BERGMAN TYPE OPERATOR ON MIXED NORM SPACES WITH APPLICATIONS

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BERGMAN TYPE OPERATOR ON MIXED NORM SPACES WITH APPLICATIONS

The authors investigate the conditions for the boundednessof Bergman type operators $ P_{s,t} $ in mixed norm space $ L_{p,q}(varphi) $ on the unit ball of $bold C^n (n geq 1)$, and obtain a sufficient condition and a necessary condition for general normal function $varphi$, and a sufficient and necessary condition for $$varphi(r)=(1-r^2)^{alpha}{log^{beta} (2(1-r)^{-1})} (alpha>0,betageq 0). $$This generalizes the resultof Forelli-Rudin$^{[3]}$ on Bergman operator in Bergman space. As applications, a more natural method is given to compute the duality of the mixed norm space, solve the Gleason's problem for mixed norm space and obtain the characterization ofmixed norm space in terms of partial derivatives. Moreover, it is proved that $ fin L_{infty,q}^{(0)}(varphi) $ iff all the functions $ {(1-|z|^2)^{|alpha|}}{frac {partial^{|alpha|}f}{partial z^{alpha}}}(z)inL_{infty,q}^{(0)}(varphi) $ for holomorphic function $f$, $ 1le qle {infty}. $