Singularities of superpositions of distributions

Donald A. Ludwig

Vol. 15 (1965), No. 1, 215–239

DOI: 10.2140/pjm.1965.15.215

Abstract

Distributions of the form

$$F\left(x,\lambda \right)=\frac{1}{\Gamma \left(\frac{\lambda +1}{2}\right)}\int \; <!--nolimits-->|f\left(x,u\right){|}^{\lambda }g\left(x,u\right)du$$

(1)

are considered, where $x$ and $u$ belong to $R^p$ and $R^n$ respectively. The parameter $\lambda$ is complex, and $F\left(x,\lambda \right)$ is evaluated for $Re\left(\lambda \right)<0$ by analytic continuation. Such integrals arise in solution formulas for partial differential equations. In case $n=1$ or $n=2$, $F$ is expressed in terms of homogeneous distributions of degree $>\lambda +\alpha$, where $\alpha$ is nonnegative and depends upon the geometry of the roots of $f$. The case of general $n$ is also treated, in case the Hessian of $f$ with respect to $u$ is different from zero. The results lead to asymptotic expansions of analogous multiple integrals.

Mathematical Subject Classification

Primary: 46.40

Milestones

Received: 6 February 1964

Published: 1 March 1965