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

Abstract

It is proved that given $-1/2<s<1/2$, for any $f\in L^2(\mathbb{R})$, there is a unique $u\in {\hat{H}}^{|s|}(\mathbb{R})$ such that $$f={\mathbf{D}}^{-s}u+{\mathbf{D}}^{s\ast }u,$$ where ${\mathbf{D}}^{-s},{\mathbf{D}}^{s\ast }$ 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.