Vol. 21, No. 3, 1967

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Properties of differential forms in n real variables

Henry B. Mann, Josephine Mitchell and Lowell Schoenfeld

Vol. 21 (1967), No. 3, 525–529

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

ℒX [f(X )] = 0

for all X in some neighborhood of the point A in the Euclidean space En. Let X = (x1,,xn) and H = (h1,,hn). Then for each p = 1,2, the homogeneous polynomial φp(H;f) defined by

                    r1   r       p
φp(H;f) =    ∑     h1-⋅⋅⋅hnn(----∂-f----)
r1+⋅⋅⋅+rn=p r1!⋅⋅⋅rn! ∂xr11 ⋅⋅⋅∂xrnn X=A

is an indefinite form, or is identically zero, and it satisfies the same differential equation H[φp(H;f)] = 0 for all H En. 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
Received: 30 December 1965
Published: 1 June 1967
Henry B. Mann
Josephine Mitchell
Lowell Schoenfeld