Deterministic k-tape and multitape Turing machines with one-way,
two-way and without a separated input tape are considered. We
investigate the classes of languages acceptable by such devices with
time bounds of the form n+r(n) where r in o(n) is a sublinear
function. It is shown that there exist infinite time hierarchies of
separated complexity classes in that range. For these classes weak
closure properties are proved. Finally, it is shown that similar
results are valid for several types of acceptors with the same
time bounds.