Some Operators On Weighted Fock Spaces

In this thesis we are interested in the Fock space where m>0,1?p<+? are parameters.Under ?·?Fmp,Fmp is a Banach space.Our research is mainly on the mapping properties of the radial derivative operator R and the extended Cesaro operator Tg between Fock spaces with differential exponentials.Here,the radial derivative operator R is defined as and for given a symbol g ? H(Cn),the extended Cesaro operator Tg is defined asThe main achievements of this master thesis are as the following:?For Fock spaces Fmp,we extend the known Littlewood-Paley formula in one variable to the higher dimensional case.Precisely,we obtain that for f?where the notation A(f)(?)B(f)means that there is some positive constant C such that for all f being considered.?We obtain some estimate on the Bergman kernel K(w,z)of Fm2.For 0<m<2,by using the asymptotic behavior of the mittag-leffler function as z??,we get the estimate on K(w,z)near the diagonal as where (?) and r0>0.Furthermore,we introduce a test function G(z,w)as and prove that there exists R>0 such that for |w|?R.?We studied the mapping properties of the radial derivative operator R and the generalized Cesaro operator Tg between Fock spaces with different ex-ponentials.With the equivalent norm we obtain on Fmp,we characterize those symbols g ? H(Cn)such that Tg:Fmp?Fmq is bounded(or compact).And also,we completely determine the values of m,p and q so that the radial deriva-tive operator R is bounded(or compact)from Fpm to Fqm.Our theory extends the main results in the references[12],[13]and[14].