The structure of Sobolev extension operators

  • Charles Fefferman

    Princeton University, United States
  • Arie Israel

    University of Texas at Austin, USA
  • Garving K. Luli

    University of California at Davis, USA


Let Lm,p(Rn)L^{m,p}(\mathbb{R}^n) denote the Sobolev space of functions whose mm-th derivatives lie in Lp(Rn)L^p(\mathbb{R}^n), and assume that p>np>n. For ERnE \subseteq \mathbb{R}^n, denote by Lm,p(E)L^{m,p}(E) the space of restrictions to EE of functions FLm,p(Rn)F \in L^{m,p}(\mathbb{R}^n). It is known that there exist bounded linear maps T ⁣:Lm,p(E)Lm,p(Rn)T \colon L^{m,p}(E) \rightarrow L^{m,p}(\mathbb{R}^n) such that Tf=fTf = f on EE for any fLm,p(E)f \in L^{m,p}(E). We show that TT cannot have a simple form called “bounded depth”.

Cite this article

Charles Fefferman, Arie Israel, Garving K. Luli, The structure of Sobolev extension operators. Rev. Mat. Iberoam. 30 (2014), no. 2, pp. 419–429

DOI 10.4171/RMI/787