  We study reproducing kernel Hilbert spaces introduced as ranges of
generalized Ces\`aro-Hardy operators, in one real variable and in one complex
variable. Such spaces can be seen as formed by absolutely continuous functions
on the positive half-line (or paths of infinite length) of fractional order, in
the real case. A theorem of Paley-Wiener type is given which connects the real
setting with the complex one. These spaces are related with fractional
operations in the context of integrated Brownian processes. We give estimates
of the norms of the corresponding reproducing kernels.
