Let C0[a,b] denote the space of
continuous functions x on [a,b] such that x(a) = 0. Recently Cameron and Storvick
defined an operator-valued Feynman integral Jq(F) of functionals F on
C0[a,b]. Let F(x) = f(x(t1),⋯,x(tn)) where a = t0< t1<⋯< tn= b. The
present authors earlier established the existence of Jq(F) for functionals F
as above under the assumption that f is factorable and bounded. In the
present paper it is shown that with the factorability assumption completely
removed, J−1(F) may fail to exist even with f required to be in Lp(Rn) for
1 ≦ p ≦∞. On the other hand it is shown that Jq(F) does exist under the
rather surprising condition that f ∈ L21⋯12 where L21⋯12 is the set of all
complex-valued measurable functions f on Rn(n ≧ 2) such that ∥f∥21⋯12≡∥f∥ < ∞
where
∥f∥
≡
1∕2.
Another positive result shows that if F is an analytic function of a finite sum of
factorable functions, then Jq(F) exists.