Leibniz proved that $D_y\cdot D^{-1}_x=D_{x}^{-1}\cdot D_y$, where $D_x=\frac{\partial}{\partial x}$. It follows that $D^n_y\cdot D^{-1}_x=D_{x}^{-1}\cdot D^n_y$ where $n \in \mathbb{N}.$
I've not been formally introduced to fractional calculus, so I'm curious as to what is the maximum set of $\alpha$ such that $$D^{\alpha}_y\cdot D^{-1}_x=D_{x}^{-1}\cdot D^{\alpha}_y$$ is true?
For what type of $\alpha,\beta$ is $$D^{\alpha}_y\cdot D^{\beta}_x=D_{x}^{\beta}\cdot D^{\alpha}_y$$ true?
I assume that $D^{\alpha}_y\cdot D^{\beta}_x=D_{x}^{\beta}\cdot D^{\alpha}_y$ immediately implies that the operators operate on functions which are $\alpha$ 'differentiable' in $x$ and $\beta$ 'differentiable' in $y$ (' ' used as if $\alpha=-1$ the functions have to be '$-1$ differentiable' $=$ integrable in $x$).