Tiến Sĩ Bilipschitz maps, analytic capacity, and the Cauchy integral

Thảo luận trong 'Khoa Học Tự Nhiên' bắt đầu bởi Củ Đậu Đậu, 2/4/14.

  1. Củ Đậu Đậu

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    Abstract
    Letϕ: C→Cbe a bilipschitz map. We prove that ifE⊂Cis compact,
    andγ(E),α(E) stand for its analytic and continuous analytic capacity respectively, thenC
    ư1γ(E)≤γ(ϕ(E))≤Cγ(E) andCư1α(E)≤α(ϕ(E))≤Cα(E),
    whereCdepends only on the bilipschitz constant ofϕ. Further, we show that
    if µis a Radon measure on Cand the Cauchy transform is bounded onL2(µ),
    then the Cauchy transform is also bounded onL2(ϕµ), whereϕµis the image
    measure ofµbyϕ. To obtain these results, we estimate the curvature of ϕµ
    by means of a corona type decomposition.

    1. Introduction
    2. Preliminaries
    3. The corona decomposition
    4. Construction of the curvesΓR, R∈Top(E)
    5. The packing condition for the top squares
    6. Estimates for the high curvature squares
    7. Estimates for the low density squares
    8. The curvature ofϕµ
     

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