The Hilbert Transform of a Measure

2008/11/05 by Alexei Poltoratski, Poltoratski, Alexei, Barry Simon +3
Mathematics · Physics and Astronomy · #26A30 #42A50 #42B25 #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.MP #msc:26A30 #msc:42A50 #msc:42B25

paper · pdf · doi:10.48550/arxiv.0811.0791

18 pages

arxiv created 2009/06/05 · arxiv updated 2009/12/01

Abstract

Let \fre be a homogeneous subset of \bbR in the sense of Carleson. Let μ be a finite positive measure on \bbR and H_μ(x) its Hilbert transform. We prove that if lim_t→∞ t \abs\fre∩\x|\absH_μ(x)>t\=0, then μ_s(\fre)=0, where μ_\s is the singular part of μ.

Related

• Measure theoretic aspects of the finite Hilbert transform
• Generalized Hilbert Operator Acting on Hardy Spaces
• Generalize Hilbert operator acting on Dirichlet spaces
• A generalized Hilbert matrix acting on Hardy spaces
• Non-extendability of the finite Hilbert transform