How important is the Atangana-Baleanu fractional derivative, the main recent development in fractional calculus? In 2016 a new definition of a fractional derivative was announced in this paper, which has since had more than 100 citations. This derivative, the Atangana-Baleanu derivative, is the main recent development in the field of fractional calculus.
How important is it? Does it merit being given the same weight as the Riemann-Liouville and Caputo fractional derivatives, and has it the potential to raise the importance of fractional calculus itself in the broader field of analysis?
 A: While the operators in question may be of some use in applied mathematics, in my opinion they bring little new to the theory of fractional calculus.
Both variants discussed in the linked paper are of the form
$$ D^\alpha f(t) = a f(t) + b f'(t) + \int_{-\infty}^t (f(t) - f(s)) k(t - s) ds $$
for some kernel $k$ and some constants $a$ and $b$, with a proper extension of $f$ from the half-line $(b,\infty)$ to all of $\mathbb{R}$ (the "boundary term"). Operators of this form are very well-studied; for example, in probability, these are generators of Lévy processes.
The usual Riemann–Liouville (or Caputo) derivative $D_R^\alpha$ has kernel $k$ with Fourier transform $\xi^\alpha$. Probabilistically, $D_R^\alpha$ corresponds to the $\alpha$-stable subordinator (a subordinator is an increasing Lévy process).
The kernel $k$ used in the paper has Laplace transform equal to
$$ \int_0^\infty e^{-\xi t} k(t) dt = \frac{\xi^\alpha}{\xi^\alpha + \tfrac{\alpha}{1 - \alpha}} ;$$
see formula (9) there. The corresponding operator $D^\alpha_{AB}$ can be written as a composition of the usual derivative $D_R$ and the $\lambda$-resolvent operator $(\lambda + D_R)^{-1}$. This is a rather standard object in the theory of strongly continuous semigroups of operators.
The operator $D^\alpha_{AB}$ indeed resembles fractional differentiation in large scales. However, at a small scale, this is really loosely related to fractional differentiation. In particular, (essentially) no regularity of $f$ is needed in order to evaluate $D^\alpha_{AB} f$. This is not a property one would expect from a "fractional derivative".
