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❙tr✐❝t s✐♥❣✉❧❛r✐t② ♦❢ ❛ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ♦♥ Hp

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯♥✐✈❡rs✐t② ♦❢ ❊❛st❡r♥ ❋✐♥❧❛♥❞ ■❲❖❚❆ ❈❤❡♠♥✐t③✱ ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✶ ✴ ✷✷

❇❛❝❦❣r♦✉♥❞ ❛♥❞ ♠♦t✐✈❛t✐♦♥

❇❛❝❦❣r♦✉♥❞ ❛♥❞ ♠♦t✐✈❛t✐♦♥

P♦♠♠❡r❡♥❦❡ ✭✶✾✼✼✮✿ ❆ ♥♦✈❡❧ ♣r♦♦❢ ♦❢ t❤❡ ❞❡❡♣ ❏♦❤♥✲◆✐r❡♥❜❡r❣ ✐♥❡q✉❛❧✐t② ✉s✐♥❣✿ ❆♥ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ♦❢ t❤❡ t②♣❡ Tgf (z) = z

✵

f (ζ)g′(ζ)dζ, z ∈ D, ✇❤❡r❡ f ❛♥❞ g ❛r❡ ❛♥❛❧②t✐❝ ✭❤♦❧♦♠♦r♣❤✐❝✮ ✐♥ t❤❡ ✉♥✐t ❞✐s❝ D = {z ∈ C : |z| < ✶} ♦❢ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ ✭f , g ∈ H(D)✮✳ g ✐s ✜①❡❞✱ t❤❡ s②♠❜♦❧ ♦❢ Tg. ❊①❛♠♣❧❡ ✶✿ g(z) = z ⇒ Tgf (z) = z

✵ f (ζ)dζ ✭t❤❡ ❝❧❛ss✐❝❛❧ ❱♦❧t❡rr❛

♦♣❡r❛t♦r✮

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✷ ✴ ✷✷

❇❛❝❦❣r♦✉♥❞ ❛♥❞ ♠♦t✐✈❛t✐♦♥

❊①❛♠♣❧❡ ✷✿ g(z) = log

✶ ✶−z ⇒ ✶ z Tgf (z) = ✶ z

z

✵ f (ζ) ✶−ζ dζ =

∞

k=✵

• ✶

k+✶

k

n=✵ an

• zk ✭t❤❡ ❈❡sàr♦ ♦♣❡r❛t♦r✮✳

❈❤❛r❛❝t❡r✐③❡ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ Tg ✐♥ t❡r♠s ♦❢ t❤❡ ✏❢✉♥❝t✐♦♥✲t❤❡♦r❡t✐❝✑ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ s②♠❜♦❧ g✳ ❆❧❡♠❛♥ ❛♥❞ ❙✐s❦❛❦✐s ✭✶✾✾✺✮✿ s②st❡♠❛t✐❝ r❡s❡❛r❝❤ ♦♥ Tg ❍❛r❞② s♣❛❝❡s Hp =

• f ∈ H(D) : f p = sup

✵≤r<✶

✶ ✷π ✷π

✵

|f (reit)|pdt ✶/p < ∞

• ,

✇❤❡r❡ ✵ < p < ∞.

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✸ ✴ ✷✷

❇❛❝❦❣r♦✉♥❞ ❛♥❞ ♠♦t✐✈❛t✐♦♥

Tg : Hp → Hp, ✶ ≤ p < ∞, ❜♦✉♥❞❡❞ ✭❝♦♠♣❛❝t✮ ✐✛ g ∈ BMOA (g ∈ VMOA)✱ ✇❤❡r❡ BMOA =

• h ∈ H(D) : h∗ = sup

a∈D

h ◦ σa − h(a)✷ < ∞

• ❛♥❞

VMOA =

• h ∈ BMOA : lim sup

|a|→✶

h ◦ σa − h(a)✷ = ✵

• ,

✇❤❡r❡ σa(z) = a−z

✶−¯ az , a, z ∈ D.

❆❧❡♠❛♥ ❛♥❞ ❈✐♠❛ ✭✷✵✵✶✮✿ s❝❛❧❡ ✵ < p < ✶.

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✹ ✴ ✷✷

❇❛❝❦❣r♦✉♥❞ ❛♥❞ ♠♦t✐✈❛t✐♦♥

❇♦✉♥❞❡❞♥❡ss✱ ❝♦♠♣❛❝t♥❡ss ❛♥❞ s♣❡❝tr❛❧ ♣r♦♣❡rt✐❡s ♦❢ Tg ❦♥♦✇♥ ✐♥ ♠❛♥② ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡s✳ ❙tr✉❝t✉r❛❧ ♣r♦♣❡rt✐❡s ♦❢ Tg ❤❛✈❡ ♥♦t ❜❡❡♥ ✇✐❞❡❧② st✉❞✐❡❞✳

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✺ ✴ ✷✷

❙tr✐❝t s✐♥❣✉❧❛r✐t② ❛♥❞ ℓp✲s✐♥❣✉❧❛r✐t②

❙tr✐❝t s✐♥❣✉❧❛r✐t② ❛♥❞ ℓp✲s✐♥❣✉❧❛r✐t②

❆ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦r L ❜❡t✇❡❡♥ ❇❛♥❛❝❤ s♣❛❝❡s X ❛♥❞ Y ✐s str✐❝t❧② s✐♥❣✉❧❛r✱ ✐❢ L r❡str✐❝t❡❞ t♦ ❛♥② ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❧♦s❡❞ s✉❜s♣❛❝❡ ♦❢ X ✐s ♥♦t ❛ ❧✐♥❡❛r ✐s♦♠♦r♣❤✐s♠ ♦♥t♦ ✐ts r❛♥❣❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✐t ✐s ♥♦t ❜♦✉♥❞❡❞ ❜❡❧♦✇ ♦♥ ❛♥② ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❝❧♦s❡❞ s✉❜s♣❛❝❡ M ⊂ X✳ ❆ ♥♦t✐♦♥ ✐♥tr♦❞✉❝❡❞ ❜② ❚✳ ❑❛t♦ ✐♥ ✬✺✽ ✭✐♥ ❝♦♥♥❡❝t✐♦♥ t♦ ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② ♦❢ ❋r❡❞❤♦❧♠ ♦♣❡r❛t♦rs✮✳ ❈♦♠♣❛❝t ♦♣❡r❛t♦rs ❛r❡ str✐❝t❧② s✐♥❣✉❧❛r✳ ❉❡♥♦t❡ ❜② S(X) t❤❡ str✐❝t❧② s✐♥❣✉❧❛r ♦♣❡r❛t♦rs ♦♥ X.

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✻ ✴ ✷✷

❙tr✐❝t s✐♥❣✉❧❛r✐t② ❛♥❞ ℓp✲s✐♥❣✉❧❛r✐t②

S(X) ✭♥♦r♠✲❝❧♦s❡❞✮ ✐❞❡❛❧ ♦❢ t❤❡ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs B(X)✿ L ∈ S(X), U ∈ B(X) ⇒ LU, UL ∈ S(X). ❊①❛♠♣❧❡✳ ❋♦r p < q, t❤❡ ✐♥❝❧✉s✐♦♥ ♠❛♣♣✐♥❣ i : ℓp ֒ → ℓq, i(x) = x, ✐s ❛ ♥♦♥✲❝♦♠♣❛❝t str✐❝t❧② s✐♥❣✉❧❛r ♦♣❡r❛t♦r✳ ❆ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦r L: X → Y ✐s ℓp−s✐♥❣✉❧❛r ✐❢ ✐t ✐s ♥♦t ❜♦✉♥❞❡❞ ❜❡❧♦✇ ♦♥ ❛♥② s✉❜s♣❛❝❡ M ✐s♦♠♦r♣❤✐❝ t♦ ℓp, ✐✳❡✳ ✐t ❞♦❡s ♥♦t ✜① ❛ ❝♦♣② ♦❢ ℓp. ❚❤❡ ❝❧❛ss ♦❢ ℓp−s✐♥❣✉❧❛r ♦♣❡r❛t♦rs ✐s ❞❡♥♦t❡❞ ❜② Sp(Hp).

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✼ ✴ ✷✷

❙tr✐❝t s✐♥❣✉❧❛r✐t② ❛♥❞ ℓp✲s✐♥❣✉❧❛r✐t②

❊①❛♠♣❧❡✳ ❚❤❡ ♣r♦❥❡❝t✐♦♥ P : Hp → Hp, P(∞

k=✵ akzk) = k=✵ a✷kz✷k ✐s ❜♦✉♥❞❡❞ ❛♥❞

s♣❛♥{z✷k} ✐s ✐s♦♠♦r♣❤✐❝ t♦ ℓ✷ ❜② P❛❧❡②✬s t❤❡♦r❡♠✳ ❍❡♥❝❡ P ∈ Sp(Hp) \ S✷(Hp). ❚❤❡ str✐❝t s✐♥❣✉❧❛r✐t② ♦❢ Tg ❛❝t✐♥❣ ♦♥ Hp?

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✽ ✴ ✷✷

▼❛✐♥ r❡s✉❧t

▼❛✐♥ r❡s✉❧t

❚❤❡♦r❡♠ ✶ ▲❡t g ∈ BMOA \ VMOA ❛♥❞ ✶ ≤ p < ∞. ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ s✉❜s♣❛❝❡ M ⊂ Hp ✐s♦♠♦r♣❤✐❝ t♦ ℓp s✳t✳ t❤❡ r❡str✐❝t✐♦♥ Tg|M : M → Tg(M) ✐s ❜♦✉♥❞❡❞ ❜❡❧♦✇✳ ✭✐✳❡✳ ❆ ♥♦♥✲❝♦♠♣❛❝t Tg : Hp → Hp ✜①❡s ❛ ❝♦♣② ♦❢ ℓp.✮ ■♥ ♣❛rt✐❝✉❧❛r✱ Tg ✐s ♥♦t str✐❝t❧② s✐♥❣✉❧❛r✳ Tg ✐s r✐❣✐❞ ✐♥ t❤❡ s❡♥s❡ t❤❛t Tg ∈ S(Hp) ⇔ Tg ∈ K(Hp) ⇔ Tg ∈ Sp(Hp). ◆♦t tr✉❡ ❢♦r ❛♥ ❛r❜✐tr❛r② ♦♣❡r❛t♦r✳

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✾ ✴ ✷✷

▼❛✐♥ r❡s✉❧t

❙tr❛t❡❣② ♦❢ t❤❡ ♣r♦♦❢

❋✐❣✉r❡✿ ❖♣❡r❛t♦rs U, V ❛♥❞ Tg

ℓp Hp Hp V U Tg ❙tr❛t❡❣②✿ ❈♦♥str✉❝t ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs U ❛♥❞ V ✿ U ❜♦✉♥❞❡❞ ❜❡❧♦✇ ✇❤❡♥ Tg ✐s ♥♦♥✲❝♦♠♣❛❝t ✭✐✳❡✳ g ∈ BMOA \ VMOA✮✳ ❚❤❡ ❞✐❛❣r❛♠ ❝♦♠♠✉t❡s✿ U = TgV Tg|M : M → Tg(M) ❜♦✉♥❞❡❞ ❜❡❧♦✇ ♦♥ M = V (ℓp) ≈ ℓp.

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✶✵ ✴ ✷✷

▼❛✐♥ r❡s✉❧t

❍♦✇ t♦ ❞❡✜♥❡ U ❛♥❞ V ? ❲❡ ✉t✐❧✐③❡ s✉✐t❛❜❧② ❝❤♦s❡♥ st❛♥❞❛r❞ ♥♦r♠❛❧✐③❡❞ t❡st ❢✉♥❝t✐♦♥s fa ∈ Hp ❞❡✜♥❡❞ ❜② fa(z) = ✶ − |a|✷ (✶ − ¯ az)✷ ✶/p , z ∈ D, ❢♦r ❡❛❝❤ a ∈ D✳ ❋♦r p = ✷✱ t❤❡ ❢✉♥❝t✐♦♥s fa ❛r❡ t❤❡ ♥♦r♠❛❧✐③❡❞ r❡♣r♦❞✉❝✐♥❣ ❦❡r♥❡❧s ♦❢ H✷✳

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✶✶ ✴ ✷✷

▼❛✐♥ r❡s✉❧t

❉❡✜♥❡ V : ℓp → Hp, V (α) =

∞

• n=✶

αnfan ❛♥❞ U : ℓp → Hp, U(α) =

∞

• n=✶

αnTgfan, ✇❤❡r❡ t❤❡ s❡q✉❡♥❝❡ (an) ∈ D, |an| → ✶ ✐s s✉✐t❛❜❧② ❝❤♦s❡♥ ❛♥❞ α = (αn) ∈ ℓp.

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✶✷ ✴ ✷✷

❚♦♦❧s

❚♦♦❧s

V ✐s ❜♦✉♥❞❡❞✱ ✐❢ |an| → ✶ ❢❛st ❡♥♦✉❣❤✳ ❍♦✇ t♦ s❤♦✇ t❤❛t U ✐s ❜♦✉♥❞❡❞ ❜❡❧♦✇❄ ❆ r❡s✉❧t ❜② ❆❧❡♠❛♥ ❛♥❞ ❈✐♠❛ ✭✷✵✵✶✮✿ ❚❤❡♦r❡♠ ✷ ▲❡t ✵ < p < ∞ ❛♥❞ t ∈ (✵, p/✷). ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t C = C(p, t) > ✵ s✳t✳ Tgfap ≥ Cg ◦ σa − g(a)t ❢♦r ❛❧❧ a ∈ D.

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✶✸ ✴ ✷✷

❚♦♦❧s

❘❡❝❛❧❧✿ g ∈ BMOA \ VMOA ⇔ lim sup|a|→✶ g ◦ σa − g(a)p > ✵ ❢♦r ❛♥② ✵ < p < ∞. ❚❤✉s ✐❢ Tg ✐s ♥♦♥✲❝♦♠♣❛❝t✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ (an) ⊂ D, an → ω ∈ T = ∂D s✳t✳ limn→∞ Tgfanp > ✵.

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✶✹ ✴ ✷✷

❚♦♦❧s

❆ ❧♦❝❛❧✐③❛t✐♦♥ r❡s✉❧t ❢♦r Tg✿ ▲❡♠♠❛ ✸ ▲❡t g ∈ BMOA, ✶ ≤ p < ∞, ❛♥❞ (ak) ⊂ D ❜❡ ❛ s❡q✉❡♥❝❡ s✉❝❤ t❤❛t ak → ω ∈ T. ❉❡✜♥❡ Aε = {eiθ : |eiθ − ω| < ε} ❢♦r ❡❛❝❤ ε > ✵✳❚❤❡♥ ✭✐✮ lim

k→∞

• T\Aε

|Tgfak|pdm = ✵ ❢♦r ❡✈❡r② ε > ✵. ✭✐✐✮ lim

ε→✵

• Aε

|Tgfak|pdm = ✵ ❢♦r ❡❛❝❤ k.

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✶✺ ✴ ✷✷

❚♦♦❧s

❯s✐♥❣ ❛ r❡s✉❧t ❜② ❆❧❡♠❛♥ ❛♥❞ ❈✐♠❛ ✭❚❤❡♦r❡♠ ✷✮ ❛♥❞ ❝♦♥❞✐t✐♦♥s ✭✐✮ ❛♥❞ ✭✐✐✮ ✐♥ ▲❡♠♠❛ ✸✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❝❤♦♦s❡ ❛ s✉❜s❡q✉❡♥❝❡ (bn) ♦❢ (an) ✭✇❤❡r❡ limn→∞ Tgfanp > ✵✮ s✳t✳ ❋✉♥❝t✐♦♥s |Tgfbn| r❡s❡♠❜❧❡ ✏❞✐s❥♦✐♥t❧② s✉♣♣♦rt❡❞ ♣❡❛❦s ✐♥ Lp(T)✑ ♥❡❛r s♦♠❡ ❜♦✉♥❞❛r② ♣♦✐♥t ω ∈ T, ✐✳❡✳ (Tgfbn) ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ♥❛t✉r❛❧ ❜❛s✐s ♦❢ ℓp. ❚❤✐s ❡♥s✉r❡s t❤❛t U(α)p =

n αnTgfbnp ≥ Cαℓp ❢♦r ❛❧❧

α = (αn) ∈ ℓp.

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✶✻ ✴ ✷✷

❚♦♦❧s

❋✐❣✉r❡✿ ❖♣❡r❛t♦rs U, V ❛♥❞ Tg

ℓp Hp Hp V U Tg f ∈ M = V (ℓp) = span{fbn} ⇒ Tgf p = U(α)p ≥ Cαℓp ≥ V (α)p = f p.

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✶✼ ✴ ✷✷

❙tr✐❝t s✐♥❣✉❧❛r✐t② ♦❢ Tg ♦♥ ❇❡r❣♠❛♥ s♣❛❝❡s ❛♥❞ ❇❧♦❝❤ s♣❛❝❡

❙tr✐❝t s✐♥❣✉❧❛r✐t② ♦❢ Tg ♦♥ ❇❡r❣♠❛♥ s♣❛❝❡s ❛♥❞ ❇❧♦❝❤ s♣❛❝❡

❙t❛♥❞❛r❞ ❇❡r❣♠❛♥ s♣❛❝❡s Ap

α = Lp(D, dAα) ∩ H(D)✱ ✇❤❡r❡

dAα(z) = (✶ − |z|✷)αdA(z), α > −✶ ❛♥❞ ✵ < p < ∞✱ ❛r❡ ✐s♦♠♦r♣❤✐❝ t♦ ℓp. S(ℓp) = K(ℓp) ⇒ t❤❡ str✐❝t s✐♥❣✉❧❛r✐t② ❛♥❞ ❝♦♠♣❛❝t♥❡ss ❛r❡ ❡q✉✐✈❛❧❡♥t ❢♦r Tg ❛❝t✐♥❣ ♦♥ Ap

α.

▲❡t B ❜❡ t❤❡ ❇❧♦❝❤ s♣❛❝❡✳ ❚❤❡♥ Tg : B → B ✐s str✐❝t❧② s✐♥❣✉❧❛r ⇒ Tg|B✵ ✐s str✐❝t❧② s✐♥❣✉❧❛r✳ ❙✐♥❝❡ t❤❡ ❧✐tt❧❡ ❇❧♦❝❤ s♣❛❝❡ B✵ ✐s ✐s♦♠♦r♣❤✐❝ t♦ c✵ ❛♥❞ S(c✵) = K(c✵)✱ t❤❡ r❡str✐❝t✐♦♥ Tg|B✵ ✐s ❝♦♠♣❛❝t✳ ❋✐♥❛❧❧②✱ t❤❡ ❜✐❛❞❥♦✐♥t (Tg|B✵)∗∗ ❝❛♥ ❜❡ ✐❞❡♥t✐✜❡❞ ✇✐t❤ Tg : B → B ❛♥❞ ❝♦♥s❡q✉❡♥t❧② Tg ❛❝t✐♥❣ ♦♥ B ✐s ❝♦♠♣❛❝t✳

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✶✽ ✴ ✷✷

❙tr✐❝t s✐♥❣✉❧❛r✐t② ♦❢ Tg ♦♥ ❇❡r❣♠❛♥ s♣❛❝❡s ❛♥❞ ❇❧♦❝❤ s♣❛❝❡

❙♣❛❝❡s BMOA ❛♥❞ VMOA

❆ r❡s✉❧t ♦❢ ▲❡✐❜♦✈ ❡♥s✉r❡s t❤❛t t❤❡r❡ ❡①✐sts ✐s♦♠♦r♣❤✐❝ ❝♦♣✐❡s ♦❢ s♣❛❝❡ c✵ ♦❢ ♥✉❧❧✲s❡q✉❡♥❝❡s ✐♥s✐❞❡ VMOA : ■❢ (hn) ⊂ VMOA ✐s ❛ s❡q✉❡♥❝❡ ♦❢ VMOA✲❢✉♥❝t✐♦♥s s✳t✳ hn∗ ≃ ✶ ❛♥❞ hn✷ → ✵, t❤❡♥ t❤❡r❡ ❡①✐sts ❛ s✉❜s❡q✉❡♥❝❡ (hnk) ✇❤✐❝❤ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ st❛♥❞❛r❞ ❜❛s✐s ♦❢ c✵. ■❢ Tg : BMOA → BMOA ✐s ♥♦♥✲❝♦♠♣❛❝t✱ t❤❡♥ ✐t t✉r♥s ♦✉t t❤❛t (Tghnk) s❛t✐s✜❡s ❝♦♥❞✐t✐♦♥s ✐♥ ▲❡✐❜♦✈✬s r❡s✉❧t ❛♥❞ ✇❡ ❝❛♥ ❛♣♣❧② ✐t ❛❣❛✐♥✳ ❈♦♥s❡q✉❡♥t❧②✱ Tg ✜①❡s ❛ ❝♦♣② ♦❢ c✵ ✐♥s✐❞❡ VMOA. ❚❤✉s str✐❝t s✐♥❣✉❧❛r✐t② ❛♥❞ ❝♦♠♣❛❝t♥❡ss ❝♦✐♥❝✐❞❡ ❢♦r Tg ❛❝t✐♥❣ ♦♥ BMOA ✭♦r VMOA✮✳

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✶✾ ✴ ✷✷

❋✉rt❤❡r r❡s✉❧ts ❛♥❞ q✉❡st✐♦♥s

❋✉rt❤❡r r❡s✉❧ts ❛♥❞ q✉❡st✐♦♥s

Tg : Hp → Hp, ✶ ≤ p < ∞, ✐s ❛❧✇❛②s ℓ✷−s✐♥❣✉❧❛r ✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ◆✐❡♠✐♥❡♥✱ ❙❛❦s♠❛♥✱ ❛♥❞ ❚②❧❧✐✮ ❆♥ ❡①❛♠♣❧❡ ♦❢ ❛♥ ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡ X ⊂ H(D) ❛♥❞ ❛ s②♠❜♦❧ g ∈ H(D) s✳t✳ Tg ∈ S(X) \ K(X)? ❲❤❛t ❛❜♦✉t t❤❡ ❝❛s❡ Tg : Hp → Hq, ✇❤❡r❡ ✶ ≤ p < q < ∞?

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✷✵ ✴ ✷✷

❋♦r ❢✉rt❤❡r r❡❛❞✐♥❣

❋♦r ❢✉rt❤❡r r❡❛❞✐♥❣

❆✳ ❆❧❡♠❛♥✱ ❆ ❝❧❛ss ♦❢ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦rs ♦♥ s♣❛❝❡s ♦❢ ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s✱ ❚♦♣✐❝s ✐♥ ❝♦♠♣❧❡① ❛♥❛❧②s✐s ❛♥❞ ♦♣❡r❛t♦r t❤❡♦r②✱ ✸✲✸✵✱ ❯♥✐✈✳ ▼á❧❛❣❛✱ ▼á❧❛❣❛✱ ✷✵✵✼✳ ❆✳●✳ ❙✐s❦❛❦✐s✱ ❱♦❧t❡rr❛ ♦♣❡r❛t♦rs ♦♥ s♣❛❝❡s ♦❢ ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s✲❛ s✉r✈❡②✱ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❋✐rst ❆❞✈❛♥❝❡❞ ❈♦✉rs❡ ✐♥ ❖♣❡r❛t♦r ❚❤❡♦r② ❛♥❞ ❈♦♠♣❧❡① ❆♥❛❧②s✐s✱ ❯♥✐✈✳ ❙❡✈✐❧❧❛ ❙❡❝r✳ P✉❜❧✳✱ ❙❡✈✐❧❧❡✱ ✷✵✵✻✱ ♣♣✳ ✺✶✲✻✽✳ ❆✳ ❆❧❡♠❛♥ ❛♥❞ ❆✳●✳ ❙✐s❦❛❦✐s✱ ❆♥ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ♦♥ Hp✱ ❈♦♠♣❧❡① ❱❛r✐❛❜❧❡s ❚❤❡♦r② ❆♣♣❧✳ ✷✽ ✭✶✾✾✺✮✱ ♥♦✳ ✷✱ ✶✹✾✲✶✺✽✳ ❆✳ ❆❧❡♠❛♥ ❛♥❞ ❏✳❆✳ ❈✐♠❛✱ ❆♥ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ♦♥ Hp ❛♥❞ ❍❛r❞②✬s ✐♥❡q✉❛❧✐t②✱ ❏✳ ❆♥❛❧②s❡ ▼❛t❤✳ ✽✺ ✭✷✵✵✶✮✱ ✶✺✼✲✶✼✻✳ ❙✳ ▼✐✐❤❦✐♥❡♥✱ ❙tr✐❝t s✐♥❣✉❧❛r✐t② ♦❢ ❛ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ♦♥ Hp✱ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❆▼❙✱ ✶✹✺ ✭✷✵✶✼✮✱ ♥♦✳ ✶✱ ✶✻✺✲✶✼✺✳

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✷✶ ✴ ✷✷

❋♦r ❢✉rt❤❡r r❡❛❞✐♥❣

❚❍❆◆❑ ❨❖❯✦

❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✷✷ ✴ ✷✷