Bergman type operators on a class of weakly pseudoconvex domains

A necessary and sufficient condition for the boundedness of the operator: $(T_{s,u,u} f)(\xi ) = h^{u + \tfrac{v}{a}} (\xi )\smallint _{\Omega _a } h^s (\xi ')K_{s,u,v} (\xi ,\xi ')f(\xi ')dv(\xi ') on L^p (\Omega _a ,dv_\lambda ),1< p< \infty $ , is obtained, where $\Omega _a = \left\{ {\xi = (z,w) \in \mathbb{C}^{n + m} :z \in \mathbb{C}^n ,w \in \mathbb{C}^m ,|z|^2 + |w|^{2/a}< 1} \right\},h(\xi ) = (1 - |z|^2 )^a - |w|^2 $ andK x,u,v (ξ,ξ′).This generalizes the works in literature from the unit ball or unit disc to the weakly pseudoconvex domain ω a . As an appli cation, it is proved thatf?L H p (ω a ,dv λ) implies $h\tfrac{{|a|}}{a} + |\beta |(\xi )D_2^a D_z^\beta f \in L^p (\Omega _a ,dv_\lambda ),1 \leqslant p< \infty $ , for any multi-indexa=(α1,?,α n and ? = (?1, —?). An interesting question is whether the converse holds.

A necessary and sufficient condition for the boundedness of the operator: (T s,u,v f)(ξ) =h u+va (ξ)∫ Ω a h s(ξ′)K s,u,v (ξ,ξ′)f(ξ′) d v(ξ′) on L p(Ω a, d v λ),1<p<∞, is obtained, where Ω a=ξ=(z,w)∈ C n+m : z∈ C n, w∈ C m, |z| 2+|w| 2/a <1, h(ξ)=(1-|z| 2) a-|w| 2 and K x,u,v (ξ,ξ′). This generalizes the works in literature from the unit ball or unit disc to the weakly pseudoconvex domain Ω a. As an application, it is proved that f∈L p H (Ω a, d v λ) implies h |α|a+|β| (ξ)D α 2D β zf∈L p(Ω a, d v λ), 1≤p<∞, for any multi index α=(α 1,...,α n) and β=(β 1,...,β n). An interesting question is whether the converse holds.