## 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 $$p$$.

### MSC:

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

### Citations:

Zbl 0008.24902; Zbl 0009.20803; Zbl 1253.46001
Full Text:

### References:

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