Speaker
Description
For a fixed analytic function $g$ in the unit disc, we consider the analytic paraproducts induced by $g$, which are defined by $T_gf(z)= \int_0^z f(\zeta)g'(\zeta)\,d\zeta$, $S_gf(z)= \int_0^z f'(\zeta)g(\zeta)\,d\zeta$, together with the multiplication operator $M_gf(z)= f(z)g(z)$. The boundedness of these operators on various spaces of analytic functions on the unit disc is well understood. The original motivation for this work is to understand the boundedness of compositions (products) of two of these operators, for example $T_g^2, \,T_gS_g,\, M_gT_g$, etc. The talk intends to present a general approach which yields a characterization of the boundedness of a large class of operators contained in the algebra generated by these analytic paraproducts acting on the classical weighted Bergman and Hardy spaces in terms of the symbol $g$. In some cases it turns out that this property is not affected by cancellation, while in others it requires stronger and more subtle restrictions on the oscillation of the symbol $g$ than the case of a single paraproduct. This is a report about joint work with C. Cascante, J. F`abrega, D. Pascua and J.A. Pel\'aez
