  Fractional operators are widely used in mathematical models describing
abnormal and nonlocal phenomena. Although there are extensive numerical methods
for solving the corresponding model problems, theoretical analysis such as the
regularity result, or the relationship between the left-side and right-side
fractional operators are seldom mentioned. In stead of considering the
fractional derivative spaces, this paper starts from discussing the image
spaces of Riemann-Liouville fractional integrals of $L_p(\Omega)$ functions,
since the fractional derivative operators that often used are all
pseudo-differential. Then high regularity situation---the image spaces of
Riemann-Liouville fractional integral operators on $W^{m,p}(\Omega)$ space are
considered. Equivalent characterizations of the defined spaces, as well as of
the intersection of the left-side and right-side spaces are given. The behavior
of the functions in the defined spaces at both the nearby boundary point/ponits
and the points in the domain are demonstrated in a clear way. Besides, tempered
fractional operators show to be reciprocal to the corresponding
Riemann-Liouville fractional operators, which is expected to make some efforts
on theoretical support for relevant numerical methods. Last, we also provide
some instructions on how to take advantage of the introduced spaces when
numerically solving fractional equations.
