  The Caputo time-derivative is usually defined pointwise for well-behaved
functions, say, for continuously differentiable functions. Accordingly, in the
theory of the partial fractional differential equations with the Caputo
derivatives, the functional spaces where the solutions are looked for are often
the spaces of the smooth functions that are too narrow. In this paper, we
introduce a suitable definition of the Caputo derivative in the fractional
Sobolev spaces and investigate it from the operator theoretic viewpoint. In
particular, some important equivalences of the norms related to the fractional
integration and differentiation operators in the fractional Sobolev spaces are
given. These results are then applied for proving the maximal regularity of the
solutions to some initial-boundary value problems for the time-fractional
diffusion equation with the Caputo derivative in the fractional Sobolev spaces.
