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SUMMARY:Daniel Girela (Universidad de Málaga)
DTSTART:20220503T130000Z
DTEND:20220503T140000Z
DTSTAMP:20260422T201236Z
UID:CAvid/73
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CAvid/73/">O
 n BMOA and the Bloch space\, normal functions\, and pointwise multipliers<
 /a>\nby Daniel Girela (Universidad de Málaga) as part of CAvid: Complex A
 nalysis video seminar\n\nLecture held in N/A.\n\nAbstract\nLet $\\mathbb D
 $ be the unit disc in $\\mathbb C$ and let ${\\rm Hol} (\\mathbb D)$ denot
 e 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 $f\\in H^1$ wh
 ose 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 $$\\sup_{z\\in \\mathbb D}(1-\\vert z\\
 vert ^2)\\frac{\\vert f^\\prime (z)\\vert }{1+\\vert f(z)\\vert ^2}<\\inft
 y .$$ We shall let $\\mathcal N$ denote the class of all normal analytic f
 unctions in $\\mathbb D$. We have that $\\mathcal B\\subset \\mathcal N$ a
 nd the inclusion is strict. In fact\, the class $\\mathcal N$ is much bigg
 er that the Bloch space. \\par Clearly\, $H^\\infty $ is an algebra\, that
  is\, the product of two $H^\\infty $-functions lies in $H^\\infty $. Howe
 ver\, if $f\\in H^\\infty $ and $g$ is a $BMOA$ function or a Bloch functi
 on\, then the product $g\\cdot f$ may not be a normal function: there exis
 t pairs of functions $(f\, g)$ with $f\\in H^\\infty $ and $g\\in \\mathca
 l B$ such that the product $f\\cdot g$ is not a normal function (or at lea
 st it is not a Bloch function). In this talk we shall present distinct exa
 mples of such pairs of functions starting with the first ones which were g
 iven in the 1960's and finishing with other which have been recently obtai
 ned. We shall reformulate these results in the language of pointwise multi
 pliers.\n
LOCATION:https://researchseminars.org/talk/CAvid/73/
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