I'm trying to wrap my head around the Riesz potential in the sense of a higher dimensional generalization of Riemann-Liouville fractional integrals but some things are coming across as somewhat unintuitive to me.
Let's just start with the definitions. Consider the R-L integral: $$R^\alpha f(x) = \frac{1}{\Gamma(\alpha)}\int_a^xf(t)(x-t)^{\alpha-1}\,dt$$ and the Riesz potential in higher dimensions: $$(Q_{\alpha}f) (x)= \frac{1}{c_\alpha} \int_{{\mathbb{R}}^n} \frac{f(y)}{| x - y |^{n-\alpha}} \, \mathrm{d}y$$
For R-L integrals, we have the very useful property $$R^{\alpha+\beta}f = R^\alpha ( R^\beta f)$$ as far as I know without restrictions on the parameters as long as the integrals all converge, but it seems according to Wikipedia and some standard literature sources (e.g. Foundations of modern potential theory) the semi-group property for Riesz potentials, i.e.
$$Q^{\alpha+\beta}f = Q^\alpha ( Q^\beta f)$$
only holds given $0 < \operatorname{Re\,} \alpha, \operatorname{Re\,} \beta < n,\quad 0 < \operatorname{Re\,} (\alpha+\beta) < n.$ For now I'm primarily interested in real valued $\alpha$ and $\beta$, so let's ignore that complication for now. I'm failing to see based on the Fourier representation $\widehat{Q_\alpha f}(\xi) = |2\pi\xi|^{-\alpha} \hat{f}(\xi)$ of Riesz potentials why this semi-group property begins to fail when the parameters exceed the dimension of the space?
My questions thus are:
If this is true and the semi-group property fails for all sensible functions for higher parameters, what is the cause of this intuitively especially with respect to the Fourier representation of Riesz potentials?
Are there special cases of functions (e.g. rapidly decaying, compactly supported, radial, whatever special class) where we can go to more general parameters $\alpha$, in particular $\alpha > n$ for which the semi-group property still holds?
Any further literature references that can address these questions for me would also be appreciated.