Derivations of higher order

Derivations of higher order

f(xy)=xf(y)+yf(x),         (x,yÎK).  (LP)

f(xy)=xf(y)+yf(x)+d_n-1(x,y),         (x,yÎK)  (D)

1. Since the composition of two derivations f and g of first order yields a derivation f o g of second order we want to describe and characterize the decomposable derivations of second order, i. e. to describe when a given derivation j of order 2 can be written in the form

   j=å_j=1^Nf_j o g_j

   where f_j and g_j are derivations of first order. This problem depends heavily on the choice of the filed K and it can be posed for derivations of higher order as well.
2. Determine the defining sets CÍK (or the uniqueness sets E) for the class of derivations of second order. These are sets C=(x_a)_aÎI such that for each (y_a)_aÎI, y_aÎK there is exactly one derivation j of second order such that j(x_a)=y_a for all aÎI. For solving this problem methods from linear algebra and the well known construction of derivations of order 2 must be applied.
3. A part of the theory of derivations of higher order deals with stability properties. It is known (cf. [42]) that the following holds.

   Theorem. Let eps>0, b:Rⁿ⁺¹ -> R and let f:R -> R be such that

   |f(x+y)-f(x)-f(y)| £eps    (x,yÎR)   (D1)

   |d_a1,...,an+1f(x)|£b(a₁,...,a_n+1)     (x,a₁,...,a_n+1ÎR).  (D2)

   Then there exists one and only one derivation D of order n, such that f-D is bounded. Moreover given a derivation D of order n and any bounded function r the function f:=D+r satisfies (D1) and (D2) for suitable eps>0 and suitable b. If b is independent of its variables, then (D1) and (D2) imply that f itself is a derivation of order n.

   Here we use the following characterization of derivations of order n: f is a derivation of order n if f is additive (i.e. f(x+y)=f(x)+f(y) for all x,yÎR) and d_a1,...,an+1f=0, where d_af(x):=f(ax)-af(x) and d_a1,...,an+1f= (d_a1 o ... o d_an+1)(f). But there are other characterizations of derivations of order n, e. g. the definition (using (D)) itself. Thus it would be interesting to investigate the validity of stability results for systems of the form

   |f(x+y)-f(x)-f(y)|£eps

   |f(xy)-xf(y)-yf(x)-d(x,y)|£d(x)

   under some additional assumptions on d.

Derivations of higher order