Norms of the Borel Transform and the Decomposition of Measures

We relate the decomposition over a; b] of a measure dd (on R) into absolutely continuous, pure point, and singular continuous pieces to the behavior of integrals b R a (Im F(x + ii)) p dx as # 0. Here F is the Borel transform of dd, that is, F(z) = R (x ? z) ?1 dd(x). x1. Introduction Given any positive measure on R with Z dd(x) 1 + jxj < 1; (1.1) one can deene its Borel transform by F(z) = Z dd(x) x ? z : (1.2) We have two goals in this note. One is to discuss the relation of the decomposition of into components (dd = dd ac + dd pp + dd sc with dd ac (x) = g(x) dx, dd pp a pure point measure, and dd sc a singular continuous measure) to integrals of powers of Im F(x + ii). This is straightforward, and global results (e.g., involving 1 R ?1 jIm F(x + ii)j 2 dx) are well-known to harmonic analysts (see, e.g., Koosis 5, pg. 157])|but there seems to be a point in writing down elementary proofs of the local results (e.g., involving b R a jIm F(x + ii)j 2 dx).