On an interpretation of Hörmander's condition of local solvability
The author re-considers the problem of the local solvability, by observing that local solvability of a partial differential operator \(P\) is essentially equivalent to the existence of a complete set of eigenfunctions for \(P\). As concrete example of this interesting point of view, the author first reviews the case of the operators with constant coefficients, the eigenfunctions being given in this case by \(\exp(ix\xi)\). As second example, attention is fixed on first-order partial differential operators of the form \[ P= p(x,D)= D_{x_n}- i \sum^{n-1}_{k=1} b_k(x) D_{x_k} \] with \(b_k(x)\) analytic real-valued functions. The author shows that, when the sufficient condition of Hörmander \(\{\text{Re p,Im p}\}< 0\) is satisfied in a neighborhood \(V\) of \((x_0,\xi_0)\), a parametrix for \(P\) can be constructed in the preceding order of ideas, microlocally in \(V\), by using eigenfunctions of the form \(\exp(if(x))\).
scientific article; zbMATH DE number 1251255
scientific article