Beltrami equations with coefficient in the fractional sobolev space Wθ, 2/θ

Antonio L. Baisón, Albert Clop, Joan Orobitg

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6 Citations (Scopus)

Abstract

© 2016 American Mathematical Society. In this paper, we look at quasiconformal solutions ϕ: ℂ → ℂ of Beltrami equations ∂z ϕ(z) = μ(z) ∂zϕ(z), where μ ∈ L∞(ℂ) is compactly supported on 𝔻, and ||μ||∞ < 1 and belongs to the fractional Sobolev space Wα, 2/α (ℂ). Our main result states that log ∂z ϕ ∈ Wα, 2/α (ℂ) whenever α ≥ 1/2. Our method relies on an n-dimensional result, which asserts the compactness of the commutator (Formula Presented) between the fractional laplacian (−Δ) β/2 and any symbol b ∈ Wβ, n/β (ℝn), provided that 1 < p < n/β.
Original languageEnglish
Pages (from-to)139-149
JournalProceedings of the American Mathematical Society
Volume145
Issue number1
DOIs
Publication statusPublished - 1 Jan 2017

Keywords

  • Beltrami equation
  • Beltrami operators
  • Fractional Sobolev spaces
  • Quasiconformal mapping

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    Baisón, A. L., Clop, A., & Orobitg, J. (2017). Beltrami equations with coefficient in the fractional sobolev space Wθ, 2/θ. Proceedings of the American Mathematical Society, 145(1), 139-149. https://doi.org/10.1090/proc/13204