Sobolev regularity of the Beurling transform on planar domains

Martí Prats

    Research output: Contribution to journalArticleResearchpeer-review

    5 Citations (Scopus)

    Abstract

    Consider a Lipschitz domain Ω and the Beurling transform of its characteristic function BχΩ(z) = −p.v. πz12 * χΩ(z). It is shown that if the outward unit normal vector N of the boundary of the domain is in the trace space of Wn,p(Ω) (i.e., the Besov space Bp,pn−1/p(∂Ω)) then BχΩ ∈ Wn,p(Ω). Moreover, when p > 2 the boundedness of the Beurling transform on Wn,p(Ω) follows. This fact has far-reaching consequences in the study of the regularity of quasiconformal solutions of the Beltrami equation.
    Original languageEnglish
    Pages (from-to)291-336
    JournalPublicacions Matematiques
    Volume61
    Issue number2
    DOIs
    Publication statusPublished - 1 Jan 2017

    Keywords

    • Beurling transform
    • David–Semmes betas
    • Lipschitz domains
    • Peter Jones’ betas
    • Quasiconformal mappings
    • Sobolev spaces

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