About thirty-five years ago H. Lewy produced an example of a first order linear partial differential equation with linear (complex) coefficients which had no solution. A complete description of differential operators which are 'solvable' remains a challenging, but unresolved issue in mathematical analysis. However, the search has led to many fundamental results about partial differential operators, especially in the context of the theory of several complex variables. One of the more interesting aspects of the work in this area concerns the question of describing the range of an 'unsolvable' operator, or, equivalently, which inhomogeneous equations can be solved? The question is reformulated in terms of the null space of the adjoint: whether the projection operator onto the null space preserves real analyticity. The tools used to investigate the question in this project involve the calculus of pseudodifferential operators defined on a manifold, the Heisenberg group. Work will be done on the specific case of the Szego projection restricted to a real analytic boundary of a bounded pseudoconvex domain of finite type in two complex dimensions. Work will also be done investigating specific one-parameter families of differential operators of second order to determine conditions under which they will be analytic hypoelliptic and on pseudodifferential operators defined on general nilpotent Lie groups.
