Simple expression for $(-\Delta)^{\frac{3}{2}}$ in the spatial domain?

Question

I can write the harmonic and biharmonic operators as:

$$-\Delta = -\sum_i \partial_{ii}, \quad (-\Delta)^2 = \sum_i\sum_j \partial_{ii}\partial_{jj}$$.

Is there such a simple expression for $(-\Delta)^{\frac{3}{2}}$? In 1D I would guess it is $-\frac{d^3}{dx^3}$. Is this correct? Would then $-\sum_i \partial_i\sum_j\partial_{jj}$ be the generalization for higher dimensions. I believe that $(-\Delta)^{\alpha}$ can be represented as $(4\pi^2|\omega|)^{\alpha}$ in the Fourier domain, but I wanted to have an expression in terms of the derivatives in the spatial domain, if such exists.

Would this then generalize to other $\alpha = \frac{n}{2}, \, n \in\mathbb{N}$?

Edit: I found this review: https://arxiv.org/abs/1801.09767 which covers quite s lot. Both of my guesses seem to have been wrong.

Answer 1

There doesn't seem to be a simple expression as Hans Engler mentioned. There are various formulations: What Is the Fractional Laplacian? beyond the spectral one, but none seems to be too useful for compactness in the spatial domain.

I also figured out that this cannot be rewritten in terms of differential operators in the spatial domain because of the modulus in the Fourier domain. So that's the main thing causing problems there. Without the modulus one gets the odd derivatives as expected.