If n > 2 and M(m1,..., xn) is a symmetric norm of the form m(x1, m(x2, m{...)...), where m is a symmetric norm on ℝ2, then m(x, y) = (|x|p + |y|p)1/p for some p ≥ 1 or else m(x, y) = max{|x|,|y|}.

The subject of this paper is the use of the theory of Schwartz distributions and approximate identities in studying the functional equation

The aj’s and b are complex-valued functions defined on a neighbourhood, U, of 0 in Rm, hj. U → Rn with hj(0) = 0 and fj, g: Rn → C for 1 ≦ j ≦ N. In most of what follows the aj's and hj's are assumed smooth and may be thought of as given. The fj‘s, b and g may be thought of as the unknowns. Typically we are concerned with locally integrable functions f1, … , fN such that, for each s in U, (1) holds for a.e. (almost every) x ∈ Rn, in the sense of Lebesgue measure.

It is known that if f is a real valued function on a rational vector space V, δ > 0,

1

and if f is unbounded then f(x + y) = f(x)f(y) for all x, y ∊ V. In response to a problem of E. Lukacs, in this paper we study the bounded solutions of (1). For example, it is shown that if f is a bounded solution of (1) then - δ ≤ f(x) ≤ (1 + (1 + 4δ)1/2)/2 for all x ∊ V and these bounds are optimal.