Gleason's problem in weighted Bergman space on egg domains

In the paper, we discuss on the egg domains: $$ \Omega_a=\left\{\xi=(z,w)\in\bold C^{n+m}: \ z\in\bold C^n, \ w\in\bold C^m, |z|^2+|w|^{2/a}<1\right\}, \qquad 0<a\le 2. $$ We show that Gleason's problem can be solved in the weight Bergman space on theegg domains. The proof will need the help of the recent work of the second named author on the weighted Bergman projections on this kind of domain. As an application, we obtain a multiplier theorem on the egg domains.