| The Qp space was first named in the paper 'On subspaces and subsets of BMOA and UBC ' by R. Aulaskari, J. Xiao and R. Zhao. They studied whether the integral condition f Qp=supw ∈DD &vbm0;f'z &vbm0;2gz,w pdAz 1/2 1 the above condition is equivalent to that f is in Bloch space. When 0 < p < 1, Qp space lies between the classical Dirichlet space (Q 0) and the analytic BMO space (Q 1 = BMOA).;In 2003, Z. Wu and C. Xie revealed the inner relation between Morrey space L2,p and Qp space in 'Q spaces and Morrey Spaces'. That is, when 0 < p < 1, Qp space can be viewed as a fractional integration of Morrey space, i.e. Qp = Iq L2,p with q = 1-p2 . Moreover they identified the predual space of Qp space by using the predual space of Morrey space (proved in the paper 'Dual Morrey Spaces' by E. A. Kalita in 1998).;BMOA can be characterized by Carleson measure, and Qp spaces can be characterized by means of modified Carleson measure. It is natural to consider a function space related to the mordified Carleson measures, namely Csp,a , defined as the set of all analytic functions g on D such that the measure |g'(z)| p(1 - |z|2)alpha dA(z) is an s-Carleson measure. We call Csp,a the Morrey type space. A different notation of this space was used earlier in 1996 by R. Zhao in the paper 'On a General Family of Function Spaces', due to a different fashion.;In this dissertation, we characterize the predual of the Morrey type space Csp,a . The technique that is used to prove the predual of Q space doesn't work for Csp,a space when p ≠ 2 (Qs is a special case of Csp,a with p = 2 and alpha = s). Using ideas in the paper 'Dual Morrey Spaces' by E. A. Kalita, we find a new method to deal with Csp,a .;Let M be the set of all nonnegative measures sigma on D with the normalized condition sigma(D) = 1. For zeta ∈ ∂ D, let Gamma(zeta) = {z ∈ D, |z - zeta| < 1 - |z|2}. For 0 ≤ s < 1, and sigma ∈ M , let osigma,s(zeta) = fGamma(zeta)(1 - |z| 2)-sdsigma( z), where zeta ∈ ∂D. Let o sigma,s(z) = f∂Dosigma, s(zeta)Pz(zeta)| dzeta|.;For -1 < alpha 1, denote by Cq,as the set of all analytic functions f on D such that f qCq,a s=inf s∈MD f'z q1- z2q 1-a+aw s,sz q-1dAz< infinity. .;The main results of this dissertation are:;Theorem 1. Let 0 ≤ s < 1, then for all sigma ∈ M , Dws,s zdmz ≤C holds if and only if mu is an s-Carleson measure.;Theorem 2. For 1 < p < infinity, -1 < alpha < 2(p - 1), and 0 ≤ s < 1. ( Cp',a s )* = Csp,a under the pairing f,gC =Df' zg'z 1-z 2dAz . Theorem 3. For 1 < p < infinity and 0 ≤ s 1 be the integer satisfying p - 2 ≤ alpha - np < 2(p - 1), and a&d5; = alpha - np. Then (In Cp',a &d5;s )* = Csp,a under the pairing f,gC =Df' zg'z 1-z 2dAz . . |
