♠❡t❤♦❞ ✐s st❛♥❞❛r❞✳

❖✉t❧✐♥❡

Pr❡❧✐♠✐♥❛r✐❡s ▼♦❞❡❧ ♣r♦❜❧❡♠ ■●❆ ❞✐s❝r❡t✐③❛t✐♦♥ ▼✉❧t✐❣r✐❞ ❛❧❣♦r✐t❤♠ ▼✉❧t✐❣r✐❞ ❢r❛♠❡✇♦r❦ ❇❛s✐s✲✐♥❞❡♣❡♥❞❡♥t s♠♦♦t❤❡r ❙♠♦♦t❤❡r ❢♦r ♦♥❡ ❞✐♠❡♥s✐♦♥ ❙♠♦♦t❤❡r ❢♦r t✇♦ ❞✐♠❡♥s✐♦♥s ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❛♥❞ ❖✉t❧♦♦❦

❖♥❡ ❞✐♠❡♥s✐♦♥

♣ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ℓ = ✶✵ ✷✷ ✷✵ ✷✵ ✷✶ ✷✶ ✷✵ ✷✵ ✷✵ ℓ = ✶✶ ✷✸ ✷✵ ✷✵ ✷✶ ✷✵ ✷✵ ✷✵ ✷✵ ℓ = ✶✷ ✷✸ ✷✵ ✷✵ ✷✵ ✷✵ ✷✵ ✷✵ ✷✵ ♣ ✾ ✶✵ ✶✷ ✶✹ ✶✻ ✶✽ ✷✵ ℓ = ✶✵ ✷✵ ✷✵ ✷✵ ✶✽ ✶✽ ✶✼ ✶✼ ℓ = ✶✶ ✷✵ ✶✾ ✶✾ ✶✽ ✶✾ ✶✽ ✶✼ ℓ = ✶✷ ✷✵ ✷✵ ✶✾ ✶✽ ✶✽ ✶✽ ✶✽ ν♣r❡ + ν♣♦st = ✶ + ✶✱ τ = ✵.✶✹ ❙t♦♣♣✐♥❣ ❝r✐t❡r✐♦♥✿ ❊✉❝❧✐❞❡❛♥ ♥♦r♠ ♦❢ t❤❡ ✐♥✐t✐❛❧ r❡s✐❞✉❛❧ ✐s r❡❞✉❝❡❞ ❜② ❛ ❢❛❝t♦r ♦❢ ǫ = ✶✵−✽✳

❚✇♦ ❞✐♠❡♥s✐♦♥s

♣ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ℓ = ✺ ✽✷ ✽✵ ✼✺ ✼✻ ✼✻ ✼✸ ✼✷ ✼✷ ℓ = ✻ ✽✸ ✽✼ ✼✻ ✼✹ ✼✺ ✼✸ ✼✷ ✼✷ ♣ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ✶✻ ℓ = ✺ ✼✵ ✼✶ ✼✵ ✻✽ ✻✾ ✻✾ ✻✻ ℓ = ✻ ✼✵ ✼✵ ✻✾ ✻✽ ✻✽ ✻✼ ✻✺ ν♣r❡ + ν♣♦st = ✶ + ✶✱ τ = ✵.✶✵ ✭❢♦r ♣ ≤ ✸✮ ❛♥❞ τ = ✵.✶✶ ✭❢♦r ♣ > ✸✮ ❙t♦♣♣✐♥❣ ❝r✐t❡r✐♦♥✿ ❊✉❝❧✐❞❡❛♥ ♥♦r♠ ♦❢ t❤❡ ✐♥✐t✐❛❧ r❡s✐❞✉❛❧ ✐s r❡❞✉❝❡❞ ❜② ❛ ❢❛❝t♦r ♦❢ ǫ = ✶✵−✽✳

❖✉t❧✐♥❡

Pr❡❧✐♠✐♥❛r✐❡s ▼♦❞❡❧ ♣r♦❜❧❡♠ ■●❆ ❞✐s❝r❡t✐③❛t✐♦♥ ▼✉❧t✐❣r✐❞ ❛❧❣♦r✐t❤♠ ▼✉❧t✐❣r✐❞ ❢r❛♠❡✇♦r❦ ❇❛s✐s✲✐♥❞❡♣❡♥❞❡♥t s♠♦♦t❤❡r ❙♠♦♦t❤❡r ❢♦r ♦♥❡ ❞✐♠❡♥s✐♦♥ ❙♠♦♦t❤❡r ❢♦r t✇♦ ❞✐♠❡♥s✐♦♥s ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❛♥❞ ❖✉t❧♦♦❦

❈♦♥❝❧✉s✐♦♥s

◮ ✭❖♥❡ ❛♥❞✮ t✇♦ ❞✐♠❡♥s✐♦♥s

❘♦❜✉st ❝♦♥✈❡r❣❡♥❝❡ r❛t❡s✱ r♦❜✉st ♥✉♠❜❡r ♦❢ s♠♦♦t❤✐♥❣ st❡♣s ❖♣t✐♠❛❧ ❝♦♠♣❧❡①✐t② ✐♥ t❤❡ s❡♥s❡✿ ✏❛s ❝♦♠♣❧❡① ❛s t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✇✐t❤ ❑ ✑ ❘✐❣♦r♦✉s ❛♥❛❧②s✐s

❈✳ ❍♦❢r❡✐t❤❡r✱ ❙✳ ❚❛❦❛❝s ❛♥❞ ❲✳ ❩✉❧❡❤♥❡r✳ ❆ ❘♦❜✉st ▼✉❧t✐❣r✐❞ ▼❡t❤♦❞ ❢♦r ■s♦❣❡♦♠❡tr✐❝ ❆♥❛❧②s✐s ✉s✐♥❣ ❇♦✉♥❞❛r② ❈♦rr❡❝t✐♦♥✳ ❙✉❜♠✐tt❡❞ ✭♣r❡♣r✐♥t✿ ●✰❙✲❘❡♣♦rt ✸✸✴✷✶✵✺✮✱ ✷✵✶✺✳ ❙✳ ❚❛❦❛❝s ❛♥❞ ❚✳ ❚❛❦❛❝s✳ ❆♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ❡st✐♠❛t❡s ❛♥❞ ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t✐❡s ❢♦r ❇✲s♣❧✐♥❡s ♦❢ ♠❛①✐♠✉♠ s♠♦♦t❤♥❡ss✳ ❙✉❜♠✐tt❡❞ ✭♣r❡♣r✐♥t✿ ❛r❳✐✈✿✶✺✵✷✳✵✸✼✸✸✮✱ ✷✵✶✺✳

❚❤❛♥❦s ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

❈♦♥❝❧✉s✐♦♥s

◮ ✭❖♥❡ ❛♥❞✮ t✇♦ ❞✐♠❡♥s✐♦♥s ◮ ❘♦❜✉st ❝♦♥✈❡r❣❡♥❝❡ r❛t❡s✱ r♦❜✉st ♥✉♠❜❡r ♦❢ s♠♦♦t❤✐♥❣

st❡♣s ❖♣t✐♠❛❧ ❝♦♠♣❧❡①✐t② ✐♥ t❤❡ s❡♥s❡✿ ✏❛s ❝♦♠♣❧❡① ❛s t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✇✐t❤ ❑ ✑ ❘✐❣♦r♦✉s ❛♥❛❧②s✐s

❈✳ ❍♦❢r❡✐t❤❡r✱ ❙✳ ❚❛❦❛❝s ❛♥❞ ❲✳ ❩✉❧❡❤♥❡r✳ ❆ ❘♦❜✉st ▼✉❧t✐❣r✐❞ ▼❡t❤♦❞ ❢♦r ■s♦❣❡♦♠❡tr✐❝ ❆♥❛❧②s✐s ✉s✐♥❣ ❇♦✉♥❞❛r② ❈♦rr❡❝t✐♦♥✳ ❙✉❜♠✐tt❡❞ ✭♣r❡♣r✐♥t✿ ●✰❙✲❘❡♣♦rt ✸✸✴✷✶✵✺✮✱ ✷✵✶✺✳ ❙✳ ❚❛❦❛❝s ❛♥❞ ❚✳ ❚❛❦❛❝s✳ ❆♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ❡st✐♠❛t❡s ❛♥❞ ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t✐❡s ❢♦r ❇✲s♣❧✐♥❡s ♦❢ ♠❛①✐♠✉♠ s♠♦♦t❤♥❡ss✳ ❙✉❜♠✐tt❡❞ ✭♣r❡♣r✐♥t✿ ❛r❳✐✈✿✶✺✵✷✳✵✸✼✸✸✮✱ ✷✵✶✺✳

❚❤❛♥❦s ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

❈♦♥❝❧✉s✐♦♥s

◮ ✭❖♥❡ ❛♥❞✮ t✇♦ ❞✐♠❡♥s✐♦♥s ◮ ❘♦❜✉st ❝♦♥✈❡r❣❡♥❝❡ r❛t❡s✱ r♦❜✉st ♥✉♠❜❡r ♦❢ s♠♦♦t❤✐♥❣

st❡♣s

◮ ❖♣t✐♠❛❧ ❝♦♠♣❧❡①✐t② ✐♥ t❤❡ s❡♥s❡✿ ✏❛s ❝♦♠♣❧❡① ❛s t❤❡

♠✉❧t✐♣❧✐❝❛t✐♦♥ ✇✐t❤ ❑ℓ✑ ❘✐❣♦r♦✉s ❛♥❛❧②s✐s

❈✳ ❍♦❢r❡✐t❤❡r✱ ❙✳ ❚❛❦❛❝s ❛♥❞ ❲✳ ❩✉❧❡❤♥❡r✳ ❆ ❘♦❜✉st ▼✉❧t✐❣r✐❞ ▼❡t❤♦❞ ❢♦r ■s♦❣❡♦♠❡tr✐❝ ❆♥❛❧②s✐s ✉s✐♥❣ ❇♦✉♥❞❛r② ❈♦rr❡❝t✐♦♥✳ ❙✉❜♠✐tt❡❞ ✭♣r❡♣r✐♥t✿ ●✰❙✲❘❡♣♦rt ✸✸✴✷✶✵✺✮✱ ✷✵✶✺✳ ❙✳ ❚❛❦❛❝s ❛♥❞ ❚✳ ❚❛❦❛❝s✳ ❆♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ❡st✐♠❛t❡s ❛♥❞ ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t✐❡s ❢♦r ❇✲s♣❧✐♥❡s ♦❢ ♠❛①✐♠✉♠ s♠♦♦t❤♥❡ss✳ ❙✉❜♠✐tt❡❞ ✭♣r❡♣r✐♥t✿ ❛r❳✐✈✿✶✺✵✷✳✵✸✼✸✸✮✱ ✷✵✶✺✳

❚❤❛♥❦s ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

❈♦♥❝❧✉s✐♦♥s

◮ ✭❖♥❡ ❛♥❞✮ t✇♦ ❞✐♠❡♥s✐♦♥s ◮ ❘♦❜✉st ❝♦♥✈❡r❣❡♥❝❡ r❛t❡s✱ r♦❜✉st ♥✉♠❜❡r ♦❢ s♠♦♦t❤✐♥❣

st❡♣s

◮ ❖♣t✐♠❛❧ ❝♦♠♣❧❡①✐t② ✐♥ t❤❡ s❡♥s❡✿ ✏❛s ❝♦♠♣❧❡① ❛s t❤❡

♠✉❧t✐♣❧✐❝❛t✐♦♥ ✇✐t❤ ❑ℓ✑

◮ ❘✐❣♦r♦✉s ❛♥❛❧②s✐s

❈✳ ❍♦❢r❡✐t❤❡r✱ ❙✳ ❚❛❦❛❝s ❛♥❞ ❲✳ ❩✉❧❡❤♥❡r✳ ❆ ❘♦❜✉st ▼✉❧t✐❣r✐❞ ▼❡t❤♦❞ ❢♦r ■s♦❣❡♦♠❡tr✐❝ ❆♥❛❧②s✐s ✉s✐♥❣ ❇♦✉♥❞❛r② ❈♦rr❡❝t✐♦♥✳ ❙✉❜♠✐tt❡❞ ✭♣r❡♣r✐♥t✿ ●✰❙✲❘❡♣♦rt ✸✸✴✷✶✵✺✮✱ ✷✵✶✺✳ ❙✳ ❚❛❦❛❝s ❛♥❞ ❚✳ ❚❛❦❛❝s✳ ❆♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ❡st✐♠❛t❡s ❛♥❞ ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t✐❡s ❢♦r ❇✲s♣❧✐♥❡s ♦❢ ♠❛①✐♠✉♠ s♠♦♦t❤♥❡ss✳ ❙✉❜♠✐tt❡❞ ✭♣r❡♣r✐♥t✿ ❛r❳✐✈✿✶✺✵✷✳✵✸✼✸✸✮✱ ✷✵✶✺✳

❚❤❛♥❦s ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

❈♦♥❝❧✉s✐♦♥s

◮ ✭❖♥❡ ❛♥❞✮ t✇♦ ❞✐♠❡♥s✐♦♥s ◮ ❘♦❜✉st ❝♦♥✈❡r❣❡♥❝❡ r❛t❡s✱ r♦❜✉st ♥✉♠❜❡r ♦❢ s♠♦♦t❤✐♥❣

st❡♣s

◮ ❖♣t✐♠❛❧ ❝♦♠♣❧❡①✐t② ✐♥ t❤❡ s❡♥s❡✿ ✏❛s ❝♦♠♣❧❡① ❛s t❤❡

♠✉❧t✐♣❧✐❝❛t✐♦♥ ✇✐t❤ ❑ℓ✑

◮ ❘✐❣♦r♦✉s ❛♥❛❧②s✐s

❈✳ ❍♦❢r❡✐t❤❡r✱ ❙✳ ❚❛❦❛❝s ❛♥❞ ❲✳ ❩✉❧❡❤♥❡r✳ ❆ ❘♦❜✉st ▼✉❧t✐❣r✐❞ ▼❡t❤♦❞ ❢♦r ■s♦❣❡♦♠❡tr✐❝ ❆♥❛❧②s✐s ✉s✐♥❣ ❇♦✉♥❞❛r② ❈♦rr❡❝t✐♦♥✳ ❙✉❜♠✐tt❡❞ ✭♣r❡♣r✐♥t✿ ●✰❙✲❘❡♣♦rt ✸✸✴✷✶✵✺✮✱ ✷✵✶✺✳ ❙✳ ❚❛❦❛❝s ❛♥❞ ❚✳ ❚❛❦❛❝s✳ ❆♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ❡st✐♠❛t❡s ❛♥❞ ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t✐❡s ❢♦r ❇✲s♣❧✐♥❡s ♦❢ ♠❛①✐♠✉♠ s♠♦♦t❤♥❡ss✳ ❙✉❜♠✐tt❡❞ ✭♣r❡♣r✐♥t✿ ❛r❳✐✈✿✶✺✵✷✳✵✸✼✸✸✮✱ ✷✵✶✺✳

❚❤❛♥❦s ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

❈♦♥❝❧✉s✐♦♥s

◮ ✭❖♥❡ ❛♥❞✮ t✇♦ ❞✐♠❡♥s✐♦♥s ◮ ❘♦❜✉st ❝♦♥✈❡r❣❡♥❝❡ r❛t❡s✱ r♦❜✉st ♥✉♠❜❡r ♦❢ s♠♦♦t❤✐♥❣

st❡♣s

◮ ❖♣t✐♠❛❧ ❝♦♠♣❧❡①✐t② ✐♥ t❤❡ s❡♥s❡✿ ✏❛s ❝♦♠♣❧❡① ❛s t❤❡

♠✉❧t✐♣❧✐❝❛t✐♦♥ ✇✐t❤ ❑ℓ✑

◮ ❘✐❣♦r♦✉s ❛♥❛❧②s✐s

❈✳ ❍♦❢r❡✐t❤❡r✱ ❙✳ ❚❛❦❛❝s ❛♥❞ ❲✳ ❩✉❧❡❤♥❡r✳ ❆ ❘♦❜✉st ▼✉❧t✐❣r✐❞ ▼❡t❤♦❞ ❢♦r ■s♦❣❡♦♠❡tr✐❝ ❆♥❛❧②s✐s ✉s✐♥❣ ❇♦✉♥❞❛r② ❈♦rr❡❝t✐♦♥✳ ❙✉❜♠✐tt❡❞ ✭♣r❡♣r✐♥t✿ ●✰❙✲❘❡♣♦rt ✸✸✴✷✶✵✺✮✱ ✷✵✶✺✳ ❙✳ ❚❛❦❛❝s ❛♥❞ ❚✳ ❚❛❦❛❝s✳ ❆♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ❡st✐♠❛t❡s ❛♥❞ ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t✐❡s ❢♦r ❇✲s♣❧✐♥❡s ♦❢ ♠❛①✐♠✉♠ s♠♦♦t❤♥❡ss✳ ❙✉❜♠✐tt❡❞ ✭♣r❡♣r✐♥t✿ ❛r❳✐✈✿✶✺✵✷✳✵✸✼✸✸✮✱ ✷✵✶✺✳

❚❤❛♥❦s ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

❈♦♥❝❧✉s✐♦♥s

◮ ✭❖♥❡ ❛♥❞✮ t✇♦ ❞✐♠❡♥s✐♦♥s ◮ ❘♦❜✉st ❝♦♥✈❡r❣❡♥❝❡ r❛t❡s✱ r♦❜✉st ♥✉♠❜❡r ♦❢ s♠♦♦t❤✐♥❣

st❡♣s

◮ ❖♣t✐♠❛❧ ❝♦♠♣❧❡①✐t② ✐♥ t❤❡ s❡♥s❡✿ ✏❛s ❝♦♠♣❧❡① ❛s t❤❡

♠✉❧t✐♣❧✐❝❛t✐♦♥ ✇✐t❤ ❑ℓ✑

◮ ❘✐❣♦r♦✉s ❛♥❛❧②s✐s

❈✳ ❍♦❢r❡✐t❤❡r✱ ❙✳ ❚❛❦❛❝s ❛♥❞ ❲✳ ❩✉❧❡❤♥❡r✳ ❆ ❘♦❜✉st ▼✉❧t✐❣r✐❞ ▼❡t❤♦❞ ❢♦r ■s♦❣❡♦♠❡tr✐❝ ❆♥❛❧②s✐s ✉s✐♥❣ ❇♦✉♥❞❛r② ❈♦rr❡❝t✐♦♥✳ ❙✉❜♠✐tt❡❞ ✭♣r❡♣r✐♥t✿ ●✰❙✲❘❡♣♦rt ✸✸✴✷✶✵✺✮✱ ✷✵✶✺✳ ❙✳ ❚❛❦❛❝s ❛♥❞ ❚✳ ❚❛❦❛❝s✳ ❆♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ❡st✐♠❛t❡s ❛♥❞ ✐♥✈❡rs❡ ✐♥❡q✉❛❧✐t✐❡s ❢♦r ❇✲s♣❧✐♥❡s ♦❢ ♠❛①✐♠✉♠ s♠♦♦t❤♥❡ss✳ ❙✉❜♠✐tt❡❞ ✭♣r❡♣r✐♥t✿ ❛r❳✐✈✿✶✺✵✷✳✵✸✼✸✸✮✱ ✷✵✶✺✳

❚❤❛♥❦s ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦