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❍②♣♦♥♦r♠❛❧ ❚♦❡♣❧✐t③ ❖♣❡r❛t♦rs ✇✐t❤ ◆♦♥✲❤❛r♠♦♥✐❝ ❙②♠❜♦❧s ❆❝t✐♥❣ ♦♥ t❤❡ ❇❡r❣♠❛♥ ❙♣❛❝❡ ▼❛tt❤❡✇ ❋❧❡❡♠❛♥ ❛♥❞ ❈♦♥st❛♥③❡ ▲✐❛✇ ❈❆❋❚ ✷✵✶✽ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❈r❡t❡ ❏✉❧② ✸✱ ✷✵✶✽

✐s s❛✐❞ t♦ ❜❡ ❤②♣♦♥♦r♠❛❧ ✐❢ ✵✳ ❚❤❛t ✐s✱ ✐❢ ❢♦r ❛❧❧ ✱ ✵ ❯s❡❞ t♦ st✉❞② ❙♣❡❝tr❛❧ ❛♥❞ ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r✐❡s ♦❢ ❍✐❧❜❡rt s♣❛❝❡ ♦♣❡r❛t♦rs ❙✐♥❣✉❧❛r ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ❙❝❛tt❡r✐♥❣ t❤❡♦r② ❙❡❧❢✲❛❞❥♦✐♥t ◆♦r♠❛❧ ❙✉❜✲♥♦r♠❛❧ ❍②♣♦♥♦r♠❛❧ ❍②♣♦♥♦r♠❛❧ ❖♣❡r❛t♦rs ▲❡t H ❜❡ ❛ ❝♦♠♣❧❡① ❍✐❧❜❡rt s♣❛❝❡ ❛♥❞ T ❜❡ ❛ ❜♦✉♥❞❡❞ ❧✐♥❡❛r ♦♣❡r❛t♦r ✇✐t❤ ❛❞❥♦✐♥t T ∗ ✳

❯s❡❞ t♦ st✉❞② ❙♣❡❝tr❛❧ ❛♥❞ ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r✐❡s ♦❢ ❍✐❧❜❡rt s♣❛❝❡ ♦♣❡r❛t♦rs ❙✐♥❣✉❧❛r ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ❙❝❛tt❡r✐♥❣ t❤❡♦r② ❙❡❧❢✲❛❞❥♦✐♥t ◆♦r♠❛❧ ❙✉❜✲♥♦r♠❛❧ ❍②♣♦♥♦r♠❛❧ ❍②♣♦♥♦r♠❛❧ ❖♣❡r❛t♦rs ▲❡t H ❜❡ ❛ ❝♦♠♣❧❡① ❍✐❧❜❡rt s♣❛❝❡ ❛♥❞ T ❜❡ ❛ ❜♦✉♥❞❡❞ ❧✐♥❡❛r ♦♣❡r❛t♦r ✇✐t❤ ❛❞❥♦✐♥t T ∗ ✳ T ✐s s❛✐❞ t♦ ❜❡ ❤②♣♦♥♦r♠❛❧ ✐❢ [ T ∗ , T ] := T ∗ T − TT ∗ ≥ ✵✳ ❚❤❛t ✐s✱ ✐❢ ❢♦r ❛❧❧ u ∈ H ✱ � [ T ∗ , T ] u , u � ≥ ✵ .

❙✐♥❣✉❧❛r ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ❙❝❛tt❡r✐♥❣ t❤❡♦r② ❙❡❧❢✲❛❞❥♦✐♥t ◆♦r♠❛❧ ❙✉❜✲♥♦r♠❛❧ ❍②♣♦♥♦r♠❛❧ ❍②♣♦♥♦r♠❛❧ ❖♣❡r❛t♦rs ▲❡t H ❜❡ ❛ ❝♦♠♣❧❡① ❍✐❧❜❡rt s♣❛❝❡ ❛♥❞ T ❜❡ ❛ ❜♦✉♥❞❡❞ ❧✐♥❡❛r ♦♣❡r❛t♦r ✇✐t❤ ❛❞❥♦✐♥t T ∗ ✳ T ✐s s❛✐❞ t♦ ❜❡ ❤②♣♦♥♦r♠❛❧ ✐❢ [ T ∗ , T ] := T ∗ T − TT ∗ ≥ ✵✳ ❚❤❛t ✐s✱ ✐❢ ❢♦r ❛❧❧ u ∈ H ✱ � [ T ∗ , T ] u , u � ≥ ✵ . ❯s❡❞ t♦ st✉❞② ❙♣❡❝tr❛❧ ❛♥❞ ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r✐❡s ♦❢ ❍✐❧❜❡rt s♣❛❝❡ ♦♣❡r❛t♦rs

❙❝❛tt❡r✐♥❣ t❤❡♦r② ❙❡❧❢✲❛❞❥♦✐♥t ◆♦r♠❛❧ ❙✉❜✲♥♦r♠❛❧ ❍②♣♦♥♦r♠❛❧ ❍②♣♦♥♦r♠❛❧ ❖♣❡r❛t♦rs ▲❡t H ❜❡ ❛ ❝♦♠♣❧❡① ❍✐❧❜❡rt s♣❛❝❡ ❛♥❞ T ❜❡ ❛ ❜♦✉♥❞❡❞ ❧✐♥❡❛r ♦♣❡r❛t♦r ✇✐t❤ ❛❞❥♦✐♥t T ∗ ✳ T ✐s s❛✐❞ t♦ ❜❡ ❤②♣♦♥♦r♠❛❧ ✐❢ [ T ∗ , T ] := T ∗ T − TT ∗ ≥ ✵✳ ❚❤❛t ✐s✱ ✐❢ ❢♦r ❛❧❧ u ∈ H ✱ � [ T ∗ , T ] u , u � ≥ ✵ . ❯s❡❞ t♦ st✉❞② ❙♣❡❝tr❛❧ ❛♥❞ ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r✐❡s ♦❢ ❍✐❧❜❡rt s♣❛❝❡ ♦♣❡r❛t♦rs ❙✐♥❣✉❧❛r ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s

❙❡❧❢✲❛❞❥♦✐♥t ◆♦r♠❛❧ ❙✉❜✲♥♦r♠❛❧ ❍②♣♦♥♦r♠❛❧ ❍②♣♦♥♦r♠❛❧ ❖♣❡r❛t♦rs ▲❡t H ❜❡ ❛ ❝♦♠♣❧❡① ❍✐❧❜❡rt s♣❛❝❡ ❛♥❞ T ❜❡ ❛ ❜♦✉♥❞❡❞ ❧✐♥❡❛r ♦♣❡r❛t♦r ✇✐t❤ ❛❞❥♦✐♥t T ∗ ✳ T ✐s s❛✐❞ t♦ ❜❡ ❤②♣♦♥♦r♠❛❧ ✐❢ [ T ∗ , T ] := T ∗ T − TT ∗ ≥ ✵✳ ❚❤❛t ✐s✱ ✐❢ ❢♦r ❛❧❧ u ∈ H ✱ � [ T ∗ , T ] u , u � ≥ ✵ . ❯s❡❞ t♦ st✉❞② ❙♣❡❝tr❛❧ ❛♥❞ ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r✐❡s ♦❢ ❍✐❧❜❡rt s♣❛❝❡ ♦♣❡r❛t♦rs ❙✐♥❣✉❧❛r ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ❙❝❛tt❡r✐♥❣ t❤❡♦r②

❍②♣♦♥♦r♠❛❧ ❖♣❡r❛t♦rs ▲❡t H ❜❡ ❛ ❝♦♠♣❧❡① ❍✐❧❜❡rt s♣❛❝❡ ❛♥❞ T ❜❡ ❛ ❜♦✉♥❞❡❞ ❧✐♥❡❛r ♦♣❡r❛t♦r ✇✐t❤ ❛❞❥♦✐♥t T ∗ ✳ T ✐s s❛✐❞ t♦ ❜❡ ❤②♣♦♥♦r♠❛❧ ✐❢ [ T ∗ , T ] := T ∗ T − TT ∗ ≥ ✵✳ ❚❤❛t ✐s✱ ✐❢ ❢♦r ❛❧❧ u ∈ H ✱ � [ T ∗ , T ] u , u � ≥ ✵ . ❯s❡❞ t♦ st✉❞② ❙♣❡❝tr❛❧ ❛♥❞ ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r✐❡s ♦❢ ❍✐❧❜❡rt s♣❛❝❡ ♦♣❡r❛t♦rs ❙✐♥❣✉❧❛r ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ❙❝❛tt❡r✐♥❣ t❤❡♦r② ❙❡❧❢✲❛❞❥♦✐♥t = ⇒ ◆♦r♠❛❧ = ⇒ ❙✉❜✲♥♦r♠❛❧ = ⇒ ❍②♣♦♥♦r♠❛❧

❚❤❡♦r❡♠ ✭❈✳❘✳ P✉t♥❛♠✱ ✶✾✼✵✮ ■❢ ✐s ❤②♣♦♥♦r♠❛❧ t❤❡♥ ❆r❡❛ ✇❤❡r❡ ❞❡♥♦t❡s t❤❡ s♣❡❝tr✉♠ ♦❢ ✳ ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ st✉❞②✐♥❣ t❤❡ st❛❜✐❧✐t② ♦❢ ❤②♣♦♥♦r♠❛❧ ♦♣❡r❛t♦rs ✉♥❞❡r ♣❡rt✉r❜❛t✐♦♥ ✐♥ ❝❡rt❛✐♥ ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡s✳ P✉t♥❛♠✬s ■♥❡q✉❛❧✐t② ❖♥❡ ♣❛rt✐❝✉❧❛r❧② ✐♥t❡r❡st✐♥❣ r❡s✉❧t ❢♦r ❤②♣♦♥♦r♠❛❧ ♦♣❡r❛t♦rs ✐s P✉t♥❛♠✬s ✐♥❡q✉❛❧✐t②✳

❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ st✉❞②✐♥❣ t❤❡ st❛❜✐❧✐t② ♦❢ ❤②♣♦♥♦r♠❛❧ ♦♣❡r❛t♦rs ✉♥❞❡r ♣❡rt✉r❜❛t✐♦♥ ✐♥ ❝❡rt❛✐♥ ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡s✳ P✉t♥❛♠✬s ■♥❡q✉❛❧✐t② ❖♥❡ ♣❛rt✐❝✉❧❛r❧② ✐♥t❡r❡st✐♥❣ r❡s✉❧t ❢♦r ❤②♣♦♥♦r♠❛❧ ♦♣❡r❛t♦rs ✐s P✉t♥❛♠✬s ✐♥❡q✉❛❧✐t②✳ ❚❤❡♦r❡♠ ✭❈✳❘✳ P✉t♥❛♠✱ ✶✾✼✵✮ ■❢ T ✐s ❤②♣♦♥♦r♠❛❧ t❤❡♥ � [ T ∗ , T ] � ≤ ❆r❡❛ ( σ ( T )) , π ✇❤❡r❡ σ ( T ) ❞❡♥♦t❡s t❤❡ s♣❡❝tr✉♠ ♦❢ T ✳

P✉t♥❛♠✬s ■♥❡q✉❛❧✐t② ❖♥❡ ♣❛rt✐❝✉❧❛r❧② ✐♥t❡r❡st✐♥❣ r❡s✉❧t ❢♦r ❤②♣♦♥♦r♠❛❧ ♦♣❡r❛t♦rs ✐s P✉t♥❛♠✬s ✐♥❡q✉❛❧✐t②✳ ❚❤❡♦r❡♠ ✭❈✳❘✳ P✉t♥❛♠✱ ✶✾✼✵✮ ■❢ T ✐s ❤②♣♦♥♦r♠❛❧ t❤❡♥ � [ T ∗ , T ] � ≤ ❆r❡❛ ( σ ( T )) , π ✇❤❡r❡ σ ( T ) ❞❡♥♦t❡s t❤❡ s♣❡❝tr✉♠ ♦❢ T ✳ ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ st✉❞②✐♥❣ t❤❡ st❛❜✐❧✐t② ♦❢ ❤②♣♦♥♦r♠❛❧ ♦♣❡r❛t♦rs ✉♥❞❡r ♣❡rt✉r❜❛t✐♦♥ ✐♥ ❝❡rt❛✐♥ ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥ s♣❛❝❡s✳

✷ ❝❛♥ ❜❡ t❤♦✉❣❤t ♦❢ ❛s ❛ s✉❜s♣❛❝❡ ♦❢ ✷ ✳ ❉❡✜♥✐t✐♦♥ ✷ ✇✐t❤ ▲❡t ❜❡ ✐♥ ✳ ❚❤❡ ❚♦❡♣❧✐t③ ♦♣❡r❛t♦r ✷ s②♠❜♦❧ ✐s ❣✐✈❡♥ ❜② ✇❤❡r❡ ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ❢r♦♠ ✷ ♦♥t♦ ✷ ✳ ❚❤❡ ❍❛r❞② ❙♣❛❝❡ ❉❡✜♥✐t✐♦♥ ❆ ❢✉♥❝t✐♦♥ f ( z ) ✱ ❛♥❛❧②t✐❝ ✐♥ D ✱ ✐s s❛✐❞ t♦ ❜❡❧♦♥❣ t♦ t❤❡ ❍❛r❞② s♣❛❝❡✱ H ✷ ✱ ✐❢ � | f ( re i θ ) | ✷ d θ < ∞ sup ✵ < r < ✶ T

❉❡✜♥✐t✐♦♥ ✷ ✇✐t❤ ▲❡t ❜❡ ✐♥ ✳ ❚❤❡ ❚♦❡♣❧✐t③ ♦♣❡r❛t♦r ✷ s②♠❜♦❧ ✐s ❣✐✈❡♥ ❜② ✇❤❡r❡ ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ❢r♦♠ ✷ ♦♥t♦ ✷ ✳ ❚❤❡ ❍❛r❞② ❙♣❛❝❡ ❉❡✜♥✐t✐♦♥ ❆ ❢✉♥❝t✐♦♥ f ( z ) ✱ ❛♥❛❧②t✐❝ ✐♥ D ✱ ✐s s❛✐❞ t♦ ❜❡❧♦♥❣ t♦ t❤❡ ❍❛r❞② s♣❛❝❡✱ H ✷ ✱ ✐❢ � | f ( re i θ ) | ✷ d θ < ∞ sup ✵ < r < ✶ T H ✷ ❝❛♥ ❜❡ t❤♦✉❣❤t ♦❢ ❛s ❛ s✉❜s♣❛❝❡ ♦❢ L ✷ ( T ) ✳

❚❤❡ ❍❛r❞② ❙♣❛❝❡ ❉❡✜♥✐t✐♦♥ ❆ ❢✉♥❝t✐♦♥ f ( z ) ✱ ❛♥❛❧②t✐❝ ✐♥ D ✱ ✐s s❛✐❞ t♦ ❜❡❧♦♥❣ t♦ t❤❡ ❍❛r❞② s♣❛❝❡✱ H ✷ ✱ ✐❢ � | f ( re i θ ) | ✷ d θ < ∞ sup ✵ < r < ✶ T H ✷ ❝❛♥ ❜❡ t❤♦✉❣❤t ♦❢ ❛s ❛ s✉❜s♣❛❝❡ ♦❢ L ✷ ( T ) ✳ ❉❡✜♥✐t✐♦♥ ▲❡t ϕ ( z ) ❜❡ ✐♥ L ∞ ( T ) ✳ ❚❤❡ ❚♦❡♣❧✐t③ ♦♣❡r❛t♦r T ϕ : H ✷ → H ✷ ✇✐t❤ s②♠❜♦❧ ϕ ✐s ❣✐✈❡♥ ❜② T ϕ f = P + ( ϕ f ) , ✇❤❡r❡ P + ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ❢r♦♠ L ✷ ( T ) ♦♥t♦ H ✷ ✳