Symmetric Decompositions of $f\in L^2(\mathbb{R})$ Via Fractional Riemann-Liouville Operators

It is proved that given $-1\mathrm{/}2<s<1\mathrm{/}2$, for any $f\in L^2(\mathbb{R})$, there is a unique $u\in \widehat{H}^{|s|}($\mathbb{R}$)$suchthat$$f={\mathit{D}}^{-s}u+{\mathit{D}}^{s\ast }u,$$where$\boldsymbol{D}^{-s}, \boldsymbol{D}^{s*}$ are fractional Riemann-Liouville operators and the fractional derivatives are understood in the weak sense. Furthermore, the regularity of $u$ is discussed, and other versions of the results are established. As an interesting consequence, the Fourier transform of elements of $L^2(\mathbb{R})$ is characterized.