A trace theorem for solutions of linear partial differential equations

Abstract

AbstractIn this paper, we prove 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 hypersuface will be just as regular as the solution itself. The proof is based on the Hörmander–Nirenberg pseudo‐differential cut‐off technique and a ‘fattening’ lemma, together with standard energy estimates.