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) is an element of D(A), has a strong solution for every continuous function f, if and only if the function integral(0)(t) integral(0)(t-8) C(tau) f(s) d tau ds, t > 0, is twice continuously differentiable for every continuous function f, that is, C(.) t;as 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.
JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES