Representation of linear functionals in \(L_ q^{m*} (\Omega)\) (Q1921765)

Let \(m\), \(n\) be natural numbers, \(n>1\) and let \(p>1\) be a real number. Denote by \(W^m_p(\Omega)\) the Banach space of \(p\)-integrable functions having weak derivatives of order \(m\) in the bounded domain \(\Omega\subset\mathbb{R}^n\), the space being endowed with the norm \[ |u|= \Biggl(\int_\Omega (|u|^2+J(u,u))^{p/2} dx\Biggr)^{1/p}, \] where \[ J(u,u)= \sum^n_{j_1=1}\cdots \sum^n_{j_m=1} \Biggl({\partial^mu\over \partial x_{j_1}\cdots\partial x_{j_m}}\Biggr)^2. \] Denote by \(L^{m*}_p(\Omega)\) the space of functionals in the dual of \(W^m_p(\Omega)\) vanishing on polynomials of degree less than \(m\). The main result of the author is a representation theorem for functionals in \(L^{m*}_p(\Omega)\).