Asgeirssonʹs Mean Value Theorem and Related Identities

Asgeirssonʹs mean value theorem states that if u satisfies the ultrahyperbolic differential equation (Δx−Δy) u=0 in a neighborhood of the convex compact set KR={(x, y); x, y∈Rν, |x|+|y|⩽R}, then 〈u, f〉=0 where f is the difference between the surface measures on the spheres {(x, 0); x∈Rν, |x|=R} and {(0, y); y∈Rν, |y|=R}. We extend this to solutions of the inhomogeneous equation by proving that f=(Δx−Δy) μR where μR=12χ(1−ν)/2+(πAR/4R2), with AR(x, y)=(|x|2−|y|2)2−2R2 (|x|2+|y|2)+R4. The distributions χa+ on R are defined by χa+(t)=ta+/Γ(a+1) when Re a>−1 and then continued analytically to all a∈C. This formula is closely related to the fundamental solution of the wave equation in Rν+1. Similar identities are given for arbitrary indefinite nonsingular real quadratic forms in R2ν. This work was originally motivated by the progressive solutions of the wave equation given by G. Friedlander and M. Riesz.