Suppose $\Omega$ is a smoothly bounded domain in $\mathbb{R}^{2n}$. Then what kind of properties does the function $h_{\alpha \bar{\beta}} (r)= \int_{S^{2n-1}}\chi_{\Omega}(\omega r) \omega^{\alpha} \bar{\omega}^{\beta} d\sigma(\omega)$ exhibit ? i.e. when can we say $h_{\alpha \bar{\beta}} (r)$ is real valued (as it is generally complex valued), when it is non-negative definite etc. ? The answers obviously depend on the geometry of $\Omega$ but I am curious about what kind of geometry implies special properties of this function.
Can someone give a reference if anyone has ever studied this kind of function ?
($\alpha, \beta$ are multi-indices here, $r$ is the radial variable and $d\sigma(\omega)$ is the Lebesgue area measure on the unit sphere $S^{2n-1}$.)