# CSASC 2013

9-13 June 2013
Koper, Slovenia
UTC timezone
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# On some bilinear problems on weighted Hardy-Sobolev spaces

Presented by Prof. Carme CASCANTE
Type: Oral presentation
Track: Several Complex Variables

## Content

If $w$ is a weight in ${\bf S}^n$, the weighted Hardy-Sobolev space $H_s^p(w)$, $0\leq s$, $0 \lt p \lt +\infty$, consists of functions $f$ holomorphic in $\B^n$ such that if ${\displaystyle f(z)=\sum_k f_k(z)}$ is its homogeneous polynomial expansion, and ${\displaystyle (I+R)^s f(z):=\sum_k (1+k)^s f_k(z)},$ we have that $${\displaystyle ||f||_{H_s^p(w)}^p:= \sup_{r\lt 1}\int_{{\bf S}^n}|(I+R)^s f(r\zeta)|^pw(\zeta)d\sigma(\zeta)<+\infty}.$$ For fixed $0 \lt s,t\lt n$, and $w$ a weight in the Muckenhoupt class ${\mathcal A}_p$, we study the positive Borel measures $\mu$ on the unit sphere of $\C^n$, ${\bf S}^n$, for which the following bilinear problem holds: There exists $C>0$ such that for any $f\in H_s^2(w)$, $g\in H_t^2(w)$, \begin{equation*} \sup_{\rho<1}\left|\int_{{\bf S}^n}f(\rho\zeta) \overline{g (\rho\zeta)}d\mu(\zeta)\right|\leq C\|f\|_{H_s^2(w)}\|g\|_{H_t^2(w)}. \end{equation*} We will give characterizations of this bilinear problem in two situations: for $s,t$ non necessarily equal, under some restrictions on $s,t$ and the weight $w$, and when $s=t$ in a more general situation. (Joint work with Joaqu\'\i n M.\ Ortega.)

## Place

Location: Koper, Slovenia

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