Properties of differential forms in n real variables

Henry B. Mann, Josephine Mitchell and Lowell Schoenfeld

Vol. 21 (1967), No. 3, 525–529

DOI: 10.2140/pjm.1967.21.525

Abstract

We prove the following theorem. Let ℒ_X be a homogeneous elliptic operator of the second order with constant coefficients. Let f be a Lebesgue integrable solution of

for all X in some neighborhood of the point A in the Euclidean space E_n. Let X = (x₁,⋯,x_n) and H = (h₁,⋯,h_n). Then for each p = 1,2,⋯ the homogeneous polynomial φ_p(H;f) defined by

is an indefinite form, or is identically zero, and it satisfies the same differential equation ℒ_H[φ_p(H;f)] = 0 for all H ∈ E_n. Analogous differential relations are true for the solutions of homogeneous hypoelliptic equations of any order. The infinite differentiability of these solutions is called upon.

Mathematical Subject Classification

Primary: 35.42

Milestones

Received: 30 December 1965

Published: 1 June 1967