AN OPERATOR EQUATION CHARACTERIZING THE LAPLACIAN

作者：Koenig H^*; Milman V

来源：St Petersburg Mathematical Journal, 2013, 24(4): 631-644.

• Laplace operator
• second order Leibniz rule
• operator functional equations

The Laplace operator on R-n satisfies the equation
Delta(fg)(x) = (Delta f)(x) g(x) + f(x)(Delta g)(x) + 2 < f'(x), g' (x)>
for all f, g is an element of C-2(R-n, R) and x is an element of R-n. In the paper, an operator equation generalizing this product formula is considered. Suppose T : C-2(R-n, R) -> C(R-n, R) and A : C-2(R-n, R) -> C(R-n, R-n) are operators satisfying the equation
(1) T(fg)(x) = (Tf)(x) g(x) + f(x)(Tg)(x) + <(Af)(x), (Ag)(x)> for all f, g is an element of C-2(R-n, R) and x is an element of R-n. Assume, in addition, that T is O(n)-invariant and annihilates the affine functions, and that A is nondegenerate. Then T is a multiple of the Laplacian on R-n, and A a multiple of the derivative,
(Tf)(x) = d(parallel to x parallel to)(2)/2 (Delta f)(x), (Af)(x) = d(parallel to x parallel to) f'(x),
where d is an element of C(R+, R) is a continuous function. The solutions are also described if T is not O(n)-invariant or does not annihilate the affine functions. For this, all operators (T, A) satisfying (1) for scalar operators A : C-2(R-n, R) -> C(R-n, R) are determined. The map A, both in the vector and the scalar case, is closely related to T and there are precisely three different types of solution operators (T, A).
No continuity or linearity requirement is imposed on T or A.

• 出版日期2013-8