Consider the group $N=\left\{n_u=\begin{pmatrix}1&u\\0&1\end{pmatrix}:u\in\mathbb R\right\}$. For every $0\neq c\in\mathbb{R}$ define an integral operator $A:C^\infty(\text{SL}(2,\mathbb R))\to C^\infty(\text{SL}(2,\mathbb R))$ by $$A_cf(g)=\int_Nf(gn_u)K_c(n_u)dn,$$ with $K_c(n_u)=(1+u^2)^{-ci}.$ Is this operator injective?
I tried to convert it into integration on $\mathbb R$ but I couldn't proceed any further. I also tried to think on $A$ as $Af=\int_{g^{-1}N} f(h)K_c(g^{-1}n_u)dn$ (like some sort of convolution but that doesn't help either). It seems like a classical question in Fourier analysis on groups but I can't figure out the solution.