Carleson Measure And Interpolation Sequences For Bloch-type Space

Let D be the open unit disk in the complex plan C, and let H(D) be the set of all analytic functions in D. For 0<α<∞, if f∈H(D) and We say that f belongs to the Bloch-type space Bα. For 0<α<∞, 0<p<∞,μis a positive Borel measure, if we sayμis a p-Bα-Carleson measure.In this thesis, we give a characterization of p-Bα-Carleson measure, geometry property and interpolation sequences for the Bloch-type space.Theorem 2.1. Suppose that 0<p<∞andμis a positive Borel measure in D, then the following assertions are satisfied.(a) Suppose 0<α<1.(1) Ifμsatisfies (1.2) thenμis a p-Bα-Carleson measure.(2) ifμis a p-Bα-Carleson measure for 2≤p<∞, then(b) Suppose 1<α<∞,μsatisfies (1.3) if and only ifμis a p-Bα-Carleson measure.Theorem 2.3. Suppose that 0<p<∞,{ak}k∞=1 is a sequence of positive numbers, andμis a positive Borel measure in D. Let where, as usual,δzk denotes the point mass at zk. (a) Let 0<α<1, there exist a positive constant C, such that for every pair of pointsλ1,λ2∈D with |λ1|=|λ2|, then the following assertions are equivalent: (1)∑k=1∞ak2αkp}<∞. (2)μsatisfies (1.2). (3)μis a p-Bα-Carleson measure.(b) Let 1<α<∞, then the following assertions are equivalent: (1)∑k=1∞ak2(α-1)kp<∞. (2)μsatisfies (1.3). (3)μis a p-Bα-Carleson measure.Theorem 3.1. IfThen the sequences is an international sequences for Bα.