It is proved that a cosine operator function C(·),
with generator A, is locally of bounded semivariation
if and only if u″(t) = Au(t)+f(t),
t>0, u(0), u′(0)∈D(A),
has a strong solution for every continuous function f, if and only if the function
∫t0∫t−s0C(τ)
f(s)dτds, t>0,
is twice continuously differentiable for
every continuous function f, that is, C(·)
has the maximal regularity property if and only if A is a bounded
operator. Some other characterisations of bounded generators of cosine operator
functions are also established in terms of their local semivariations.