Recall that the extension of function from $u:\mathbb{R}^n\to \mathbb{R}$ can be defined using the Poisson Kernel as follows:
$$u^{\mathrm{e}}(\mathbf{x}):=\gamma\_{n} \int\_{\mathbb{R}^{n}} \frac{x\_{n+1} u(y)}{\left(|x-y|^{2}+x\_{n+1}^{2}\right)^{\frac{n+1}{2}}} \mathrm{~d} y \quad \text { for } \mathbf{x}=\left(x, x\_{n+1}\right) \in \mathbb{R}\_{+}^{n+1}$$
where $\gamma\_n>0$ is some dimension dependent constant. This is also called the Harmonic Extension since $\Delta u^e = u$ on the half-space and is equal to the function $u$ on the boundary.


I came across a paper where the author mentioned that the Fourier transform of the above kernel is $\exp(-2\pi |\xi| x\_{n+1})$. So I am wondering what happens in the fractional case, that is when


\begin{align\*}
 u^e(x,x\_{n+1}) = c\_{n,s} \int \frac{x\_{n+1}^{2s}u(y)}{(|x-y|^2+x\_{n+1}^2)^{\frac{n+2s}{2}}}dy
\end{align\*}


for $s\in (0,1)$. Here $\operatorname{div}(x\_{n+1}^{1-2s}\nabla u^e)=0$ on the half space. Is there an explicit expression (or $L^2$ and $L^\infty$ estimates) known for the Fourier Transform of the kernel $$P=\frac{x\_{n+1}^{2s}}{(|x|^2+|x\_{n+1}|^2)^{n/2+s}}?$$



Fourier transform of the fractional Poisson kernel
Yes indeed, for fractional $s$ it is related to Modified Bessel Functions of 2nd Kind (Macdonald Function). For $r=|\mathbf{x}|,\ a=|x\_{n+1}|$ $$P(r,a)=\frac{a^{2s}}{(r^2+a^2)^{n/2+s}}$$ The Fourier Transform of an n-dimensional radial function $f(r)$ is a radial function in the trasformed space $\mathcal{F}(|\xi|)$. It is given by the following Hankel Transform. $$\mathcal{F}(|\xi|)=\frac{1}{|\xi|^{\frac{n}{2}-1}}\int\_0^\infty r^{\frac{n}{2}-1}\left[r\,J\_{\frac{n}{2}-1}(|\xi|\cdot r)\right]\cdot f(r)\,dr$$ See, for instance, Sec 11 pg. 63-65 in
*Sneddon, Ian N.*, Fourier transforms, New York: McGraw-Hill Book Company, Inc. (1950). [ZBL0038.26801](https://zbmath.org/?q=an:0038.26801). for a proof.


Also, *Sneddon, Ian N.*, A note on some relations between Fourier and Hankel transforms, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 9, 799-806 (1961). [ZBL0100.31501](https://zbmath.org/?q=an:0100.31501). Therefore $$\mathcal{F}\_P(|\xi|,a)=\frac{a^{2s}}{|\xi|^{\frac{n}{2}-1}}\int\_0^\infty J\_{\frac{n}{2}-1}(|\xi|\cdot r)\cdot \frac{r^{\frac{n}{2}}\,dr}{(r^2+a^2)^{\frac{n}{2}+s}}$$ But integral is formula 6.565.4 in 7-th Edition of


*Gradshteyn, I. S.; Ryzhik, I. M.*, Table of integrals, series, and products. Ed. by Alan Jeffrey. CD-ROM version 1. 0 for PC, MAC, and UNIX computers., San Diego, CA: Academic Press. (1996). [ZBL0918.65001](https://zbmath.org/?q=an:0918.65001).


It reads (adapted to our notation), for $s>0$ $$\int\_0^\infty J\_{\frac{n}{2}-1}(|\xi|\cdot r)\cdot \frac{r^{\frac{n}{2}}\,dr}{(r^2+a^2)^{\frac{n}{2}+s}}=\left(\frac{|\xi|}{2}\right)^{\frac{n}{2}+s-1}\cdot\frac{K\_s(a\,|\xi|)}{a^s\,\Gamma(\frac{n}{2}+s)}$$ Being $K\_s(\cdot)$ the Modified Bessel Function of 2nd Kind. Therefore, $$\mathcal{F}\_P(|\xi|,a)=\frac{(a\,|\xi|)^sK\_s(a\,|\xi|)}{2^{\frac{n}{2}+s-1}\Gamma(\frac{n}{2}+s)}$$
Since $K\_\frac{1}{2}(z)=\sqrt{\frac{\pi}{2z}}\cdot e^{-z}$, for $s=\frac{1}{2}$ we get $$\mathcal{F}\_P(|\xi|,a)=\frac{\sqrt{\pi}}{2^\frac{n}{2}\Gamma\left(\frac{n+1}{2}\right)}\cdot e^{-a\,|\xi|}$$ Note: Dimension constant and $2\pi$ in exponent are matter of Fourier Transform's normalization and convention used respectively. By 'Rosetta Table' Formulae 504 at the end of [this Wiki page](https://en.wikipedia.org/wiki/Fourier_transform#Formulas_for_general_n-dimensional_functions) this can be fitted accordingly.


Using question convention -*FT* unitary, ordinary frequency, $\mathcal{F}^\*$- and notation we get,


$$P(|x|,x\_{n+1})=\frac{2^{s-\frac{1}{2}}\Gamma\left(\frac{n}{2}+s\right)}{\pi^{\frac{n+1}{2}}}\cdot\frac{x\_{n+1}^{2s}}{(|x|^2+|x\_{n+1}|^2)^{n/2+s}}$$ producing this amazingly simple expression $$\mathcal{F}\_P^\*(|\xi|,x\_{n+1})=\sqrt{\frac{2}{\pi}}\cdot \left(2\pi\,x\_{n+1}\,|\xi|\right)^s\cdot K\_s\left(2\pi\, x\_{n+1}\,|\xi|\right)$$


This gives $\exp(-2\pi |\xi| x\_{n+1})$ for $s=1/2$.


Being so simple, this result should be added as new Formulae 505 in [Wiki's Table](https://en.wikipedia.org/wiki/Fourier_transform#Formulas_for_general_n-dimensional_functions) Fourier Transforms of n-dimensional functions.


I leave this for further reading, a nice article on Fourier Transforms of Fractional Poison Kernels and Fractional Laplacian (See Section 2.8) at


*Kwaśnicki, Mateusz*, [**Ten equivalent definitions of the fractional Laplace operator**](http://dx.doi.org/10.1515/fca-2017-0002), Fract. Calc. Appl. Anal. 20, No. 1, 7-51 (2017). [ZBL1375.47038](https://zbmath.org/?q=an:1375.47038).


