Sobolev extension by linear operators (Q2862635)

For fixed positive integers \(m,n\) and \(n<p<\infty\), the authors consider the homogeneous Sobolev space \(\mathbb{X}=L^{m,p}(\mathbb{R}^n)\) endowed with the seminorm NEWLINE\[NEWLINE\|F\|_{X}=\max_{|\alpha|=m}\left(\int_{\mathbb{R}^n}\left|\partial^{\alpha}F(x)\right|^pdx\right)^{1/p}.NEWLINE\]NEWLINE For \(E\subset \mathbb{R}^n\), \(\mathbb{X}(E)\) is defined as the space formed by the restrictions to \(E\) of functions of \(\mathbb{X}\) endowed with the natural quotient seminorm NEWLINE\[NEWLINE\|f\|_{\mathbb{X}(E)}=\inf\{\|F\|_{\mathbb{X}}:F\in\mathbb{X},\;F|_E=f\}.NEWLINE\]NEWLINE The problem studied by the authors is when it is possible to get a linear extension operator from \(\mathbb{X}(E)\) to \(\mathbb{X}\) and, in the case where \(E\) is finite, how the norm of the extension of \(f\in \mathbb{X}(E)\) can be estimated. The results which they get are the following: {\parindent=6mm \begin{itemize}\item[(a)] There exists a continuous linear \(T:\mathbb{X}(E)\to \mathbb{X}\) such that \(T(f)=f\) on \(E\) (i.e., \(T\) is a linear extension operator). \item[(b)] If \(E\) is assumed to be finite (say of size \(N\)), then there exist constants \(c,C>0\) and linear functionals \(\psi_1,\dots,\psi_L:\mathbb{X}(E)\to \mathbb{R}\) such that \(L\leq CN\) and NEWLINE\[NEWLINEc\sum_{l=1}^{L}|\psi_l(f)|^p \leq \|f\|_{\mathbb{X}(E)}^{p}\leq C \sum_{l=1}^{L}|\psi_l(f)|^pNEWLINE\]NEWLINE for each \(f\in \mathbb{X}(E)\). NEWLINENEWLINE\end{itemize}} The classical Whitney extension theorem is the analogue of (a) above for \(\mathbb{X}=C^m(\mathbb{R}^n)\). A lot of references about recent research on good computation of the norm for the smooth case are given in the paper, including important work of the first author.NEWLINENEWLINEThe authors also prove that, in the case where \(E\) is finite, there exists a collection of functionals \(\Omega=\{\omega_1,\dots,\omega_S\}\) such that in (b) \(T\) and the functionals \(\psi_1,\dots,\psi_L\) can be taken to have \(\Omega\)-assisted bounded depth. The significance of this condition is related to a bound in the number of computations needed to evaluate \(T\) and \(\psi_1,\dots,\psi_L\). They conjecture that the coefficients arising in the formulae for the so-called assists and the functionals could be computed in an efficient way in order to give algorithms for \(X=L^{m,p}(\mathbb{R}^n)\) analogous to those given by \textit{C. Fefferman} and \textit{B. Klartag} for \(X=C^m(\mathbb{R}^n)\) [Ann. Math. (2) 169, No.~1, 315--346 (2009; Zbl 1175.41001); Rev. Mat. Iberoam. 25, No.~1, 49--273 (2009; Zbl 1170.65006); erratum ibid. 28, No.~4, 1193 (2012; Zbl 1253.65017)].NEWLINENEWLINEMoreover, the authors indicate that for the non-homogeneous case \(\mathbb{X}=W^{m,p}(\mathbb{R}^n)\) (a Banach space) the analogous results are also true.NEWLINENEWLINEThe result (a) is first proved for a finite subset \(E\subset \mathbb{R}^n\). From a careful examination of the proof, (b) is obtained and the result about bounded depth. The general theorem (a) for infinite \(E\) is obtained taking a Banach limit.