Sobolev scales with arbitrary origin (Q1347836)

The author recalls the definition of a Sobolev scale \((W^m(E))\) \((m \in \mathbb{Z}),\) of origin \(E\), when \(E\) is a subspace of the L. Schwartz's space of distributions \(\mathcal{D}'(\mathbb{R}^n).\) Recently, M. Wojciechowski proved that \(W^m(E)\) is not such a scale when \(E=L^1(T^2).\) In the present paper, the authors obtain the same result for either \(E=L^1(\mathbb{R}^n)\) or \(E=L^{ \infty}(\mathbb{R}^n)),\) with \(n\geq{2}.\) They show that, in these cases, \(E\) is a proper subspace of \(W^1(W^{-1}(E)),\) and \(W^{-1}(W^1(E))\) is a proper subspace of \(E.\) A method by duality, and explicit counter-examples are investigated.