Colloquium: "Factorization and weak factorization in functional spaces of holomorphic functions"

Hosts: Guido Weiss & Brett Wick
Tea @ 3:45pm in Cupples I, Room 200

Abstract:  We will begin with Blaschke Factorization in Hardy spaces for the unit disc and the upper half plane: a holomorphic function $f\in H^{p}$ can be written as a product of functions respectively in $H^{pr}$ and  $H^{pr'}$, with $r$ and $r'$ conjugate exponents:
                          $f=gh, \qquad \|f\|_p=\|g\|_{pr}\|h\|_{pr'}.$
   This has many applications, among which  Nehari's Theorem, which characterizes symbols of bounded Hankel operators. When dealing with the unit ball of $\mathbb C^n$ with $n>1$, counter-examples (by Rudin, Rosay,...) prove that  the factorization is no more possible. Nevertheless Nehari Theorem is still valid and comes from the  weak factorization of the Hardy space $H^1$, which has been proved by Coifman,Rochberg and Weiss in the eighties. Namely, a function $f\in H^1$ can be written as:
                          $f=\sum_jg_jh_j, \qquad \|f\|_p\approx\sum_j\|g_j\|_{pr}\|h_j\|_{pr'}.$
   The same has been proved for the polydisc by Ferguson and Lacey much more recently.  The question of weak factorization of $H^p$, for $p>1$, is open, except  for an isolated result on the polydisc, that is, the fact that functions of $H^p$ can be weakly factorized as products of functions of $H^{2p}$ when $p<3/2$.
   The same kind of questions can be addressed when Hardy spaces are replaced by Bergman spaces. In this case Pau and Zhao proved the weak factorization for all $p>1$ for the unit ball. Their method generalizes widely.
   We will also discuss the characterization of symbols of bounded Hankel operators on the Hardy space $H^1$ and related factorizations.