Two theorems on removable singularities (Q1204623)

The author proves two theorems on removable singularities for linear partial differential equations having a special structure with respect to groups of variables. Let \(\Omega\) be a domain in the space of variables \((x,y)\in\mathbb{R}^ m \oplus \mathbb{R}^ n\). Theorem 1: Let \(P=\Delta_ x^ k+ P_ 1(x,y, \partial_ x, \partial_ y)\), where \(\Delta_ x\) is the Laplacian with respect to the \(x\) variables, \(k\) is a positive integer, \(P_ 1\) has smooth coefficients, and \(\deg_ x P_ 1<2k\). Suppose \(Pu=0\) in \(\Omega\setminus A\), where \(A\) is a closed subset of \(\Omega\), and suppose \(u\in W^{2\ell}_{\infty,\text{loc}} (\Omega)\), where \(\ell\) is a positive integer, and \(\ell<k\). Let \(\gamma=m-2 (k-\ell)\geq 0\), and let \(A_ x\) be the projection of \(A\) onto the plane \(y=0\). If for every compact set \(K\subset\Omega\), the intersection \(K\cap A_ x\) has zero capacity in \(\mathbb{R}^ m\) with respect to the potential \(| x|^{-\gamma}\) (or \(-\ln| x|\) if \(\gamma=0\)), then \(Pu=0\) in all of \(\Omega\). Theorem 2: Suppose \(m\) is even, and let \(L=L_ m(x,\partial_ x)+ L_ 1(x,y, \partial_ x, \partial_ y)\) have smooth coefficients, where \(L_ m(0, \partial_ x)\) is a homogeneous operator of order \(m\), elliptic with respect to \(x\), and \(\deg_ x L_ 1\leq m-1\). Suppose \(Lu=0\) in \(\Omega\setminus \{x=0\}\) and \(| u(x,y)|=o(\ln| x|)\) locally uniformly in \((x,y)\) as \(| x|\to 0\). If \(\int_{|\xi|=1} (L_ m(0,\xi))^{-1} d\sigma_ \xi \neq 0\), then \(Lu=0\) in all of \(\Omega\).