Date:	Mon, March 26, 2012
Time:	17:15
Place:	Research II Lecture Hall

Abstract: The Leibniz product rule and the chain rule D(f · g) = D(f) · g + f · D(g) ; D(f \circ g) = D(f)\circ g · D(g) are basic formulas for calculating derivatives D(f) of functions f . We consider the simple question which operators T : C^1(R) \to C(R) besides the derivative satisfy T(f · g) = T(f) · g + f · T(g) ; T(f \circ g) = T(f) \circ g · T(g) for all f,g \in C^1(R) . Here C(R) and C^1(R) denote the continuous resp. continuously differentiable functions from R to R. It turns out that there are fewer solution operators than one might think. We also consider analogues of these rules for the second derivative and give a characterization of the Laplace operator by such equations and a standard invariance property. The second order Leibniz rule type equation has the form T(f · g) = T(f) · g + f · T(g) + Af · A(g) for unknown operators T,A : C^2(R) \to C(R). This is joint work with Vitali Milman.