I apologize for not being able to motivate the question below; it would go into technicalities.

Let $n=d+1\ge2$ be the space-time dimension, and $$H(y,t):=\left(\frac{t^2}{(t^2+|y|^2)^{1+d/2}}\right)^{\frac1d}.$$

Is there a close form for the Fourier transform $\hat H(\xi,\tau)$ ?

Mind that $H$ is homogeneous of degree $-1$. The singularity at the origin being integrable, $H$ might be viewed, at worst, as a tempered distribution. Thus $\hat H$ is, at least, a tempered distribution, homogeneous of degree $1-n=-d$. Of course, it is isotropic in $\xi$.

Actually, I should be very happy if $H$ was known to be the solution of $$P(D_{y,t})H=\delta_{y=0,t=0},$$ where $P$ is a pseudo-differential operator, of course homogeneous of degree $d$. One has then $$P(\xi,\tau)=\frac1{\hat H(\xi,\tau)}\,.$$

I apologize for not being able to motivate the question below; it would go into technicalities.

Let $n=d+1\ge2$ be the space-time dimension, and $$H(y,t):=\left(\frac{t^2}{(t^2+|y|^2)^{1+d/2}}\right)^{\frac1d}.$$

Is there a closed form for the Fourier transform $\hat H(\xi,\tau)$ ?

Mind that $H$ is homogeneous of degree $-1$. The singularity at the origin being integrable, $H$ might be viewed, at worst, as a tempered distribution. Thus $\hat H$ is, at least, a tempered distribution, homogeneous of degree $1-n=-d$. Of course, it is isotropic in $\xi$.

Actually, I should be very happy if $H$ was known to be the solution of $$P(D_{y,t})H=\delta_{y=0,t=0},$$ where $P$ is a pseudo-differential operator, of course homogeneous of degree $d$. One has then $$P(\xi,\tau)=\frac1{\hat H(\xi,\tau)}\,.$$