An Operator Equation Generalizing the Leibniz Rule for the Second Derivative

Abstract

We determine all operators $T : {C}^{2}(\mathbb{R}) \rightarrow C(\mathbb{R})$ and $A : {C}^{1}(\mathbb{R}) \rightarrow C(\mathbb{R})$ which satisfy the equation

$$T(f \cdot g) = (Tf) \cdot g + f \cdot (Tg) + (Af) \cdot (Ag)\ ;\quad f,g \in {C}^{2}(\mathbb{R}).$$

(1)

This operator equation models the second order Leibniz rule for (f ⋅g)′ with $Af = \sqrt{2}f'$. Under a mild regularity and non-degeneracy assumption on A, we show that the operators T and A have to be of a very restricted type. In addition to the operator solutions S of the Leibniz rule derivation equation corresponding to A = 0,

$$S(f \cdot g) = (Sf) \cdot g + f \cdot (Sg)\ ;\quad f,g \in {C}^{2}(\mathbb{R})\text{ or }{C}^{1}(\mathbb{R})\,$$

(2)

which are of the form

$$Sf = bf' + af\ln \vert f\vert,\quad a,b \in C(\mathbb{R}),$$

T and A may be of the following three types

$$\begin{array}{lll} Tf & = \frac{1} {2}{d}^{2}f'' &,\ Af = d\,f' \\ Tf & = \frac{1} {2}{d}^{2}f{(\ln \vert f\vert )}^{2} &,\ Af = d\,f\ln \vert f\vert \\ Tf & = {d}^{2}f(\epsilon \vert f{\vert }^{p} - 1)&,\ Af = d\,f(\epsilon \vert f{\vert }^{p} - 1) \end{array}$$

for suitable continuous functions d, c and p and where ε is either 1 or sgnf and p ≥ − 1. The last operator solution is degenerate in the sense that T is a multiple of A. We also determine all solutions of (1) if T and A operate only on positive ${C}^{2}(\mathbb{R})$-functions or ${C}^{2}(\mathbb{R})$-functions which are nowhere zero.

Supported in part by the Alexander von Humboldt Foundation by ISF grant 387/09 BSF grant 2006079.