Radon problems for hyperboloids

We offer a variant of Radon transforms for a pair $\mathcal{X}$ and $\mathcal{Y}$ of hyperboloids in ${\Bbb R}^3$ defined by $[x,x]=1$ and $[y,y]=-1, y_1\geqslant 1$, respectively, here $[x,y]=-x_1y_1+x_2y_2+x_3y_3$. For a kernel of these transforms we take $\delta([x,y])$, $\delta(t)$ being the Dirac delta function. We obtain two Radon transforms $\mathcal{D}(\mathcal{X}) \to C^{\infty}(\mathcal{Y})$ and $\mathcal{D}(\mathcal{Y})\to C^{\infty}(\mathcal{X})$. We describe kernels and images of these transforms. For that we decompose a sesqui-linear form with the kernel $\delta([x,y])$ into inner products of Fourier components.