Bergman type operators on a class of weakly pseudoconvex domains |
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Authors: | Shi Jihuai |
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Institution: | (1) Department of Mathematics, University of Science and Technology of China, 230026 Hefei, China |
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Abstract: | 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 ß = (ß1, —ß). An interesting question is whether the converse holds. |
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Keywords: | Bergman type operator weakly pseudoconvex domain weighted Bergman space |
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