Integral operators mapping into the space of bounded analytic functions

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Integral operators mapping into the space of bounded analytic functions

Manuel D. Contreras

Departamento de Matem ´atica Aplicada II and IMUS Universidad de Sevilla

Work in progress with J.A. Peláez, Ch. Pommerenke, J. Rättyä.

Workshop on Banach Spaces, Granada 2015, on the occasion of the 60th birthday of Rafael Pay ´a

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... on the occasion of the 60th birthday of Rafael Pay ´a

I was lucky to attend Rafa’s courses on Complex Analysis, Functional Analysis and Measure Theory.

With you, I learned many theorems and mathematical techniques but, certainly, my best memory is the passion and enthusiasm that one can feel when he is trying to tackle a problem.

I learned with you how to enjoy doing maths!

Thank you very much for everything you taught me and for being my

“Maestro”.

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... on the occasion of the 60th birthday of Rafael Pay ´a

“Maestro”.

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... on the occasion of the 60th birthday of Rafael Pay ´a

“Maestro”.

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... on the occasion of the 60th birthday of Rafael Pay ´a

“Maestro”.

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Integral operators mapping into the space of bounded analytic functions

Manuel D. Contreras

Departamento de Matem ´atica Aplicada II and IMUS Universidad de Sevilla

Work in progress with J.A. Peláez, Ch. Pommerenke, J. Rättyä.

Workshop on Banach Spaces, Granada 2015, on the occasion of the 60th birthday of Rafael Pay ´a

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A bit of history Boundedness on H^∞ Weak compactness on H^∞ Compactness on H^∞

Definition

Notation: D unit disk in the complex plane C H(D):= {f : D → C : analytic}.

For g analytic on D, we define T_g(f )(z) =

Z z 0

f (ξ)g⁰(ξ)d ξ, f ∈ H(D).

T_gis calledintegral operator,Volterra operator,Pommerenke operator, ...

The goal of this talk:

The purpose is to show some recent results about T_g on H^∞.

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Definition

Notation: D unit disk in the complex plane C H(D):= {f : D → C : analytic}.

For g analytic on D, we define T_g(f )(z) =

Z z 0

f (ξ)g⁰(ξ)d ξ, f ∈ H(D).

T_gis calledintegral operator,Volterra operator,Pommerenke operator, ...

The goal of this talk:

The purpose is to show some recent results about Tg on H^∞.

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Example: Ces `aro operator.

If g(z) = log

1 1−z

and f (z) =

∞

X

n=0

a_kz^k,then

Tg(f )(z) = Z z

0

f (ζ) 1

1 − ζ d ζ = z

∞

X

n=0

1 n + 1

n

X

k =0

an

! zⁿ.

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Example: Integration operator on simply connected domains.

Let Ω be a simply connected domain in C, Ω 6= C, with 0 ∈ Ω:

I_Ω(f )(z) = Z z

0

f (ζ) d ζ, for all f ∈ H(Ω).

If h : D → Ω is a Riemann map, with h(0) = 0, then C_h : H(Ω) → H(D) given by Ch(f ) = f ◦ h is one-to-one, onto and

C_h◦ IΩ◦ C_h⁻¹(f )(z) = Z z

0

f (ζ)h⁰(ζ)d ζ = T_h(f )(z), for all f ∈ H(D). So the integration operator on Ω is the integral operator with symbol h.

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Example: Integration operator on simply connected domains.

Let Ω be a simply connected domain in C, Ω 6= C, with 0 ∈ Ω:

I_Ω(f )(z) = Z z

0

f (ζ) d ζ, for all f ∈ H(Ω).

If h : D → Ω is a Riemann map, with h(0) = 0, then C_h: H(Ω) → H(D) given by Ch(f ) = f ◦ h is one-to-one, onto and

C_h◦ IΩ◦ C_h⁻¹(f )(z) = Z z

0

f (ζ)h⁰(ζ)d ζ = T_h(f )(z), for all f ∈ H(D).

So the integration operator on Ω is the integral operator with symbol h.

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The first result about T

_g

Theorem (Pommerenke, 1977) g ∈ BMOA ⇐⇒

There exists a constant A > 0 s.t.

exp(Ag) ∈ H².

⇐⇒







There exists a univalent function h s.t.

g = a log h⁰ with a ∈ C \ {0}

and ∂h(D) a quasi − smooth Jordan curve.

Pommerenke’s proof is based on the next property of Tg: T_g :H²→ H²⇐⇒ g ∈ BMOA.

With the same ideas, it is possible to obtain

Tg :H²→ H²is compact ⇐⇒ g ∈ VMOA.

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The first result about T

_g

exp(Ag) ∈ H².

⇐⇒







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The first result about T

_g

exp(Ag) ∈ H².

⇐⇒







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Basic questions about T

_g

Let X , Y ⊂ H(D) be two Banach spaces.

Consider the restriction of T_gto X .

1 For which g is Tg :X → Y bounded?, compact?,

weakly compact?, ...

2 Spectral properties of T_g?

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Some spaces of analytic functions: Hardy spaces.

For 1 ≤ p < ∞, the spaceH^p consists of those analytic functions in f : D → C such that

||f ||^p_p:=sup

r <1

1 2π

Z 2π 0

|f (re^iθ)|^pd θ < +∞.

H^∞denotes the space of bounded analytic functions in D with the supremum norm: ||f ||∞:=sup_z∈D|f (z)|.

Thedisk algebra Aconsists of those functions inH^∞that are continuous up to the boundary of D.

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Some spaces of analytic functions: Hardy spaces.

For 1 ≤ p < ∞, the spaceH^p consists of those analytic functions in f : D → C such that

||f ||^p_p:=sup

r <1

1 2π

Z 2π 0

|f (re^iθ)|^pd θ < +∞.

H^∞denotes the space of bounded analytic functions in D with the supremum norm: ||f ||∞:=sup_z∈D|f (z)|.

Thedisk algebra Aconsists of those functions inH^∞that are continuous up to the boundary of D.

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Some spaces of analytic function: Hardy spaces.

AlemanandSiskakis(p = 2 is due toPommerenke):

1 ≤ p < ∞

T_g :H^p→ H^p ⇐⇒ g ∈ BMOA and

Tg:H^p → H^pis compact ⇐⇒ g ∈ VMOA.

Laitila,Miihkinen, andNieminen:

T_g:H¹→ H¹is weakly compact ⇐⇒ g ∈ VMOA.

AlemanandCimastudied Tg :H^p→ H^q, 1 ≤ p, q < +∞.

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Some spaces of analytic functions: Mean oscillation.

A function f ∈ H²belongs to thespace of analytic functions of bounded mean oscillation BMOAif

there exists C > 0 s.t. Z

R(I)

|f⁰(z)|²(1 − |z|²)dA(z)

≤ C|I|, for any arc I ⊂ ∂D, where R(I) is the Carleson rectangle

R(I) :=

re^iθ∈ D : 1 − |I|

2π <r < 1 and e^iθ∈ I

.

|I| denotes the length of I and

dA(z) the Lebesgue measure on D.

The corresponding BMOA norm is kf k_BMOA:= |f (0)| + sup

I⊂∂D

1

|I| Z

R(I)

|f⁰(z)|²(1 − |z|²)dA(z)

!1/2

.

VMOA:= (

f ∈ BMOA : lim

|I|→0

1

|I| Z

R(I)

|f⁰(z)|²(1 − |z|²)dA(z) = 0 )

.

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Some spaces of analytic functions: Mean oscillation.

A function f ∈ H²belongs to thespace of analytic functions of bounded mean oscillation BMOAif there exists C > 0 s.t.

Z

R(I)

|f⁰(z)|²(1 − |z|²)dA(z)≤ C|I|, for any arc I ⊂ ∂D, where R(I) is the Carleson rectangle

R(I) :=

re^iθ∈ D : 1 − |I|

.

|I| denotes the length of I and dA(z) the Lebesgue measure on D.

I⊂∂D

1

|I| Z

R(I)

|f⁰(z)|²(1 − |z|²)dA(z)

!1/2

.

VMOA:= (

f ∈ BMOA : lim

|I|→0

1

|I| Z

R(I)

|f⁰(z)|²(1 − |z|²)dA(z) = 0 )

.

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Some spaces of analytic functions: Mean oscillation.

Z

R(I)

R(I) :=

re^iθ∈ D : 1 − |I|

.

I⊂∂D

1

|I|

Z

R(I)

|f⁰(z)|²(1 − |z|²)dA(z)

!1/2

.

VMOA:= (

f ∈ BMOA : lim

|I|→0

1

|I| Z

R(I)

|f⁰(z)|²(1 − |z|²)dA(z) = 0 )

.

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Some spaces of analytic functions: Mean oscillation.

Z

R(I)

R(I) :=

re^iθ∈ D : 1 − |I|

.

I⊂∂D

1

|I|

Z

R(I)

|f⁰(z)|²(1 − |z|²)dA(z)

!1/2

.

VMOA:=

(

f ∈ BMOA : lim

|I|→0

1

|I|

Z

R(I)

|f⁰(z)|²(1 − |z|²)dA(z) = 0 )

.

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Some spaces of analytic functions: Mean oscillation.

SiskakisandZhao:

T_g :BMOA → BMOA ⇐⇒ T_g :VMOA → VMOA

⇐⇒ g ∈ BMOA_log.

T_g is compact on BMOA ⇐⇒ T_g is compact on VMOA

⇐⇒ g ∈ VMOA_log.

Laitila,Miihkinen, andNieminen:

T_gis weakly compact on BMOA ⇐⇒ T_gis compact on BMOA

⇐⇒ T_g(BMOA) ⊂ VMOA. This result was obtained independently byBlasco,C.,

D´ıaz-Madrigal,Mart´ınez,Papadimitrakis, andSiskakis.

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Some spaces of analytic functions: Mean oscillation.

SiskakisandZhao:

T_g :BMOA → BMOA ⇐⇒ T_g :VMOA → VMOA

⇐⇒ g ∈ BMOA_log.

T_g is compact on BMOA ⇐⇒ T_g is compact on VMOA

⇐⇒ g ∈ VMOA_log. Laitila,Miihkinen, andNieminen:

T_g is weakly compact on BMOA ⇐⇒ T_gis compact on BMOA

⇐⇒ T_g(BMOA) ⊂ VMOA.

This result was obtained independently byBlasco,C., D´ıaz-Madrigal,Mart´ınez,Papadimitrakis, andSiskakis.

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Some spaces of analytic functions: Bloch spaces.

A function f ∈ H(D) belongs to theBloch space B if

||f ||B := |f (0)| + sup

z∈D

(1 − |z|²)|f⁰(z)| < +∞.

Thelittle Bloch space B₀is defined by the fact that

|z|→1lim (1 − |z|²)|f⁰(z)| = 0.

Yonedacharacterized the boundedness and compactness of Tg

on B and B₀.

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Some spaces of analytic functions: Bloch spaces.

A function f ∈ H(D) belongs to theBloch space B if

||f ||B := |f (0)| + sup

z∈D

(1 − |z|²)|f⁰(z)| < +∞.

Thelittle Bloch space B₀is defined by the fact that

|z|→1lim (1 − |z|²)|f⁰(z)| = 0.

Yonedacharacterized the boundedness and compactness of Tg

on B and B₀.

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Other classical spaces of analytic functions.

T_ghas been studied between many other spaces:

AlemanandSiskakisstudied Tg betweenA^pα.

PauandPel ´aezstudied Tg betweenA^p(ω)with ω a rapidly decreasing weight.

Galanopoulos,Girela, andPel ´aezstudied Tg between Dirichlet spacesD^p_α.

BonetandTaskinenstudied Tg betweenHv(C). ...

But almost nothing was known about H^∞. Problem

What happens with H^∞? and A?

Boundedness of Tg :X → H^∞? and Tg :X → A?

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Other classical spaces of analytic functions.

T_ghas been studied between many other spaces:

AlemanandSiskakisstudied Tg betweenA^pα.

PauandPel ´aezstudied Tg betweenA^p(ω)with ω a rapidly decreasing weight.

Galanopoulos,Girela, andPel ´aezstudied Tg between Dirichlet spacesD^p_α.

BonetandTaskinenstudied Tg betweenHv(C). ...

But almost nothing was known about H^∞. Problem

What happens with H^∞? and A?

Boundedness of Tg :X → H^∞? and Tg :X → A?

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Boundedness on H

^∞

g(z) = g(0) + Z z

0

1g⁰(ξ)d ξ = g(0) + Tg(1)(z).

So Remark

g ∈ H(D) and Tg :X → H^∞⇒ g ∈ H^∞.

The converse is not true. First non-trivial results: Example (Anderson, Jovovic, Smith, 2014)

1 There is a univalent function g ∈ A such that T_g is not bounded on H^∞.

2 If g is an interpolating Blaschke product with zero

sequence {a_k} contained in (0, 1), then Tg is not bounded on H^∞.

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Boundedness on H

^∞

g(z) = g(0) + Z z

0

1g⁰(ξ)d ξ = g(0) + Tg(1)(z).

So Remark

g ∈ H(D) and Tg :X → H^∞⇒ g ∈ H^∞. The converse is not true. First non-trivial results:

Example (Anderson, Jovovic, Smith, 2014)

1 There is a univalent function g ∈ A such that T_g is not bounded on H^∞.

2 If g is an interpolating Blaschke product with zero

sequence {a_k} contained in (0, 1), then Tg is not bounded on H^∞.

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Boundedness on H

^∞

: a geometric sufficient condition

Theorem

Let g ∈ B. Suppose that there is M > 0 and a dense set E ⊂ ∂D such that,

for all ζ ∈ E , there exists a Jordan arc Γ_ζin D ∪ {ζ} from 0 to ζ

with Z

Γ_ζ

|g⁰(s)| |ds| ≤ M.

Then T_g :H^∞→ H^∞is bounded.

The space of analytic functions with bounded radial variation is BRV:=

(

g ∈ H(D) : sup

0≤θ≤2π

Z 1 0

|g⁰(te^iθ)|dt < ∞ )

.

Corollary (Anderson, Jovovic, Smith, 2014)

g ∈ BRV ⇒ T_g :H^∞→ H^∞is bounded.

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Boundedness on H

^∞

: a geometric sufficient condition

Theorem

with Z

Γ_ζ

|g⁰(s)| |ds| ≤ M.

The space of analytic functions with bounded radial variation is BRV:=

(

g ∈ H(D) : sup

0≤θ≤2π

Z 1 0

|g⁰(te^iθ)|dt < ∞ )

.

Corollary (Anderson, Jovovic, Smith, 2014)

g ∈ BRV ⇒ T_g :H^∞→ H^∞is bounded.

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Boundedness on H

^∞

: a geometric sufficient condition

Anderson, Clunie, and Pommerenke:

LetCbe a circle or a line withC∩ ∂D 6= ∅ and letGbe a domain with

G⊂ D \C, A:= ∂G\C⊂ D.

If h ∈ B, then

dist(h(z), h(A)) ≤ e γ

2 sin γkhkB (z ∈G),

where γ is the angle betweenCand ∂D (like in the pictures).

The point is that to control |h| on G we have to know a bound of |h| only in Aand not onC.

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Boundedness on H

^∞

: a geometric sufficient condition

Anderson, Clunie, and Pommerenke:

LetCbe a circle or a line withC∩ ∂D 6= ∅ and letGbe a domain with

G⊂ D \C, A:= ∂G\C⊂ D.

If h ∈ B, then

dist(h(z), h(A)) ≤ e γ

2 sin γkhkB (z ∈G),

where γ is the angle betweenCand ∂D (like in the pictures).

The point is that to control |h| on G we have to know a bound of |h| only in Aand not onC.

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Boundedness on H

^∞

: a geometric sufficient condition

Theorem

with Z

Γζ

|g⁰(s)| |ds| ≤ M.

Step 1: f ∈ H^∞and g ∈ B ⇒ h := Tg(f ) ∈ B.

Step 2: Using that ζ ∈ E , h is uniformly bounded on Γ_ζ.

(Steps 1 and 2 are easy). Fix γ = π/4.

Step 3: Using the result of Anderson-Clunie- Pommerenke, h is uniformly bounded onΩ. Step 4: Using the density of the set E , h is bounded on D.

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Boundedness on H

^∞

: a geometric sufficient condition

Theorem

with Z

Γζ

|g⁰(s)| |ds| ≤ M.

(Steps 1 and 2 are easy).

Fix γ = π/4.

Step 3: Using the result of Anderson-Clunie- Pommerenke, h is uniformly bounded onΩ. Step 4: Using the density of the set E , h is bounded on D.

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Boundedness on H

^∞

: a geometric sufficient condition

Theorem

with Z

Γζ

|g⁰(s)| |ds| ≤ M.

Step 3: Using the result of Anderson-Clunie- Pommerenke, h is uniformly bounded onΩ.

Step 4: Using the density of the set E , h is bounded on D.

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Boundedness on H

^∞

: a geometric sufficient condition

Theorem

with Z

Γζ

|g⁰(s)| |ds| ≤ M.

Step 3: Using the result of Anderson-Clunie- Pommerenke, h is uniformly bounded onΩ.

Step 4: Using the density of the set E , h is bounded on D.

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T

_g

from quotients of L

^p

into H

^∞

Theorem

Let (Ω, µ) be a finite measure space. Let K : D × Ω → C be s. t.

1 The function z ∈ D 7→ K (z, w) is holomorphic for all w ∈ Ω;

2 The function w ∈ Ω 7→ K (z, w ) is measurable for all z ∈ D.

Write

P(h)(z) = Z

Ω

h(w )K (z, w ) d µ(w ).

Take g ∈ H^∞and X a Banach space of analytic functions.

Consider 1 < p ≤ ∞, ¹_p+ _p¹0 =1. If P : L^p(Ω) →X is bounded and onto, then T_g :X → H^∞is bounded if and only if

sup

z∈D

Z

Ω

Z z 0

g⁰(ζ)K (ζ, w ) d ζ

p⁰

d µ(w ) < ∞.

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T

_g

from quotients of L

^∞

into H

^∞

: B

Bergman projection P : L^∞(D) → B is given by P(h)(z) =

Z

D

h(w ) 1

(1 − zw )²dA(w ), (z ∈ D).

Bloch space:

TFAE

1 T_g : B →H^∞is bounded;

2 Tg : B₀→ H^∞is bounded;

3

sup

z∈D

Z

D

Z z 0

g⁰(ζ) 1

(1 − ζw )²d ζ

dA(w ) < ∞.

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T

_g

from quotients of L

^∞

into H

^∞

: BMOA

Szeg ¨o projection P : L^∞(∂D) → BMOA is given by P(h)(z) = 1

2π Z 2π

0

h(θ)

1 − ze^−iθ d θ, (z ∈ D).

BMOA:

TFAE

1 Tg :BMOA → H^∞is bounded;

2 Tg :VMOA → H^∞is bounded;

3

sup

z∈D

Z 2π 0

Z z 0

g⁰(ζ) 1 − e^−iθζ d ζ

d θ < ∞.

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T

_g

from quotients of L

^p

into H

^∞

: H

^p

, A

^p_α

, Dirichlet, ...

1 ≤ p < +∞ and α > −1:

A^p_α:=

f ∈ H(D) : Z

D

|f (z)|^p(1 − |z|²)^αdA(z) < +∞

. Bergman projection P : L^p(D) → A^pα is given by

P(h)(z) = Z

D

h(w ) 1

(1 − zw )^2+α(1 − |w |²)^αdA(w ), (z ∈ D).

Bergman space:

Let 1 < p < ∞, _p¹+_p¹0 =1, −1 < α < ∞.

T_g:A^p_α → H^∞is bounded if and only if

sup

z∈D

Z

D

Z z 0

g⁰(ζ) 1

(1 − ζw )^2+αd ζ

p⁰

(1 − |w |²)^αdA(w ) < ∞.

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T

_g

from quotients of L

^p

into H

^∞

: H

^p

, A

^p_α

, Dirichlet, ...

Bergman space:

Let 1 < p < ∞, −1 < α < ∞.

Tg:A^pα → H^∞is bounded if and only if

sup

z∈D

Z

D

Z z 0

g⁰(ζ) 1

(1 − ζw )^2+αd ζ

p⁰

(1 − |w |²)^αdA(w ) < ∞.

Let us take g(z) = z.

Z

D

Z z 0

1

(1 − ζw )^2+αd ζ

p⁰

(1 − |w |²)^αdA(w ) Z

D

(1 − |w |²)^α

|1 − zw |^(1+α)p⁰ dA(w )

sup

z∈D

Z

D

(1 − |w |²)^α

|1 − zw |^(1+α)p⁰ dA(w ) < +∞ ⇔ α <p − 2. Corollary

Let 1 < p < ∞, −1 < α < p − 2 and g is a polynomial. Then T_g:A^p_α→ H^∞is bounded.

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T

_g

from quotients of L

^p

into H

^∞

: H

^p

, A

^p_α

, Dirichlet, ...

Bergman space:

Let 1 < p < ∞, −1 < α < ∞.

sup

z∈D

Z

D

Z z 0

g⁰(ζ) 1

(1 − ζw )^2+αd ζ

p⁰

(1 − |w |²)^αdA(w ) < ∞.

Z

D

Z z 0

1

(1 − ζw )^2+αd ζ

p⁰

(1 − |w |²)^αdA(w ) Z

D

(1 − |w |²)^α

|1 − zw |^(1+α)p⁰ dA(w ) sup

z∈D

Z

D

(1 − |w |²)^α

|1 − zw |^(1+α)p⁰ dA(w ) < +∞ ⇔

α <p − 2. Corollary

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T

_g

from quotients of L

^p

into H

^∞

: H

^p

, A

^p_α

, Dirichlet, ...

Bergman space:

Let 1 < p < ∞, −1 < α < ∞.

sup

z∈D

Z

D

Z z 0

g⁰(ζ) 1

(1 − ζw )^2+αd ζ

p⁰

(1 − |w |²)^αdA(w ) < ∞.

Z

D

Z z 0

1

(1 − ζw )^2+αd ζ

p⁰

(1 − |w |²)^αdA(w ) Z

D

(1 − |w |²)^α

|1 − zw |^(1+α)p⁰ dA(w ) sup

z∈D

Z

D

(1 − |w |²)^α

|1 − zw |^(1+α)p⁰ dA(w ) < +∞ ⇔ α <p − 2.

Corollary

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T

_g

from quotients of L

^p

into H

^∞

: H

^p

, A

^p_α

, Dirichlet, ...

Bergman space:

Let 1 < p < ∞, −1 < α < ∞.

sup

z∈D

Z

D

Z z 0

g⁰(ζ) 1

(1 − ζw )^2+αd ζ

p⁰

(1 − |w |²)^αdA(w ) < ∞.

Z

D

Z z 0

1

(1 − ζw )^2+αd ζ

p⁰

(1 − |w |²)^αdA(w ) Z

D

(1 − |w |²)^α

|1 − zw |^(1+α)p⁰ dA(w ) sup

z∈D

Z

D

(1 − |w |²)^α

|1 − zw |^(1+α)p⁰ dA(w ) < +∞ ⇔ α <p − 2.

Corollary

Let 1 < p < ∞, −1 < α < p − 2 and g is a polynomial.

Then T_g:A^p_α→ H^∞is bounded.

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T

_g

from quotients of L

^p

into H

^∞

: H

^p

, A

^p_α

, Dirichlet, ...

Bergman space:

Let 1 < p < ∞, −1 < α < ∞.

T_g:A^p_α→ H^∞is bounded if and only if

sup

z∈D

Z

D

Z z 0

g⁰(ζ) 1

(1 − ζw )^2+αd ζ

p⁰

(1 − |w |²)^αdA(w ) < ∞.

What if α ≥ p − 2?

To simplify p = 2 (Hilbert case) and α = 0: g(z) =P∞

n=1anzⁿ6= 0 (assuming g(0) = 0) sup

z

Z

D

Z z 0

g⁰(ζ) 1 (1 − ζw )²d ζ

2

dA(w ) ≥ ... ≥

≥ sup

r <1

∞

X

k =0

1 k + 1

∞

X

n=0

|a_n+1|²r²⁽ⁿ⁺¹⁾

!

r^2k ≈ kgk_H2sup

r <1

∞

X

k =0

1

k + 1r^2k = ∞.

(48)

T

_g

from quotients of L

^p

into H

^∞

: H

^p

, A

^p_α

, Dirichlet, ...

Bergman space:

Let 1 < p < ∞, −1 < α < ∞.

sup

z∈D

Z

D

Z z 0

g⁰(ζ) 1

(1 − ζw )^2+αd ζ

p⁰

(1 − |w |²)^αdA(w ) < ∞.

What if α ≥ p − 2? To simplify p = 2 (Hilbert case) and α = 0:

g(z) =P∞

z

Z

D

Z z 0

g⁰(ζ) 1 (1 − ζw )²d ζ

2

dA(w )

≥ ... ≥

≥ sup

r <1

∞

X

k =0

1 k + 1

∞

X

n=0

|a_n+1|²r²⁽ⁿ⁺¹⁾

!

r^2k ≈ kgk_H2sup

r <1

∞

X

k =0

1

k + 1r^2k = ∞.

(49)

T

_g

from quotients of L

^p

into H

^∞

: H

^p

, A

^p_α

, Dirichlet, ...

Bergman space:

Let 1 < p < ∞, −1 < α < ∞.

sup

z∈D

Z

D

Z z 0

g⁰(ζ) 1

(1 − ζw )^2+αd ζ

p⁰

(1 − |w |²)^αdA(w ) < ∞.

What if α ≥ p − 2? To simplify p = 2 (Hilbert case) and α = 0:

g(z) =P∞

z

Z

D

Z z 0

g⁰(ζ) 1 (1 − ζw )²d ζ

2

dA(w ) ≥ ... ≥

≥ sup

r <1

∞

X

k =0

1 k + 1

∞

X

n=0

|a_n+1|²r²⁽ⁿ⁺¹⁾

!

r^2k ≈ kgk_H2sup

r <1

∞

X

k =0

1

k + 1r^2k = ∞.

(50)

T

_g

from quotients of L

^p

into H

^∞

: H

^p

, A

^p_α

, Dirichlet, ...

Then Tg :A^pα→ H^∞is bounded.

Theorem

1 Assume that 1 < p < ∞ and α ≥ p − 2.

If Tg:A^pα → H^∞, then g is constant.

2 Assume that α > −1.

If Tg:A¹_α → H^∞, then g is constant.

(51)

T

_g

from quotients of L

^p

into H

^∞

: H

^p

, A

^p_α

, Dirichlet, ...

Then Tg :A^pα→ H^∞is bounded.

Theorem

1 Assume that 1 < p < ∞ and α ≥ p − 2.

If Tg:A^pα → H^∞, then g is constant.

2 Assume that α > −1.

If Tg:A¹_α → H^∞, then g is constant.

(52)

Weak Compactness on H

^∞

Theorem

If Tg :H^∞→ H^∞is weakly compact, then g ∈ A.

Corollary

Assume that Tg :H^∞→ H^∞is bounded. TFAE:

1 T_g :H^∞→ H^∞is weakly compact;

2 T_g :H^∞→ A;

3 T_g :A → A is weakly compact. Working in A is easier:

T_g:A → A is weakly compact if and only if given a bounded sequence in A, (f_n), that goes to zero uniformly on compacta in D, we have that (Tg(fn))goes to zero pointwise in ∂D.

(53)

Weak Compactness on H

^∞

Theorem

Corollary

2 T_g :H^∞→ A;

3 T_g :A → A is weakly compact.

Working in A is easier:

(54)

Weak Compactness on H

^∞

Theorem

Corollary

2 T_g :H^∞→ A;

3 T_g :A → A is weakly compact.

Working in A is easier: