Closed range composition operators on BMOA Maria Tjani University - PowerPoint PPT Presentation

Closed range composition operators on BMOA Maria Tjani University of Arkansas Joint work with Kevser Erdem University of Arkansas CAFT 2018 University of Crete July 2-6, 2018 Maria Tjani C ϕ : BMOA → BMOA

Notation D = { z ∈ C : | z | < 1 } T = { z ∈ C : | z | = 1 } ϕ : D → D analytic self map of D α q ( z ) = q − z 1 − qz , q ∈ D Mobius transformation D ( q , r ) = { z ∈ D : | α q ( z ) | < r } , r ∈ (0 , 1) H ( D ) is the set of analytic functions on D Maria Tjani C ϕ : BMOA → BMOA

Composition operators C ϕ f = f ◦ ϕ C ϕ is a linear operator Let ϕ be a non constant self map of D By the Open Mapping Theorem for analytic functions, ϕ ( D ) is an open subset of D C ϕ : H ( D ) → H ( D ) is one to one Maria Tjani C ϕ : BMOA → BMOA

The Hardy space H 2 H 2 is the Hilbert space of analytic functions f on D ∞ ∞ | a n | 2 < ∞ a n z n , || f || 2 � � f ( z ) = H 2 = n =0 n =0 Maria Tjani C ϕ : BMOA → BMOA

The Hardy space H 2 H 2 is the Hilbert space of analytic functions f on D ∞ ∞ | a n | 2 < ∞ a n z n , || f || 2 � � f ( z ) = H 2 = n =0 n =0 an equivalent norm on H 2 is given by � H 2 ≍ | f (0) | 2 + || f || 2 (1 − | z | 2 ) | f ′ ( z ) | 2 dA ( z ) D Maria Tjani C ϕ : BMOA → BMOA

BMOA BMOA is the Banach space of analytic functions on D || f || G = sup || f ◦ α q − f ( q ) || H 2 < ∞ q ∈ D Maria Tjani C ϕ : BMOA → BMOA

BMOA BMOA is the Banach space of analytic functions on D || f || G = sup || f ◦ α q − f ( q ) || H 2 < ∞ q ∈ D � | f ′ ( z ) | 2 (1 − | α q ( z ) | 2 ) dA ( z ) || f || 2 ∗ = sup q ∈ D D Maria Tjani C ϕ : BMOA → BMOA

BMOA BMOA is the Banach space of analytic functions on D || f || G = sup || f ◦ α q − f ( q ) || H 2 < ∞ q ∈ D � | f ′ ( z ) | 2 (1 − | α q ( z ) | 2 ) dA ( z ) || f || 2 ∗ = sup q ∈ D D The norm we will use in BMOA is: || f || BMOA = | f (0) | + || f || ∗ Maria Tjani C ϕ : BMOA → BMOA

C ϕ f = f ◦ ϕ Composition operators on BMOA C ϕ is always a bounded operator on BMOA Maria Tjani C ϕ : BMOA → BMOA

C ϕ f = f ◦ ϕ Composition operators on BMOA C ϕ is always a bounded operator on BMOA � || f ◦ ϕ || 2 | ( f ◦ ϕ ) ′ ( z ) | 2 (1 − | α q ( z ) | 2 ) dA ( z ) = sup ∗ q ∈ D D � | f ′ ( ϕ ( z )) | 2 | ϕ ′ ( z ) | 2 (1 − | α q ( z ) | 2 ) dA ( z ) = sup q ∈ D D Maria Tjani C ϕ : BMOA → BMOA

C ϕ f = f ◦ ϕ Composition operators on BMOA C ϕ is always a bounded operator on BMOA � || f ◦ ϕ || 2 | ( f ◦ ϕ ) ′ ( z ) | 2 (1 − | α q ( z ) | 2 ) dA ( z ) = sup ∗ q ∈ D D � | f ′ ( ϕ ( z )) | 2 | ϕ ′ ( z ) | 2 (1 − | α q ( z ) | 2 ) dA ( z ) = sup q ∈ D D � | f ′ ( ζ ) | 2 � (1 − | α q ( z ) | 2 ) dA ( ζ ) = sup q ∈ D D ϕ ( z )= ζ Maria Tjani C ϕ : BMOA → BMOA

Counting functions on BMOA For each q ∈ D we define the BMOA counting function by � (1 − | α q ( z ) | 2 ) N q ,ϕ ( ζ ) = ϕ ( z )= ζ if ζ �∈ ϕ ( D ), N q ,ϕ ( ζ ) = 0 Maria Tjani C ϕ : BMOA → BMOA

Counting functions on BMOA For each q ∈ D we define the BMOA counting function by � (1 − | α q ( z ) | 2 ) N q ,ϕ ( ζ ) = ϕ ( z )= ζ if ζ �∈ ϕ ( D ), N q ,ϕ ( ζ ) = 0 � || C ϕ f || 2 | f ′ ( ζ ) | 2 N q ,ϕ ( ζ ) dA ( ζ ) = sup ∗ q ∈ D D const . || f || 2 ≤ ∗ Maria Tjani C ϕ : BMOA → BMOA

C ϕ bounded below, closed range on BMOA C ϕ is bounded below on BMOA ⇔ ∃ C > 0 such that ∀ f ∈ BMOA || f || BMOA ≤ C || f ◦ ϕ || BMOA Maria Tjani C ϕ : BMOA → BMOA

C ϕ bounded below, closed range on BMOA C ϕ is bounded below on BMOA ⇔ ∃ C > 0 such that ∀ f ∈ BMOA || f || BMOA ≍ || f ◦ ϕ || BMOA Maria Tjani C ϕ : BMOA → BMOA

C ϕ bounded below, closed range on BMOA C ϕ is bounded below on BMOA ⇔ ∃ C > 0 such that ∀ f ∈ BMOA || f || BMOA ≍ || f ◦ ϕ || BMOA C ϕ is bounded below on BMOA ⇔ ∀ f ∈ BMOA || f ◦ ϕ || ∗ ≍ || f || ∗ Maria Tjani C ϕ : BMOA → BMOA

C ϕ bounded below, closed range on BMOA C ϕ is bounded below on BMOA ⇔ ∃ C > 0 such that ∀ f ∈ BMOA || f || BMOA ≍ || f ◦ ϕ || BMOA C ϕ is bounded below on BMOA ⇔ ∀ f ∈ BMOA || f ◦ ϕ || ∗ ≍ || f || ∗ We say that C ϕ is closed range in BMOA if C ϕ ( BMOA ) is closed in BMOA C ϕ is closed range ⇔ C ϕ is bounded below Maria Tjani C ϕ : BMOA → BMOA

C ϕ closed range C ϕ : H 2 → H 2 J. Cima, J. Thomson, W. Wogen (1973) ν ( E ) := m ( ϕ − 1 ( E )), for E any Borel subset of T C ϕ closed range on H 2 ⇔ d ν dm essentially bounded away from 0 Maria Tjani C ϕ : BMOA → BMOA

C ϕ closed range C ϕ : H 2 → H 2 J. Cima, J. Thomson, W. Wogen (1973) ν ( E ) := m ( ϕ − 1 ( E )), for E any Borel subset of T C ϕ closed range on H 2 ⇔ d ν dm essentially bounded away from 0 N. Zorboska (1994), K. Luery (2013) and P. Ghatage, MT (2014) Maria Tjani C ϕ : BMOA → BMOA

Sampling sets in BMOA We define H ⊆ D to be a sampling set for BMOA if for all f ∈ BMOA � | f ′ ( z ) | 2 (1 − | α q ( z ) | 2 ) dA ( z ) ≍ || f || 2 sup ∗ . q ∈ D H Maria Tjani C ϕ : BMOA → BMOA

Sampling sets in BMOA We define H ⊆ D to be a sampling set for BMOA if for all f ∈ BMOA � | f ′ ( z ) | 2 (1 − | α q ( z ) | 2 ) dA ( z ) ≍ || f || 2 sup ∗ . q ∈ D H For each ε > 0 and q ∈ D G ε, q = { ζ : N q ,ϕ ( ζ ) > ε (1 − | α q ( ζ ) | 2 ) } C ϕ is closed range on BMOA ⇒ ∃ ε > 0 such that ∪ q ∈ D G ε, q is a sampling set for BMOA Maria Tjani C ϕ : BMOA → BMOA

Sampling sets in BMOA We define H ⊆ D to be a sampling set for BMOA if for all f ∈ BMOA � | f ′ ( z ) | 2 (1 − | α q ( z ) | 2 ) dA ( z ) ≍ || f || 2 sup ∗ . q ∈ D H For each ε > 0 and q ∈ D G ε, q = { ζ : N q ,ϕ ( ζ ) > ε (1 − | α q ( ζ ) | 2 ) } C ϕ is closed range on BMOA ⇒ ∃ ε > 0 such that ∪ q ∈ D G ε, q is a sampling set for BMOA ∩ q ∈ D G ε, q is a sampling set for BMOA ⇒ C ϕ is closed range on BMOA Maria Tjani C ϕ : BMOA → BMOA

Carleson measures Let µ be a finite positive Borel measure on D . We say that µ is a (Bergman space) Carleson measure on D if there exists c > 0 such that for all f ∈ A 2 � � | f ( z ) | 2 d µ ( z ) ≤ c | f ( z ) | 2 dA ( z ) D D Let 0 < r < 1. Then µ is a Carleson measure if and only if there exists c r > 0 such that for all q ∈ D , µ ( D ( q , r )) ≤ c r A ( D ( q , r )) Maria Tjani C ϕ : BMOA → BMOA

Carleson measures The Berezin symbol of µ is � q ( z ) | 2 d µ ( z ) , | α ′ µ ( q ) = ˜ q ∈ D . D µ is Carleson measure if and only if ˜ µ is a bounded function on D Maria Tjani C ϕ : BMOA → BMOA

Carleson measures The Berezin symbol of µ is � q ( z ) | 2 d µ ( z ) , | α ′ µ ( q ) = ˜ q ∈ D . D µ is Carleson measure if and only if ˜ µ is a bounded function on D µ q ′ , q ′ ∈ D , is a collection of uniformly Carleson measures if and only if the Berezin symbols of µ q ′ , for all q ′ ∈ D , are uniformly bounded in D . Recall � || C ϕ f || 2 | f ′ ( ζ ) | 2 N q ,ϕ ( ζ ) dA ( ζ ) ≤ const . || f || 2 = sup ∗ ∗ q ∈ D D The measures N q ,ϕ ( ζ ) dA ( ζ ) are uniformly Carleson measures Maria Tjani C ϕ : BMOA → BMOA

Dan Luecking and the RCC Let µ be a finite positive Carleson measure on D . We say that µ satisfies the reverse Carleson condition if ∃ r ∈ (0 , 1) such that µ ( D ( q , r )) ≍ A ( D ( q , r )) , q ∈ D G ⊂ D satisfies the reverse Carleson condition if the Carleson measure χ G ( z ) dA ( z ) satisfies the reverse Carleson condition. Luecking ⇔ � � | f ( z ) | 2 dA ( z ) ≤ C | f ( z ) | 2 dA ( z ) , ∀ f ∈ A 2 D G ⇔ A ( G ∩ D ( q , r )) ≍ A ( D ( q , r )), q ∈ D Maria Tjani C ϕ : BMOA → BMOA

G ε, q ′ , q A subset H of D satisfies the reverse Carleson condition if and only if H is a sampling set for BMOA . Maria Tjani C ϕ : BMOA → BMOA

G ε, q ′ , q A subset H of D satisfies the reverse Carleson condition if and only if H is a sampling set for BMOA . For each ε > 0 and q , q ′ ∈ D G ε, q ′ , q = { ζ : N q ′ ,ϕ ( ζ ) > ε (1 − | α q ( ζ ) | 2 ) } Maria Tjani C ϕ : BMOA → BMOA

G ε, q ′ , q A subset H of D satisfies the reverse Carleson condition if and only if H is a sampling set for BMOA . For each ε > 0 and q , q ′ ∈ D G ε, q ′ , q = { ζ : N q ′ ,ϕ ( ζ ) > ε (1 − | α q ( ζ ) | 2 ) } ∃ k > 0 ∀ q ∈ D || α q ◦ ϕ || ∗ ≥ k ⇔ ∃ ε > 0, r ∈ (0 , 1) such that ∀ q ∈ D , ∃ q ′ ∈ D such that | G ε, q ′ , q ∩ D ( q , r ) | ≍ 1 . (1) | D ( q , r ) | Maria Tjani C ϕ : BMOA → BMOA

RCC - Geometry of disks - Luecking 1981 Given a measurable set F , the following are equivalent: ∃ δ > 0, r ∈ (0 , 1) such that ∀ disks D with centers on T , | F ∩ D | > δ | D ∩ D | . ∃ δ 0 > 0, η ∈ (0 , 1) such that ∀ q ∈ D | F ∩ D ( q , η (1 − | q | )) | > δ 0 | D ( q , η (1 − | q | )) | . ∃ δ 1 > 0, r ∈ (0 , 1) such that ∀ q ∈ D | F ∩ D ( q , r ) | > δ 1 | D ( q , r ) | . Maria Tjani C ϕ : BMOA → BMOA