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

dense in

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 Cn and Bn easily. 1955 Heins. Exists a Blaschke product universal in unit ball of

H(D)

1979 Chee. There exists a universal f in unit ball of

H(Dn)

and

H(Bn)

2005 Xiao Jie & Xiao Shan. Exists a universal inner f in unit ball of

H(Bn)

2007 Aron, Richard; Gorkin, Pamela. Universality is generic 2007 Bayart, Frédéric; Gorkin, Pamela. Not just Bn 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,

U \ Ω ∅ ⇒ γ(U \ Ω) > 0;

(1) (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 ⊂ Rn is Greenian. The following are equivalent: (i) for each u ∈ S (Ω), there are v j ∈ S (Ω) upper bounded,

vj ↘ u and vj = u eventually on compacta.

(ii) for each bounded open U ⊂ Rn, 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 Cn, n > 1 Homogeneous expansion f holomorphic at 0 ∈ Cn

f(z) =

∞

∑

j=0

h j(z), |z| < r,

where hj 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

sup

z∈K

• J

∑

j=0

h j(z) − p(z)

• < ϵ.

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 Rn. 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

• n

∂B ⇒ u ≤ h

• n

B. Ω open set in Cn. 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

• n

∂D ⇒ u ≤ h

• n

D.

For F = R (respectively F = C),

S c(Fn) =

• cont. subharm. (respectively, plurisubharm.) functions
• n Rn (resp, on Cn).

2007 Gauthier-Pouryayevali. There exists a universal f ∈ S c(Fn). Namely, its translates f(z+a), a ∈ Fn, are dense in S c(Fn).

7

Ω ⊂ Rn. u : Ω → [−∞, +∞) uppersemicontinuous is

subharmonic iff for each ball B ⊂ Ω and h continuous

• n B and harmonic on B,

lim sup

x∈B,x→y

u(x) ≤ h(y) ∀y ∈ ∂B ⇒ u ≤ h

• n

B. Ω ⊂ Cn. u : Ω → [−∞, +∞) uppersemicontinuous is

plurisubharmonic iff for each complex disc D ⊂ Ω and function h continuous on D and harmonic on D,

lim sup

z∈D,z→ζ

u(z) ≤ h(ζ) ∀ζ ∈ ∂D ⇒ u ≤ h

• n

D.

For sequence {uj} in S (Fn) and u ∈ S (Fn), we write

uj ↘ u

if u j → u pointwise and for each compact K ⊂ Fn, there is a jK such that,

u j(x) ≥ u j+1(x), ∀ j ≥ jK

and

x ∈ K.

2007 Gauthier-Pouryayevali. There is u ∈ S c(Fn) univer- sal in S (Fn) : for each v ∈ S (Fn), there is a sequence

aj ∈ Fn, such that u(· + aj) ↘ v(·).

Universal functions on arbitrary Stein manifolds

X Stein manifold. Bn ball in Cn. φ biholomorphic mapping

• f neighborhood of Bn into X.

B = φ(Bn) is a parametric ball in X.

Let Φ(X) be the family of all φ such that

φ : Bn → X is a parametric ball in X. f ∈ H(X) universal with respect to H(Bn) if the family f ◦ Φ(X) dense in H(Bn).

2005 Gauthier-Pouryayevali For each Stein manifold X of dimension n, most f ∈ H(X) are universal with respect to H(Bn). 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 → Rm such that |h| < |f| on

∂Ω. Then, µ(0, f, Ω) = µ(0, f + h, Ω).

8

THANKYOU!

9