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

Characterization of Image Spaces of Riemann-Liouville Fractional Integral Operators on Sobolev Spaces $W^{m,p}(\Omega)$