Extension of Cm,ωC^{m, \omega}-Smooth Functions by Linear Operators

  • Charles Fefferman

    Princeton University, United States


Let Cm,ω(Rn)C^{m, \omega} ( \mathbb{R}^n) be the space of functions on Rn\mathbb{R}^n whose mthm^{\sf th} derivatives have modulus of continuity ω\omega. For ERnE \subset \mathbb{R}^n, let Cm,ω(E)C^{m , \omega} (E) be the space of all restrictions to EE of functions in Cm,ω(Rn)C^{m , \omega} ( \mathbb{R}^n). We show that there exists a bounded linear operator T:Cm,ω(E)Cm,ω(Rn)T: C^{m , \omega} ( E ) \rightarrow C^{m , \omega } ( \mathbb{R}^n) such that, for any fCm,ω(E)f \in C^{m , \omega} ( E ), we have Tf=fT f = f on EE.

Cite this article

Charles Fefferman, Extension of Cm,ωC^{m, \omega}-Smooth Functions by Linear Operators. Rev. Mat. Iberoam. 25 (2009), no. 1, pp. 1–48

DOI 10.4171/RMI/568