On BMOA and the Bloch space, normal functions, and pointwise multipliers

Abstract: Let $\mathbb D$ be the unit disc in $\mathbb C$ and let ${\rm Hol} (\mathbb D)$ denote the space of all holomorphic functions in $\mathbb D$. In this talk we shall be concerned with a number of subspaces of ${\rm Hol}(\mathbb D)$, especially with the space $H^\infty $ of all bounded analytic functions in $\mathbb D$, the space $BMOA$ which consists of those $f\in H^1$ whose boundary values have bounded mean oscillation on $\partial \mathbb D$, and the Bloch space $\mathcal B$ which consists of those $f\in{\rm Hol} (\mathbb D)$ for which $$\sup_{z\in \mathbb D}(1-\vert z\vert ^2)\vert f^\prime (z)\vert <\infty .$$ It is well known that $H^\infty \subset BMOA\subset \mathcal B$, and that these inclusions are strict. \par A function $f$, analytic in $\mathbb D$, is a normal function (in the sense of Lehto-Virtanen) if $$\sup_{z\in \mathbb D}(1-\vert z\vert ^2)\frac{\vert f^\prime (z)\vert }{1+\vert f(z)\vert ^2}<\infty .$$ We shall let $\mathcal N$ denote the class of all normal analytic functions in $\mathbb D$. We have that $\mathcal B\subset \mathcal N$ and the inclusion is strict. In fact, the class $\mathcal N$ is much bigger that the Bloch space. \par Clearly, $H^\infty $ is an algebra, that is, the product of two $H^\infty $-functions lies in $H^\infty $. However, if $f\in H^\infty $ and $g$ is a $BMOA$ function or a Bloch function, then the product $g\cdot f$ may not be a normal function: there exist pairs of functions $(f, g)$ with $f\in H^\infty $ and $g\in \mathcal B$ such that the product $f\cdot g$ is not a normal function (or at least it is not a Bloch function). In this talk we shall present distinct examples of such pairs of functions starting with the first ones which were given in the 1960's and finishing with other which have been recently obtained. We shall reformulate these results in the language of pointwise multipliers.

complex variablesdynamical systems

Audience: researchers in the topic

CAvid: Complex Analysis video seminar

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