The distributional product of Dirac's delta in a hypercone

摘要

Let P be a quadratic form in n variables and signature (p,q). The hypersurface P=0 is a hypercone with a singular point (the vertex) at the origin. We know that the kth derivative of Dirac's delta in P there exists under some condition depending on n. This is due to the fact that the cone P(x)=0 has a critical point at the origin. In our study, the main purpose is to relate distribution product of the Dirac delta with the coefficient corresponding to the double pole of the expansion in the Laurent series of P+λ+μ. From this we can arrive at a formula in terms of the ultrahyperbolic operator defined in the paper.