We prove the following
theorem. Let ℒX 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 En. Let
X = (x1,⋯,xn) and H = (h1,⋯,hn). Then for each p = 1,2,⋯ the homogeneous
polynomial φp(H;f) defined by
is an indefinite form, or is identically zero, and it satisfies the same differential
equation ℒH[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.
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