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 E 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. CR Subject Classification (1998): F.1.3, F.1.1, F.4.3