A Trace Theorem for Solutions of Linear Partial Differential Equations

This paper proves a trace regularity theorem for the solutions of general linear partial differential equations with smooth coefficients. Our result shows that by imposing additional microlocal smoothness along certain directions. the trace of the solution on a codimension one hypersurface will be just as regular as the solution itself. The proof is based on the Hoermander-Nirenberg pseudo-differential cut-off technique and a fattening lemma, together with standard energy estimates.