Local solvability of analytic pseudodifferential complexes in top degree
Let \(\Omega\) be an open set in \({\mathbb{R}}^{n+1}\) and \(\Psi(\Omega)\) be the ring of pseudodifferential operators on \(\Omega\). A left ideal \({\mathcal P}\) in \(\Psi(\Omega)\) is said to be a Frobenius ideal of rank \(n\) if, for every point \(y\in\Omega\), there exists an open neighborhood \(U\) of \(y\) and \(n\) first-order pseudodifferential operators \(P_1,\ldots,P_n\in{\mathcal P}\) such that \(P_1,\ldots,P_n\) are linearly independent over \(\Psi(U)\), and for all \(P\in{\mathcal P}\), there exist \(A_1,\ldots,A_n\in\Psi(U)\) such that \(P=\sum_{i=1}^n A_iP_i\) and there exist \(c_{jk}^l\in\Psi^0(\Omega)\) such that \([P_j,P_k]=\sum_l c_{jk}^l P_l\). The characteristic set of \({\mathcal P}\) is \(\text{ char} {\mathcal P}={\text{char}}P_1\cap\cdots\cap{\text{char}}P_n\). \({\mathcal P}\) is elliptic at \(0\) if \({\text{char}} {\mathcal P}|_0=\{0\}\). Here it is assumed that \({\mathcal P}\) is nonelliptic at \(0\) and has simple real characteristics. A phase function \(S(y,\eta)\) of \({\mathcal P}\) is defined by the equations \(p_{j,0}(y,S_y)=0\), \(j=1,2,\ldots,n\), \(S_y(0)=\eta\), where \(\eta\in{\text{char}} {\mathcal P}|_0\) is fixed and \(p_{j,0}\) is the principal symbol of \(P_j\). A condition \((\Psi)_{n-1}\) at \(0\) is introduced as follows: For any open neighborhood \(U\) of \(0\), there exists another open neighborhood \(V\) of \(0\), \(V\subset U\) such that the level set \(\{y\in V: {\text{Re}} S=u_0, {\text{Im}} S\geq v_0\}\) has no compact components whatever \(u_0\), \(v_0\in{\mathbb{R}}\). With this notation, the equation \(Pu:=P_1u_1+P_2u_2+\cdots P_nu_n=f\) is locally solvable at \(0\) if for any \(f\in C^\infty(U)\) there exist \(u_1,\ldots u_n \in{\mathcal E}'(U)\) such that \(Pu=f\) is satisfied in \(V\subset U\). The author proves the following theorem: Equation \(Pu=f\) is locally solvable at \(0\) if and only if \({\mathcal P}\) satisfies condition \((\Psi)_{n-1}\) at \(0\).
scientific article; zbMATH DE number 1011421