The derivation equation for C ^k-functions: stability and localization

Abstract

For a given integer k ∈ ℕ we determine the possible forms of operators T: C ^k(ℝ) → C(ℝ) satisfying a generalized Leibniz rule operator equation T(f · g)(x) = Tf(x) · g(x)+f(x) · Tg(x)+S(f, g)(x), f,g ∈ C ^k(ℝ), x ∈ ℝ for two different types of perturbations S(f, g). In the first case, S is given by a function B in localized form

$$S(f,g)(x) = B(x,({f^{(j)}}(x))_{j = 0}^{k - 1},({g^{(j)}}(x))_{j = 0}^{k - 1})$$

involving only derivatives of lower order. We show that, if in addition T annihilates the polynomials of degree ≤ k − 1, T is a multiple of the k-th derivative. For k = 2 and functions on ℝⁿ, we give a characterization of the Laplacian by a similar equation, orthogonal invariance and annihilation of affine functions. In the second setting, we assume S to have the form S(f, g)(x) = Af(x) · Ag(x) where A: C ^k(ℝ) → C(ℝ) is a general operator. Thus here, S has a product form, but the factor Af(x) is not assumed to depend only on the jet of f at x. We describe the possible forms of T and A satisfying the generalized Leibniz rule; T and A turn out to be closely related. Here, T and A need not to be localized, i.e., Tf(x) and Af(x) may depend on values f(y) for y ≠ x.