Sharp Fronts of Paired Oscillatory Integrals

Extending the work of Lax [6], using the theory of oscillatory integrals and Fourier integral operators, Duistermaat and Hormander [3] constructed global parametrices of fundamental solutions of strongly hyperbolic differential operators with smooth coefficients. The singular spectra of these objects are certain Lagrangean manifolds. The explicit formulas make it possible at least in principle to see how such a parametrix behaves close to its singular support. In particular, it should be possible to investigate the existence and non-existence of sharp fronts. These are defined as follows. Let Y be a closed subset of a manifold X containing the singular support of a distribution F and let U be a component of X\Y whose boundary conatins a point XQ of Y. We say that F is sharp or has a sharp front at XQ from U if there is a neighborhood V of XQ such that F has a C°° extension from U to U fl V. When this does not happen, -F is said to be diffuse or to have a diffuse front at XQ from U. For instance, the distributions in one variable (x ± z'O) ~ are diffuse from both sides of the origin but their difference 2m8(x) = (x — z'O)" — (.r-fz'O)" is sharp from both sides. The singular support of the forward fundamental solution of a second order hyperbolic differential operator is the corresponding forward light-cone. The fundamental solution is sharp from the outside for the trivial reason that it vanishes outside the cone. On the other hand, the classical parametrix construction by Hadamard shows that it is diffuse and sharp from inside the cone according as the number n of variables is odd or even. In the latter case, when n^>2 and the operator is homogeneous with constant coefficients, the inside of the light-cone is a lacuna, i.e. the fundamental solution vanishes there.