Monge-Amp` ere Measures and Poletsky-Stessin Hardy Spaces on - PowerPoint PPT Presentation

Monge-Amp` ere Measures and Poletsky-Stessin Hardy Spaces on Bounded Hyperconvex Domains Sibel S .ahin Sabancı University, ˙ Istanbul Poletsky-Stessin Hardy Spaces

Outline Preliminaries Poletsky-Stessin Hardy Spaces on Domains Bounded by An Analytic Jordan Curve Composition Operators on Poletsky-Stessin Hardy Spaces Poletsky-Stessin Hardy Spaces

Preliminaries Classical Hardy Spaces-Unit Disc Hardy Spaces on the unit disc are defined for 1 ≤ p < ∞ as [4] : � 2 π ( 1 1 H p ( D ) = { f ∈ O ( D ) : sup | f ( re i θ ) | p d θ ) p < ∞} 2 π 0 < r < 1 0 Poletsky-Stessin Hardy Spaces

Preliminaries Classical Hardy Spaces-Polydisc Hardy spaces on the unit polydisc of C n are defined for 1 ≤ p < ∞ as [5] : 1 � 1 H p ( D n ) = { f ∈ O ( D n ) : sup T n | f ( rz ) | p d µ ) p < ∞} ( (2 π ) n 0 < r < 1 where T n is torus and µ is the usual product measure on the torus. Poletsky-Stessin Hardy Spaces

Preliminaries Classical Hardy Spaces-Unit Ball Hardy spaces on the unit ball of C n are defined for 1 ≤ p < ∞ as [1] : � H p ( B ) = { f ∈ O ( B ) : sup | f ( z ) | p d µ < ∞} 0 < r < 1 S ( r ) where S ( r ) is the sphere with center 0 and radius r and µ is the usual surface area measure on the sphere. Poletsky-Stessin Hardy Spaces

Preliminaries Hyperconvex Domains A connected open subset Ω of C n is called hyperconvex if there exists a plurisubharmonic function g : Ω → [ −∞ , 0) such that { z ∈ Ω : g ( z ) < c } is relatively compact for each c < 0. Here g is called an exhaustion function for Ω. Poletsky-Stessin Hardy Spaces

Preliminaries Monge-Amp` ere Measures Let Ω be a hyperconvex domain in C n and ϕ : Ω → [ −∞ , 0) be a negative, continuous, plurisubharmonic exhaustion for Ω. Define pseudoball: B ϕ ( r ) = { z ∈ Ω : ϕ ( z ) < r } , r ∈ [ −∞ , 0) , and pseudosphere: S ϕ ( r ) = { z ∈ Ω : ϕ ( z ) = r } , r ∈ [ −∞ , 0) , and set ϕ r = max { ϕ, r } , r ∈ ( −∞ , 0) . Poletsky-Stessin Hardy Spaces

Preliminaries Monge-Amp` ere Measures In 1985, Demailly ([4]) introduced the Monge-Amp` ere measures in the sense of currents as : µ r = ( dd c ϕ r ) n − χ Ω \ B ϕ ( r ) ( dd c ϕ ) n r ∈ ( −∞ , 0) which is supported on S ϕ ( r ) Monge-Amp` ere Mass ere mass of an exhaustion function u on Ω ⊂ C n The Monge-Amp` is defined as: � ( dd c u ) n MA ( u ) = Ω Poletsky-Stessin Hardy Spaces

Preliminaries Pluricomplex Green Function Pluricomplex Green function of Ω ⊂ C n is defined as : g Ω ( z , w ) = sup u ( z ) where u ∈ PSH (Ω) (including u ≡ −∞ ), u is non-positive and the function t → u ( t ) − log | t − w | is bounded from above in a neighborhood of w . Pluricomplex Green function g Ω ( z , w ) is a negative plurisubharmonic function with a logarithmic pole at w . Poletsky-Stessin Hardy Spaces

Preliminaries Lelong-Jensen Formula, (Demailly ’87) Let r < 0 and φ be a plurisubharmonic function on Ω then for any negative, continuous, plurisubharmonic exhaustion function u � � � φ ( dd c u ) n = ( r − u ) dd c φ ( dd c u ) n − 1 (1) φ d µ u , r − S u ( r ) B u ( r ) B u ( r ) Poletsky-Stessin Hardy Spaces, (Poletsky & Stessin ’08) H p ϕ (Ω), p > 0, is the space of all holomorphic functions f on Ω such that � | f | p d µ ϕ, r < ∞ lim sup r → 0 − Ω The norm on these spaces is given by: � 1 � � p | f | p d µ ϕ, r � f � H p ϕ = lim r → 0 − Ω Poletsky-Stessin Hardy Spaces

Preliminaries PS-Hardy Norm Let Ω be a hyperconvex domain in C n with an exhaustion function u such that the set L ( u ) = { z ∈ Ω | u ( z ) = −∞} is finite. If f is a holomorphic function on Ω then � � | f | p ( dd c u ) n + ( − u ) dd c | f | p ∧ ( dd c u ) n − 1 � f � p u (Ω) = H p Ω Ω Basic Facts H p ϕ (Ω) are Banach spaces ([2]). When exhaustion function is chosen as the Pluricomplex Green function, the Poletsky-Stessin Hardy classes coincide with the classical Hardy spaces in unit disc, polydisc and unitball cases. Poletsky-Stessin Hardy Spaces

Domains Bounded by An Analytic Jordan Curve in C Theorem Let Ω be a domain in C containing 0 and bounded by an analytic Jordan curve. Suppose ϕ is a continuous, negative, subharmonic exhaustion function for Ω such that ϕ is harmonic out of a compact set K ⊂ Ω. Then for a holomorphic function f ∈ O (Ω), ϕ (Ω) if and only if | f | p has a harmonic majorant. f ∈ H p Corollary Let Ω be a domain in C containing 0 and bounded by an analytic Jordan curve. Suppose g Ω ( z , w ) is the Green function of Ω with the logarithmic pole at w ∈ Ω. Then H p (Ω) = H p g Ω (Ω) for p ≥ 1. Poletsky-Stessin Hardy Spaces

Domains Bounded by An Analytic Jordan Curve in C Exhaustion with Finite Monge-Amp` ere Mass There exists an exhaustion function u with finite Monge-Amp` ere mass such that the Hardy space H 1 u ( D ) � H 1 ( D ). Boundary Monge-Amp` ere Measure (Demailly ’87) Let ϕ : Ω → [ −∞ , 0) be a continuous plurisubharmonic exhaustion function for Ω. Suppose that the total Monge-Amp` ere mass of ϕ is finite, i.e. � ( dd c ϕ ) n < ∞ (2) Ω Then as r tends to 0, the measures µ r converge to a positive Ω ( dd c ϕ ) n and � measure µ weak*-ly on Ω with total mass supported on ∂ Ω. This limit measure µ is called the boundary Monge-Amp` ere measure associated with the exhaustion ϕ . Poletsky-Stessin Hardy Spaces

Domains Bounded by An Analytic Jordan Curve in C Remark In the case of the unit disc, using ϕ 1 ( z ) = log | z | as the exhaustion function we get µ ϕ 1 = d θ (3) where d θ is the usual Lebesgue measure on the circle. For the unit ball of C n when we use ϕ 2 ( z ) = log � z � as the exhaustion function we obtain 1 µ ϕ 2 = σ ( S ) d σ (4) which is the normalized surface area measure on the sphere. Now consider the polydisc D n ⊂ C n with ϕ 3 ( z ) = log(max | z j | ) as the exhaustion function, we have 1 µ ϕ 3 = (2 π ) n d θ 1 d θ 2 . . . d θ n (5) which is the usual product measure on the torus. Poletsky-Stessin Hardy Spaces

Domains Bounded by An Analytic Jordan Curve in C Let Ω be a domain in C containing 0 and bounded by an analytic Jordan curve and u be a continuous, negative, subharmonic exhaustion function for Ω with finite Monge-Amp` ere mass. Since we have H p u (Ω) ⊂ H p (Ω), any holomorphic function f ∈ H p u (Ω) has the boundary value function f ∗ from the classical theory ([3]). Boundary Value Characterization Let f ∈ H p (Ω) be a holomorphic function and u be a continuous, negative, subharmonic exhaustion function for Ω. Then f ∈ H p u (Ω) if and only if f ∗ ∈ L p ( d µ u ) for 1 ≤ p < ∞ . Moreover � f ∗ � L p ( d µ u ) = � f � H p u (Ω) . Poletsky-Stessin Hardy Spaces

Domains Bounded by An Analytic Jordan Curve in C Factorization Let f ∈ H p u (Ω), 1 ≤ p < ∞ , where u is a continuous exhaustion function with finite Monge-Amp` ere mass. Then f can be factored as f = IF where I is Ω-inner and F is Ω-outer. Moreover I ∈ H p u (Ω) and F ∈ H p u (Ω). Let A (Ω) be the algebra of the functions which are holomorphic in Ω and continuous on ∂ Ω. Using functional analysis techniques we have : Approximation The algebra A (Ω) is dense in H p u (Ω), 1 ≤ p < ∞ . Poletsky-Stessin Hardy Spaces

Domains Bounded by An Analytic Jordan Curve in C Composition Operators In 2003, Shapiro and Smith showed that on a domain Ω which is bounded by an analytic Jordan curve, every holomorphic self map φ of Ω induces a bounded composition operator on the classical Hardy space H p (Ω). However this is not the case when Poletsky-Stessin Hardy classes are concerned. Counter Example 1 ∈ H 1 The function 4 / u ( D ). Consider the composition operator 3 ( z − 1) C φ ( f ) = f ◦ φ with symbol φ ( z ) = ze i π 2 , then the function 1 4 is in H 1 ∈ H 1 u ( D ) but C φ ( f ) / u ( D ). f ( z ) = 3 ( z − i ) Poletsky-Stessin Hardy Spaces

Domains Bounded by An Analytic Jordan Curve in C Let ψ be a continuous, subharmonic, exhaustion function for D and ϕ : D → D be a holomorphic function then the generalized Nevanlinna function ([2]) is given as � ( − ψ ) dd c log | ϕ − w | N ϕ ψ ( w ) = D Theorem Let ϕ : D → D be a holomorphic function with ϕ (0) = 0 and suppose that ψ is a continuous, subharmonic exhaustion function N ϕ ψ ( w ) 1 2 dd c ψ < ∞ and lim sup | w |→ 1 for D . If � − ψ ( w ) < ∞ then p D (1 −| ϕ | 2 ) C ϕ is bounded on H p ψ ( D ), p ≥ 1. Poletsky-Stessin Hardy Spaces

Alexandru Aleman, Nathan S. Feldman, William T. Ross, The Hardy Space of a Slit Domain , Frontiers in Mathematics, Birkh¨ auser Basel, 2009. Fausto Di Biase, Bert Fischer, Boundary Behavior of H p Functions on Convex Domains of Finite Type in C n , Pacific Journal of Mathematics, Vol. 183, No:1, (1998). Urban Cegrell, Pluricomplex Energy , Acta Math. 180, 187217, (1998). Jean-Pierre Demailly, Mesures de Monge-Amp` ere et ebraiques Affines , Caract´ erisation G´ eom´ etrique des Vari´ et´ es Alg´ M´ emoire de la Soci´ et´ e Math´ ematique de France, 19 , 1-24, (1985). Poletsky-Stessin Hardy Spaces