Spectral synthesis in Sobolev weighted spaces
The author announces the following result: Theorem 1. Let \(f\) belong to the weighted Sobolev space \(W^m_p (\mathbb{R}^N, w)\), where \(1< p< \infty\) and the weight \(w\) belongs to the Muckenhoupt's class \(A_p\). In addition, let \(K\subset \mathbb{R}^N\) be a closed set and \(D^\alpha f|_K=0\) for \(0\leq |\alpha |\leq m-1\). Then for any \(\varepsilon>0\) there exists a function \(\omega\), \(0\leq \omega\leq 1\), such that \(\omega (x)=1\) in a neighborhood of \(K\) and \(|\omega f\mid W_p^m (\mathbb{R}^N, w)|\leq \varepsilon\). As a corollary she gets a uniqueness theorem for the Dirichlet problem of the polyharmonic equation \(\Delta_m n=0\) in weighted Sobolev spaces.
scientific article; zbMATH DE number 830714