POLYNOMIALLY BOUNDED SOLUTIONS OF THE LOEWNER DIFFERENTIAL EQUATION IN SEVERAL COMPLEX VARIABLES

We determine the form of polynomially bounded solutions to the Loewner differential equation that is satisfied by univalent subordination chains of the form f (z, t) = e(integral 0t A(tau)d tau) Z + ..., where A : [0, infinity] -> L(C-n, C-n) is a locally Lebesgue integrable mapping and satisfying the condition sup(s >= 0)integral(infinity)(0) parallel to exp {integral(t)(s) [A(tau) - 2m(A(tau))I-n]d tau}parallel to dt < infinity, and m(A(t)) > 0 for t >= 0, where m(A) = min {Re <(A(z), z > : parallel to z parallel to = 1}. We also give sufficient conditions for g(z, t) = M (f (z, t)) to be polynomially bounded, where f(z,t) is an A(t)-normalized polynomially bounded Loewner chain solution to the Loewner differential equation and M is an entire function. On the other hand, we show that all A(t)-normalized polynomially bounded solutions to the Loewner differential equation are Loewner chains.