I'm reading Lieb and Frank's paper 'Sharp constants in several inequalities on the Heisenberg group' these days. And I am really confused about the Proposition 4.1 there.
They define the 'Laplacian' $\mathcal L_s$ on the Heisenberg group by
$$ \mathcal{L}_s:=|2 \mathcal{T}|^s \frac{\Gamma\left(|2 \mathcal{T}|^{-1} \mathcal{L}+\frac{1+s}{2}\right)}{\Gamma\left(|2 \mathcal{T}|^{-1} \mathcal{L}+\frac{1-s}{2}\right)} $$where $\mathcal L=-\cfrac 14 \sum_{i=1}^n (X_i^2+Y_i^2)$ and $\mathcal T=\partial_t$ with $X_i, Y_i$ are the left ivariant vector fileds on Heisenberg group $\mathbb H^n$.
More precisely, $$ \left(\mathcal{L}_s^{-1} \delta_0\right)(u)=a_s|u|^{-Q+2 s}, \quad a_s=\frac{2^{n-s-1} \Gamma^2\left(\frac{Q-2 s}{4}\right)}{\pi^{n+1} \Gamma(s)}, $$ where $\delta_0$ denotes a Dirac delta at the point $0$ . And then they said:
For given $\lambda$, we abbreviate $s:=(Q-\lambda) / 2$ and define $$ k:=a_s^{-1 / 2} \mathcal{L}_s^{-1 / 2} \delta_0 $$ Since $\mathcal{L}_s^{-1 / 2} \mathcal{L}_s^{-1 / 2}=\mathcal{L}_s^{-1}$, this function satisfies $$ \int_{\mathbb{H}^n} k\left(w^{-1} u\right) k\left(v^{-1} w\right) d w=|u^{-1} v|^{-\lambda}. $$
It seems that they just admitted that $k$ is a funtion. But I can't see this.
In my concerns, the equality $$ k:=a_s^{-1/2}\mathcal L_s^{-1/2}\delta_0 $$ is in the sense of distribution.
Of course, when we deal with $-\Delta$ on $\mathbb R^n$, we know that $$ (-\Delta)^{-1/2}\delta_0= |x|^{1-n} $$ is indeed a function. But how can I see the defined $k$ is realy a function.
Thanks in advance! I think it could be thought in the sense of convolution, since we know $$ ((-\Delta)^{-1}\delta_0) (f)=f*|x|^{-n+2} $$ for functions in $\mathbb R^n$.