A new proof of Poltoratskii's theorem

摘要

We provide a new simple proof to the celebrated theorem of Poltoratskii concerning ratios of Borel transforms of measures. That is, we show that for any complex Borel measure mu on R and any f is an element of L-1 (R, dmu), lim(epsilon-->0)(F-fu(E + iepsilon)/F-mu(E + iepsilon)) = f (E) a.e. w.r.t. mu(sing), where mu(sing) is the part of p which is singular with respect to Lebesgue measure and F denotes a Borel transform, namely, F-fmu(z) = f (x - z)(-1f) (x) d(mu)(x) and F-mu(z) = f (x - z)(-1) d(mu)(x). (C) 2004 Elsevier Inc. All rights reserved.