Continuous linear right inverses for partial differential operators of order 2 and fundamental solutions in half spaces
At the beginning of the fifties L. Schwartz formulated the problem to characterize those linear partial differential operators \(P(D)\) that admit a (continuous linear) right inverse on the Fréchet space \(C^\infty(\Omega)\), \(\Omega\subset\mathbb{R}^n\) is open. This problem was solved by the authors in an earlier paper. In the present paper, the authors give a more detailed characterization of the differential operators \(P(D)\) of order 2 that admit a right inverse on \(C^\infty(\mathbb{R}^n)\). The first step is the reduction of general quadratic polynomials to certain normal forms. A necessary and sufficient condition is proved for the operator \(P(D)\), which corresponds to a normal form, to have a right inverse on \(C^\infty(\mathbb{R}^n)\). Among other results the following equivalence result is proved: For each non-constant polynomial \(P\in\mathbb{C}[z_1,\dots,z_n]\) of degree 2 the following assertions are equivalent: 1. \(\mathbb{R}^n\) is \(P\)-convex with bounds, this means, \(P(D)\) has a right inverse on \(C^\infty(\mathbb{R}^n)\) if and only if \(P(D)\) has a right inverse on \(D'(\mathbb{R}^n)\); 2. there exists a basis \(\{N_1,\dots,N_n\}\) of \(\mathbb{R}^n\) and fundamental solutions \(E^{\pm}_1,\dots, E^{\pm}_n\) for \(P(D)\) satisfying \(\text{supp }E^{\pm}_j\subset \overline{H_{\pm}(N_j)}\) for \(1\leq j\leq n\), \(H_{\pm}(N):= \{x\in\mathbb{R}^n:\pm(x,N)>0\}\); 3. there exists a basis \(\{N_1,\dots,N_n\}\) of \(\mathbb{R}^n\) for which \(H_{\pm}(N_j)\) is \(P\)-convex with bounds for all \(j\); 4. there exists an open half space \(H\) which is \(P\)-convex with bounds; 5. there exists a bounded convex open set \(\Omega\) in \(\mathbb{R}^n\) which is \(P\)-convex with bounds. Finally, an example shows that this result cannot be generalized to operators of third order.
scientific article; zbMATH DE number 980707