A bounded linear extension operator for \(L^{2,p}(\mathbb R^2)\). (English) Zbl 1315.46034

The author considers the problem of a smooth extension of real-valued functions. In a general form, this problem is formulated as follows. Let \(E\) be an arbitrary subset of \({\mathbb R}^n\). Let \(\mathbb X\) be a space of real-valued smooth functions on \({\mathbb R}^n\). Given a function \(f : E \to {\mathbb R}\), how can we tell whether there exists a smooth function \(F \in \mathbb X\) such that \(F = f\) on \(E\)? The function \(F\) is called a smooth extension of \(f\). If a smooth extension exists, can we take it to depend linearly on \(f\)? There are many definitions of “smooth” functions and for each such definition we have a variation of the above question.
The study of the extension problem was initiated by H. Whitney in two classical papers [“Analytic extensions of differentiable functions defined in closed sets”, Trans. Am. Math. Soc. 36, 63–89 (1934; Zbl 0008.24902)] and “[Differentiable functions defined in closed sets. I”, ibid. 36, 369–387 (1934; Zbl 0009.20803)]. Whitney considered extensions of differentiable functions from closed subsets and solved the problem for the space \(C^m ({\mathbb R})\). Further results for the space \(C^m ({\mathbb R^n})\) were obtained by Y. Brudnyi, P. Shvartsman, C. Fefferman and others. For more information on the history of the subject, see the monograph of A. Brudnyi and Y. Brudnyi [Methods of geometric analysis in extension and trace problems. Vol. 1. Basel: Birkhäuser (2012; Zbl 1253.46001)].
Let \({\mathbb X}=L^{m,p}({\mathbb R}^n)\) be the Sobolev space of real-valued functions with \(m\)th derivatives belonging to \(L^p\), \(n<p<\infty\), with the seminorm \(\|F\|_{\mathbb X}:=\left( \int_{{\mathbb R}^n}|\nabla^m F(x)|^p\right)^{1/p}\). Define the trace space \({\mathbb X}|_E= \{F|_E ; F\in {\mathbb X}\}\) with the trace seminorm \(\|f\|_{{\mathbb X}|_E} := \inf \{\|F\|_{\mathbb X}: F\in {\mathbb X}\), \(F|_E=f \} \) for \(f\in {\mathbb X}|_E\).
The results on the extension problem for \({\mathbb X}=L^{m,p}({\mathbb R}^n)\) are modest compared with those for \({\mathbb X}=C^m(\mathbb R^n)\). The solutions are known only for some special cases: for \({\mathbb X}=L^{m,p}({\mathbb R})\), \({\mathbb X}=L^{1,p}({\mathbb R}^n)\) and some others. In the paper under review, the author gives the solution for \({\mathbb X}=L^{2,p}({\mathbb R}^2)\), \(p>2\). The case of arbitrary \(E\) is reduced to the case of a finite subset \(E\). The main result is the following theorem.
Theorem 1. Let \(2 < p < \infty\). Suppose that \(E \subset {\mathbb R}^2\) has cardinality \(\#E=N<\infty\). Then there exists a bounded linear extension operator \(T: L^{2,p}({\mathbb R}^2)|_E\to L^{2,p}({\mathbb R}^2)\) with norm \(C\). Moreover, there exist linear functionals \(\lambda_1, \lambda_2,\dots, \lambda_K\), where \(K<CN^2\), such that \(M(f)=\left(\sum^K_{k=1}|\lambda_k(f)|^p\right)^{1/p}\) satisfies \(C^{-1}M(f)\leq \|f\|_{L^{2,p}({\mathbb R}^2)|_E} \leq CM(f)\) for all \(f:E\to {\mathbb R}\). Here, the constant \(C > 1\) depends only on \(p\).


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26B05 Continuity and differentiation questions
Full Text: DOI arXiv


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