$L^2$ estimates for the eigenfunctions corresponding to real eigenvalues of the Tricomi operator

We introduce a family ${\cal F}$ of normal Tricomi domains $\Om_{\a,\b}$, $\a>0>\b$, and we show that its elements are $D$-star-shaped with respect to the vector field $D=-3x\partial_x-2y\partial_y$ if and only if $\a\ge1/2$. Provided that the underlying domain $\Om$ belongs to ${\cal F}$ for some $\a\ge1/2$, we then establish $L^2$ estimates for the eigenfunctions corresponding to real eigenvalues of the Tricomi operator. In particular, our result highlights a dependency of these estimates on the values of $\a$ and $\b$ and the parabolic diameter of $\Om$.