Notation and definitions Morrey spaces Semigroups of composition operators Known results Semigroups on Morrey spaces Strong continuity of semigroups of composition operators on Morrey spaces Noel Merchán Universidad de Málaga, Spain Joint work with Petros Galanopoulos and Aristomenis G. Siskakis New Developments in Complex Analysis and Function Theory Heraklion, Greece. 2-6 July 2018 Noel Merchán noel@uma.es Strong continuity of semigroups of composition operators on Morre
Notation and definitions Morrey spaces Semigroups of composition operators Known results Semigroups on Morrey spaces Index Notation and definitions 1 Morrey spaces 2 Semigroups of composition operators 3 Strong continuity Infinitesimal generator Examples of semigroups Denjoy-Wolff point Motivation Known results 4 Semigroups on Morrey spaces 5 Noel Merchán noel@uma.es Strong continuity of semigroups of composition operators on Morre
Notation and definitions Morrey spaces Semigroups of composition operators Known results Semigroups on Morrey spaces Spaces of analytic functions in the unit disc D = { z ∈ C : | z | < 1 } , the unit disc. H ol ( D ) is the space of all analytic functions in D . Automorphisms on D We consider Aut ( D ) = { ϕ : D → D : ϕ is conformal } . It is known that Aut ( D ) = { λσ a : | λ | = 1 , a ∈ D } z − a where σ a : D → D is the Möbius map σ a ( z ) = 1 − az . Noel Merchán noel@uma.es Strong continuity of semigroups of composition operators on Morre
Notation and definitions Morrey spaces Semigroups of composition operators Known results Semigroups on Morrey spaces Hardy spaces If 0 < r < 1 and f ∈ H ol ( D ) , we set � 1 / p � 2 π � 1 | f ( re it ) | p dt M p ( r , f ) = , 0 < p < ∞ , 2 π 0 M ∞ ( r , f ) = sup | f ( z ) | . | z | = r If 0 < p ≤ ∞ , we consider the Hardy spaces H p , � � H p = f ∈ H ol ( D ) : � f � H p def = sup M p ( r , f ) < ∞ . 0 < r < 1 Noel Merchán noel@uma.es Strong continuity of semigroups of composition operators on Morre
Notation and definitions Morrey spaces Semigroups of composition operators Known results Semigroups on Morrey spaces Bergman spaces If 0 < p < ∞ , we consider the Bergman spaces A p , � � � A p = | f ( z ) | p dA ( z ) < ∞ f ∈ H ol ( D ) : . D BMOA f ∈ H 1 : f � � e i θ � � BMOA = ∈ BMO . H ∞ ⊂ BMOA ⊂ � H p . 0 < p < ∞ Noel Merchán noel@uma.es Strong continuity of semigroups of composition operators on Morre
Notation and definitions Morrey spaces Semigroups of composition operators Known results Semigroups on Morrey spaces Bloch space ( 1 − | z | 2 ) | f ′ ( z ) | < ∞} . B = { f ∈ H ol ( D ) : sup z ∈ D H ∞ ⊂ BMOA ⊂ B . Noel Merchán noel@uma.es Strong continuity of semigroups of composition operators on Morre
Notation and definitions Morrey spaces Semigroups of composition operators Known results Semigroups on Morrey spaces Morrey spaces For 0 < λ < 1 we define the Morrey space L 2 ,λ as � � L 2 ,λ = f ∈ H 2 : sup 1 − λ ( 1 − | a | 2 ) 2 � f ◦ σ a − f ( a ) � H 2 < ∞ . a ∈ D We also define for 0 < λ < 1 the little Morrey spaces L 2 ,λ as 0 � � f ∈ L 2 ,λ : 1 − λ L 2 ,λ | a |→ 1 ( 1 − | a | 2 ) 2 � f ◦ σ a − f ( a ) � H 2 = 0 = lim . 0 For 0 < λ < 1 BMOA = L 2 , 1 ⊂ L 2 ,λ ⊂ L 2 , 0 = H 2 . Noel Merchán noel@uma.es Strong continuity of semigroups of composition operators on Morre
Notation and definitions Morrey spaces Semigroups of composition operators Known results Semigroups on Morrey spaces ∞ z 2 n ∈ B \ L 2 ,λ � f ( z ) = 0 < λ < 1 . n = 0 f ( z ) = ( 1 − z ) − 1 − λ ∈ L 2 ,λ \ B 0 < λ < 1 . 2 Growth in Morrey spaces For 0 < λ < 1 there exists a constant C such that if f ∈ L 2 ,λ then C | f ( z ) | ≤ z ∈ D . 1 − λ ( 1 − | z | ) 2 It follows that L 2 ,λ ⊂ B 3 − λ 0 < λ < 1 . 2 Noel Merchán noel@uma.es Strong continuity of semigroups of composition operators on Morre
Notation and definitions Morrey spaces Semigroups of composition operators Known results Semigroups on Morrey spaces α -Bloch spaces If α > 0 we can consider the spaces B α = { f ∈ H ol ( D ) : sup ( 1 − | z | 2 ) α | f ′ ( z ) | < ∞} . z ∈ D B = B 1 ⊂ B α 1 ⊂ B α 2 , 1 ≤ α 1 ≤ α 2 . Noel Merchán noel@uma.es Strong continuity of semigroups of composition operators on Morre
Notation and definitions Strong continuity Morrey spaces Infinitesimal generator Semigroups of composition operators Examples of semigroups Known results Denjoy-Wolff point Semigroups on Morrey spaces Motivation Semigroups of analytic functions A semigroup ( ϕ t ) for t ≥ 0 consists of analytic functions on D with ϕ t ( D ) ⊂ D which satisfies the following: ϕ 0 is the identity in D . ϕ t + s = ϕ t ◦ ϕ s , for all t , s ≥ 0. ϕ t → ϕ 0 , as t → 0, uniformly on compact subsets of D . Semigroups of composition operators Each semigroup ( ϕ t ) gives rise to a semigroup ( C t ) consisting on composition operators on H ol ( D ) , C t ( f ) = f ◦ ϕ t , f ∈ H ol ( D ) . Noel Merchán noel@uma.es Strong continuity of semigroups of composition operators on Morre
Notation and definitions Strong continuity Morrey spaces Infinitesimal generator Semigroups of composition operators Examples of semigroups Known results Denjoy-Wolff point Semigroups on Morrey spaces Motivation We are going to be interested in the restriction of ( C t ) to certain linear subspaces of H ol ( D ) . Definition Given a Banach space X ⊂ H ol ( D ) and a semigroup ( ϕ t ) , we say that ( ϕ t ) generates a semigroup of operators on X if ( C t ) is a well defined strongly continuous semigroup of bounded operators in X . This means that for every f ∈ X , we have C t ( f ) ∈ X for all t ≥ 0 and t → 0 + � C t ( f ) − f � X = 0 . lim Noel Merchán noel@uma.es Strong continuity of semigroups of composition operators on Morre
Notation and definitions Strong continuity Morrey spaces Infinitesimal generator Semigroups of composition operators Examples of semigroups Known results Denjoy-Wolff point Semigroups on Morrey spaces Motivation Definition For a semigroup ( ϕ t ) we define the infinitesimal generator G of ( ϕ t ) as ϕ t ( z ) − z G ( z ) = lim , z ∈ D . t t → 0 + This convergence holds uniformly on compact subsets of D so G ∈ H ol ( D ) . Moreover G ( ϕ t ( z )) = ∂ϕ t ( z ) = G ( z ) ∂ϕ t ( z ) , z ∈ D , t ≥ 0 . ∂ t ∂ z Noel Merchán noel@uma.es Strong continuity of semigroups of composition operators on Morre
Notation and definitions Strong continuity Morrey spaces Infinitesimal generator Semigroups of composition operators Examples of semigroups Known results Denjoy-Wolff point Semigroups on Morrey spaces Motivation Examples of semigroups Some examples of semigroups are: ϕ t ( z ) = z , t ≥ 0 G ( z ) = 0 (Trivial semigroup). ϕ t ( z ) = e − t z , t ≥ 0 G ( z ) = − z . ϕ t ( z ) = e it z , t ≥ 0 G ( z ) = iz . Noel Merchán noel@uma.es Strong continuity of semigroups of composition operators on Morre
Notation and definitions Strong continuity Morrey spaces Infinitesimal generator Semigroups of composition operators Examples of semigroups Known results Denjoy-Wolff point Semigroups on Morrey spaces Motivation Representation of the infinitesimal generator G has a unique representation G ( z ) = ( bz − 1 )( z − b ) P ( z ) , z ∈ D , where b ∈ D and P ∈ H ol ( D ) with Re P ( z ) ≥ 0 for all z ∈ D . If G �≡ 0, ( b , P ) is uniquely determined from ( ϕ t ) . The point b is called Denjoy-Wolff point of the semigroup. Noel Merchán noel@uma.es Strong continuity of semigroups of composition operators on Morre
Notation and definitions Strong continuity Morrey spaces Infinitesimal generator Semigroups of composition operators Examples of semigroups Known results Denjoy-Wolff point Semigroups on Morrey spaces Motivation Denjoy-Wolff point in the disc Studying the semigroup in the case b ∈ D can be reduced by renormalization to the case b = 0. Then ϕ t ( z ) = h − 1 � e − ct h ( z ) � , where h : D → h ( D ) = Ω is a univalent function with Ω a spirallike domain, h ( 0 ) = 0, Re c ≥ 0 and ω e − ct ∈ Ω for each ω ∈ Ω , t ≥ 0. Noel Merchán noel@uma.es Strong continuity of semigroups of composition operators on Morre
Notation and definitions Strong continuity Morrey spaces Infinitesimal generator Semigroups of composition operators Examples of semigroups Known results Denjoy-Wolff point Semigroups on Morrey spaces Motivation Denjoy-Wolff point in the boundary If b ∈ ∂ D it may be reduced to b = 1. Then ϕ t ( z ) = h − 1 ( h ( z ) + ct ) , where h : D → h ( D ) = Ω is a univalent function with Ω a close-to-convex domain, h ( 0 ) = 0, Re c ≥ 0 and ω + ct ∈ Ω for each ω ∈ Ω , t ≥ 0. Noel Merchán noel@uma.es Strong continuity of semigroups of composition operators on Morre
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