Some examples of \(C^\infty\) extension by linear operators (Q664908)

The work is devoted to construct linear continuous operators extending \(\mathcal{C}^{\infty}\) functions defined in two different kinds of sets \(X\subseteq\mathbb{R}^{n}\), to \(\mathcal{C}^{\infty}\) functions defined on \(\mathbb{R}^{n}\). Firstly, the extension operator is found when \(X=\{\lambda_{n}:n\in\mathbb{N}\}\), with \((\lambda_{n})_{n\in\mathbb{N}}\) being a sequence of positive real numbers, monotonically converging to 0 such that \(\lambda_{n}-\lambda_{n+1} \sim n^{-a-1}\) for some \(a>0\). Also, the extension operator is found for the sets of the form \(X=X_{1}\cup(-X_{2})\), where \(X_{1},X_{2}\) are chosen as before. The proof rests on an interpolating argument by means of the families of Lagrange polynomials asociated to a finite number of elements of \(X\), and by fixing a partition of unity. This first extension result is applied in the second result, where the authors construct an extension operator departing from sets of the form \(\{(x,y):y=\lambda_{n}x\), \(x>0\), \(\lambda_n\in X\}\subset\mathbb{R}^{2}\), where \(X\) is chosen as above.