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❙tr✐❝t s✐♥❣✉❧❛r✐t② ♦❢ ❛ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ♦♥ H p ❙❛♥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♣❡ � z f ( ζ ) g ′ ( ζ ) d ζ, z ∈ D , T g f ( z ) = ✵ ✇❤❡r❡ f ❛♥❞ g ❛r❡ ❛♥❛❧②t✐❝ ✭❤♦❧♦♠♦r♣❤✐❝✮ ✐♥ t❤❡ ✉♥✐t ❞✐s❝ D = { z ∈ C : | z | < ✶ } ♦❢ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ ✭ f , g ∈ H ( D ) ✮✳ g ✐s ✜①❡❞✱ t❤❡ s②♠❜♦❧ ♦❢ T g . � z ❊①❛♠♣❧❡ ✶✿ g ( z ) = z ⇒ T g f ( z ) = ✵ f ( ζ ) d ζ ✭t❤❡ ❝❧❛ss✐❝❛❧ ❱♦❧t❡rr❛ ♦♣❡r❛t♦r✮ ❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✷ ✴ ✷✷

❇❛❝❦❣r♦✉♥❞ ❛♥❞ ♠♦t✐✈❛t✐♦♥ � z f ( ζ ) ✶ − z ⇒ ✶ ✶ z T g f ( z ) = ✶ ❊①❛♠♣❧❡ ✷✿ g ( z ) = log ✶ − ζ d ζ = z ✵ � � z k ✭t❤❡ ❈❡sàr♦ ♦♣❡r❛t♦r✮✳ � ∞ � k ✶ n = ✵ a n k = ✵ k + ✶ ❈❤❛r❛❝t❡r✐③❡ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ T g ✐♥ t❡r♠s ♦❢ t❤❡ ✏❢✉♥❝t✐♦♥✲t❤❡♦r❡t✐❝✑ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ s②♠❜♦❧ g ✳ ❆❧❡♠❛♥ ❛♥❞ ❙✐s❦❛❦✐s ✭✶✾✾✺✮✿ s②st❡♠❛t✐❝ r❡s❡❛r❝❤ ♦♥ T g ❍❛r❞② s♣❛❝❡s � ✶ � ✷ π � ✶ / p � � H p = | f ( re it ) | p dt f ∈ H ( D ) : � f � p = sup < ∞ , ✷ π ✵ ≤ r < ✶ ✵ ✇❤❡r❡ ✵ < p < ∞ . ❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✸ ✴ ✷✷

❇❛❝❦❣r♦✉♥❞ ❛♥❞ ♠♦t✐✈❛t✐♦♥ T g : H p → H p , ✶ ≤ p < ∞ , ❜♦✉♥❞❡❞ ✭❝♦♠♣❛❝t✮ ✐✛ g ∈ BMOA ( g ∈ VMOA ) ✱ ✇❤❡r❡ � � BMOA = h ∈ H ( D ) : � h � ∗ = sup � h ◦ σ a − h ( a ) � ✷ < ∞ a ∈ D ❛♥❞ � � VMOA = h ∈ BMOA : lim sup � h ◦ σ a − h ( a ) � ✷ = ✵ , | a |→ ✶ ✇❤❡r❡ σ a ( 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 ♦❢ T g ❦♥♦✇♥ ✐♥ ♠❛♥② ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡s✳ ❙tr✉❝t✉r❛❧ ♣r♦♣❡rt✐❡s ♦❢ T g ❤❛✈❡ ♥♦t ❜❡❡♥ ✇✐❞❡❧② st✉❞✐❡❞✳ ❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✺ ✴ ✷✷

❙tr✐❝t s✐♥❣✉❧❛r✐t② ❛♥❞ ℓ p ✲s✐♥❣✉❧❛r✐t② ❙tr✐❝t s✐♥❣✉❧❛r✐t② ❛♥❞ ℓ p ✲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 ❛♥❞ ℓ p ✲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 : ℓ p ֒ → ℓ q , i ( x ) = x , ✐s ❛ ♥♦♥✲❝♦♠♣❛❝t str✐❝t❧② s✐♥❣✉❧❛r ♦♣❡r❛t♦r✳ ❆ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦r L : X → Y ✐s ℓ p − s✐♥❣✉❧❛r ✐❢ ✐t ✐s ♥♦t ❜♦✉♥❞❡❞ ❜❡❧♦✇ ♦♥ ❛♥② s✉❜s♣❛❝❡ M ✐s♦♠♦r♣❤✐❝ t♦ ℓ p , ✐✳❡✳ ✐t ❞♦❡s ♥♦t ✜① ❛ ❝♦♣② ♦❢ ℓ p . ❚❤❡ ❝❧❛ss ♦❢ ℓ p − s✐♥❣✉❧❛r ♦♣❡r❛t♦rs ✐s ❞❡♥♦t❡❞ ❜② S p ( H p ) . ❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✼ ✴ ✷✷

❙tr✐❝t s✐♥❣✉❧❛r✐t② ❛♥❞ ℓ p ✲s✐♥❣✉❧❛r✐t② ❊①❛♠♣❧❡✳ ❚❤❡ ♣r♦❥❡❝t✐♦♥ k = ✵ a ✷ k z ✷ k ✐s ❜♦✉♥❞❡❞ ❛♥❞ P : H p → H p , P ( � ∞ k = ✵ a k z k ) = � s♣❛♥ { z ✷ k } ✐s ✐s♦♠♦r♣❤✐❝ t♦ ℓ ✷ ❜② P❛❧❡②✬s t❤❡♦r❡♠✳ ❍❡♥❝❡ P ∈ S p ( H p ) \ S ✷ ( H p ) . ❚❤❡ str✐❝t s✐♥❣✉❧❛r✐t② ♦❢ T g ❛❝t✐♥❣ ♦♥ H p ? ❙❛♥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 ⊂ H p ✐s♦♠♦r♣❤✐❝ t♦ ℓ p s✳t✳ t❤❡ r❡str✐❝t✐♦♥ T g | M : M → T g ( M ) ✐s ❜♦✉♥❞❡❞ ❜❡❧♦✇✳ ✭✐✳❡✳ ❆ ♥♦♥✲❝♦♠♣❛❝t T g : H p → H p ✜①❡s ❛ ❝♦♣② ♦❢ ℓ p . ✮ ■♥ ♣❛rt✐❝✉❧❛r✱ T g ✐s ♥♦t str✐❝t❧② s✐♥❣✉❧❛r✳ T g ✐s r✐❣✐❞ ✐♥ t❤❡ s❡♥s❡ t❤❛t T g ∈ S ( H p ) ⇔ T g ∈ K ( H p ) ⇔ T g ∈ S p ( H p ) . ◆♦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 ❛♥❞ T g ℓ p V U H p H p T g ❙tr❛t❡❣②✿ ❈♦♥str✉❝t ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs U ❛♥❞ V ✿ U ❜♦✉♥❞❡❞ ❜❡❧♦✇ ✇❤❡♥ T g ✐s ♥♦♥✲❝♦♠♣❛❝t ✭✐✳❡✳ g ∈ BMOA \ VMOA ✮✳ ❚❤❡ ❞✐❛❣r❛♠ ❝♦♠♠✉t❡s✿ U = T g V T g | M : M → T g ( M ) ❜♦✉♥❞❡❞ ❜❡❧♦✇ ♦♥ M = V ( ℓ p ) ≈ ℓ p . ❙❛♥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 f a ∈ H p ❞❡✜♥❡❞ ❜② � ✶ − | a | ✷ � ✶ / p f a ( z ) = , z ∈ D , az ) ✷ ( ✶ − ¯ ❢♦r ❡❛❝❤ a ∈ D ✳ ❋♦r p = ✷✱ t❤❡ ❢✉♥❝t✐♦♥s f a ❛r❡ t❤❡ ♥♦r♠❛❧✐③❡❞ r❡♣r♦❞✉❝✐♥❣ ❦❡r♥❡❧s ♦❢ H ✷ ✳ ❙❛♥t❡r✐ ▼✐✐❤❦✐♥❡♥✱ ❯❊❋ ❱♦❧t❡rr❛✲t②♣❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ✶✹✲✶✽ ❆✉❣✉st ✷✵✶✼ ✶✶ ✴ ✷✷