I got a bit familiar with fractional calculus, and it seems that the Riemann-Liouville differintegral formula (and all other similar formulas) are defined for a function $f$ that is one dimensional, and they mean a repeated amount of integration/differentiation with fractional power. So if there is an operator $D$ that performs the integration/differentiation $n$ times, then one should apply $D(D(D(\dots D(f))))$, $n$ times. The full theory extends this $n$ to non-integer numbers.
My question is not exactly related to this, but rather I'm looking for a definition of a fractional integral where the power $a$ of the integration defines the "dimensionality" of the problem, e.g. $a = 2$ would mean a $2$-dimensional integral instead of integrating twice.
I would be interested if there is any definition of $\int dx^{\alpha} f(x)$, where $f(x)$ is not strictly a one-dimensional function, but it can be "arbitrary", or not even a function but some other "function-placeholder" (to call it like that).