Vol. 15, No. 1, 1965

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Singularities of superpositions of distributions

Donald A. Ludwig

Vol. 15 (1965), No. 1, 215–239
Abstract

Distributions of the form

F(x,λ) = 1 Γ(λ+1 2 )|f(x,u)|λg(x,u)du (1)

are considered, where x and u belong to Rp and Rn respectively. The parameter λ is complex, and F(x,λ) is evaluated for Re(λ) < 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 > λ + α, where α 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
Authors
Donald A. Ludwig