Variation of the fractional derivatives

Question

$\DeclareMathOperator\AC{AC}\DeclareMathOperator\Lip{Lip}$Suppose we have $f\in L^1(\mathbb{R})\cap \AC(\mathbb{R})\cap \Lip(\mathbb{R})$ and $f$ piecewise linear function, bounded and $|f|\leqslant \frac{1}{x\ln^2(x)}, |x|\geqslant 2$. It is known (Samko Fractional Integrals (6.34), (6.14)) that for the fractional derivative $D^\alpha f$ there is next a.e. inequality $|D^\alpha f|\leqslant c\ln(2+|x|){(1+|x|)^{-1-\alpha}}$, where $c$ doesn't depend on $x$. Are there some results about variation of the function $D^\alpha f$? In my case $D^\alpha f$ is absolute continuous on each bounded interval $D^\alpha f\in \AC[a,b]$, but if we need $D^\alpha f$ is abcolute continuous on the whole real line $D^\alpha f$ has to additionally have bounded variation on $\mathbb{R}$.