The Properties Of Random Functions On Hardy-Sobolev Spaces And Fock Spaces

Let C be the complex plane and let D be the unit disk in C.Let f(z)=?n=0? anzn be a formal series.Let Rf(z)=?n=0? an?nzn be the the randomization of f,where?n=±1.Let Hp(D)be the classical Hardy spaces and H?p(D)be Hardy-Sobolev spaces over the unit disk D,where p>0,? ? R.In 1930,Littlewood proved that if f(z)=?n=0? anzn ? H2(D),then Rf(z)=?n=0?an?nzn ?Hp(D)almost surely for any p>0.When f(?)H2(D),for almost every choice of signs,Rf has a radial limit almost nowhere,i.e.,R f(?)Hp(D)almost surely.The third chapter of this paper mainly describes that when f ?H?p(D),after adding some conditions,Rf is almost surely in L?q(D).The most important tool for proving is the mixed norm space.The fourth chapter mainly studies the properties of the entire function after randomization,and whether the entire function almost surely belongs to Fock space after randomization.Then to establish a relationship between the randomized function and the Fock space.