The Operator And Duality On Mixed Norm Spaces With Small Exponent

This dissertation will deal with the dual spaces and boundedness of the Bergman type operator on mixed norm spaces,mainly with small exponent.The dual space of Hp(B) for 0 < p < 1 has been studied by several authors.In the one dimensional case Romberg[1] described the dual of Hp for all p ∈ (0, 1) except p = 1/k+1 (where k is any positive integer). In[2], Duren, Romberg and Shields completed the characterization of the dual of Hp for all 0 < p < 1 in the case n = 1. For a large class of weight functions , Which go to zero at least as fast as some power of (1 - r) but no faster than some other power of (1 - r), A.L.Shields and D.L.Williams[3] studied certain weighted spaces of analytic functions, and they showed that:(1)A0()* ≈ A1(), (2)A1()* ≈A∞(). We also consider the duality problem for weighted Bergman spaces with small exponent.For 0 < p < l,we let Ap() denote the weighted Bergman spaces consisting of analytic function f in the disk D such thatwhere{,} is a normal pair.Our main result is Theoreml.l stated below:Theoreml.l Let{,} be a normal pair and = (1 - r)α. To each bounded linear functional F on AP(),0 < p < 1, there corresponds a unique function g ∈ H (D) such thatand g(z) = O(1/(r)), |z| = r. conversely, for any g ∈ H(D) with g(z) = O(1/(r)) ,the above formula defines a bounded linear functional for all / ∈ Ap(). For s ∈ R, t > 0,let Ps,tbe the Bergman type operator defined bywhere f is the measurable complex function on ball B .For s = 0, Forelli.F,Rudin.W[4] first obtained a sufficient and necessary condition on the boundedness of P0,t forLP(B)(1 < p < ∞),and subsequently G.B.Ren[5] discussed the boundedness of the P0)t on Lp(B) in the case 0 < p < 1 and obtained that P0,t projects Lp(B) ∩ h(B) onto Lp(B)∩H(B) under some conditions. Meanwhile, G.B.Ren and J.H.Shi in [6] investigate the condition for the boundedness of Bergman type operators P,,t in mixed norm space Lp,9() on the unit ball of Cn(n > 1) and show that :If t > b > a > -s, then Ps,t is a bounded operator of Lp,q() into Lp,g()(l < p < ∞, 1 < q < ∞). Recently, Y.M.Liu[7] has obtained a sufficient condition for the boundedness of the operator Ps,t on Lp,q()(0 < p < 1,1 < q < ∞), and the result is :If t > b > a > - s,then Ps,t also is a bounded operator of Lp,q() into Lp,q( b > a > -s and (p is normal function ,then Ps,t: Hp,q -Lp,q() is a bounded operator.Theorem2.3 If 0 < p < 1,t+n(1-1/p) > a+1/p then P0,t projects Lp(dva)∩(B) onto Lp(dvQ)∩H(B).Theorem2.4 For any integer m > 1, there exists bounded linear operators Aa on Hp,q()(0 ,l < q <∞). In the Chapters, we discuss the duality of Hq()(0 < q < 1,q < n/n+1-a) with reference to [8]. Our main result is :Theorems. 1 Let be a normal function and 0 < q < 1, q < .To each bounded linear functional F on Hq(), there corresponds a unique g ∈ H(Bn). Such thatand g(w) = O, where a and b are the positivenumber of normal function's definition and a < n+1. Conversely, for any g ∈ H(Bn) with g the above formula defines a bounded linear functional on...