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 language | English |
---|---|
Pages (from-to) | 291-336 |
Journal | Publicacions Matematiques |
Volume | 61 |
Issue number | 2 |
DOIs | |
Publication status | Published - 1 Jan 2017 |
Keywords
- Beurling transform
- David–Semmes betas
- Lipschitz domains
- Peter Jones’ betas
- Quasiconformal mappings
- Sobolev spaces