H ( ) = holomorphic functions on Aut ( ) = automorphism group of - PDF document

Universality in Several Complex Variables Paul Gauthier ∗ , Extinguished professor Université de Montréal Informal Analysis Seminar focusing on Universality Kent State University, April 11-13, 2014 1

Ω domain in C n H ( Ω ) = holomorphic functions on Ω Aut ( Ω ) = automorphism group of Ω f ∈ H ( Ω ) is universal in A ⊂ H ( Ω ) , if dense in { f ◦ φ : φ ∈ Aut ( Ω ) } A 2

1929 Birkhoff. There exists a universal f ∈ H ( C ) 1941 Seidel-Walsh. There exists a universal f ∈ H ( D ) These results extend to C n and B n easily. 1955 Heins. Exists a Blaschke product universal in unit ball of H ( D ) 1979 Chee. There exists a universal f in H ( D n ) H ( B n ) unit ball of and 2005 Xiao Jie & Xiao Shan. Exists a universal inner f in H ( B n ) unit ball of 2007 Aron, Richard; Gorkin, Pamela. Universality is generic 2007 Bayart, Frédéric; Gorkin, Pamela. Not just B n 2008 Gorkin, Pamela; León-Saavedra, Fernando; Mor- tini, Raymond. Characterize such universal functions 3

Approximation by bounded functions Mortini posed problem in Oberwolfach 2007. Need ap- peared in 2008 Gorkin León-Savedra Mortini. H ∞ ( Ω ) denotes bounded functions in H ( Ω ) . 2009 Gauthier-Melnikov For Ω open in C , the following are equivalent: (i) H ∞ ( Ω ) is dense in H ( Ω ); (ii) for each open U ⊂ C , with ∂ U ⊂ Ω , (1) U \ Ω � ∅ ⇒ γ ( U \ Ω ) > 0; (iii) for each open U ⊂ Ω , (1) holds. Here, γ ( E ) = analytic capacity of E . Sets of analytic ca- pacity zero are precisely removable sets for bounded holomorphic functions. 4

Potential theoretic analogue 2008 Sylvain Roy. Suppose Ω ⊂ R n is Greenian. The following are equivalent: (i) for each u ∈ S ( Ω ) , there are v j ∈ S ( Ω ) upper bounded, v j ↘ u and v j = u eventually on compacta. (ii) for each bounded open U ⊂ R n , with ∂ U ⊂ Ω , and also each open U ⊂ C , with ∂ U ⊂ Ω , if n = 2 , U \ Ω � ∅ ⇒ c ( U \ Ω ) > 0 . Here, c ( E ) = capacity. Sets of capacity zero are pre- cisely removable sets for bounded harmonic functions. Greenian domains are precisely those admitting non-constant upper bounded subharmonic functions. 5

Universal series in C n , n > 1 Homogeneous expansion f holomorphic at 0 ∈ C n ∞ ∑ f ( z ) = h j ( z ) , | z | < r , j = 0 where h j homogeneous polynomials. Theorem For r ≥ 0 , there exists a homogeneous series which converges for | z | < r and for each compact convex K outside | z | ≤ r , and each polynomial p and each ϵ > 0 , there is a J such that � � J � � � � ∑ � � < ϵ. sup h j ( z ) − p ( z ) � � � � z ∈ K � � j = 0 � � This theorem is due to: R. Clouätre when r = 0 . Bull. CMS, 2011. N. J. Daras; V. Nestoridis when r > 0 . ArXiv, February 2013. Additional result: M. Manolaki. Unpublished, April 2014. 6

Universal Plurisubharmonic Functions Ω open set in R n . A continuous function u : Ω → R is subharmonic iff for each closed ball B ⊂ Ω and function h continuous on B and harmonic on B , u ≤ h on ∂ B ⇒ u ≤ h on B . Ω open set in C n . A continuous function u : Ω → R is plurisubharmonic iff for each closed complex disc D ⊂ Ω and function h continuous on D and harmonic on D , u ≤ h on ∂ D ⇒ u ≤ h on D . For F = R (respectively F = C ), S c ( F n ) = cont. subharm. (respectively, plurisubharm.) functions on R n (resp, on C n ). 2007 Gauthier-Pouryayevali. There exists a universal f ∈ S c ( F n ) . Namely, its translates f ( z + a ) , a ∈ F n , are dense in S c ( F n ) . 7

Ω ⊂ R n . u : Ω → [ −∞ , + ∞ ) uppersemicontinuous is subharmonic iff for each ball B ⊂ Ω and h continuous on B and harmonic on B , lim sup u ( x ) ≤ h ( y ) ∀ y ∈ ∂ B ⇒ u ≤ h on B . x ∈ B , x → y Ω ⊂ C n . u : Ω → [ −∞ , + ∞ ) uppersemicontinuous is plurisubharmonic iff for each complex disc D ⊂ Ω and function h continuous on D and harmonic on D , on lim sup u ( z ) ≤ h ( ζ ) ∀ ζ ∈ ∂ D ⇒ u ≤ h D . z ∈ D , z → ζ For sequence { u j } in S ( F n ) and u ∈ S ( F n ) , we write u j ↘ u if u j → u pointwise and for each compact K ⊂ F n , there is a j K such that, u j ( x ) ≥ u j + 1 ( x ) , ∀ j ≥ j K and x ∈ K . 2007 Gauthier-Pouryayevali. There is u ∈ S c ( F n ) univer- sal in S ( F n ) : for each v ∈ S ( F n ) , there is a sequence a j ∈ F n , such that u ( · + a j ) ↘ v ( · ) .

Universal functions on arbitrary Stein manifolds X Stein manifold. B n ball in C n . φ biholomorphic mapping of neighborhood of B n into X . B = φ ( B n ) is a parametric ball in X . Let Φ ( X ) be the family of all φ such that φ : B n → X is a parametric ball in X . f ∈ H ( X ) universal with respect to H ( B n ) if the family f ◦ Φ ( X ) dense in H ( B n ) . 2005 Gauthier-Pouryayevali For each Stein manifold X of dimension n , most f ∈ H ( X ) are universal with respect to H ( B n ) . The main topological notion for an equidimensional map- ping of a domain Ω is the degree or index of the mapping with respect to a point y which we denote by µ ( y , f , Ω ) . Rouché type theorem. Ω compact domain in an m - dimensional orientable real manifold X and let f and h be continuous mappings Ω → R m such that | h | < | f | on ∂ Ω . Then, µ (0 , f , Ω ) = µ (0 , f + h , Ω ) . 8

THANKYOU! 9