Geometrical Toeplitz operators and Carleson embeddings over smoothly bounded convex domains of finite type in $C^n$

For a smoothly bounded convex domain $\Omega\subset\mathbb{C}^n$ of finite type, let $A^p(\Omega)$ be the Bergman space on $\Omega$ with its reproducing kernel $K(\cdot,\cdot)$. We geometrically characterize such a nonnegative Borel measure $\mu$ that the Toeplitz operator $T_\mu~f(z)=\int_\Omega~f(w)K(z,w)~{d}\mu(w)$ is: (i) bounded from $A^p(\Omega)$ to $A^q(\Omega)$, (ii) compact from$A^p(\Omega)$ to $A^q(\Omega)$, and (iii) in the Schatten class on $A^2(\Omega)$. Meanwhile, we can geometrically characterize the boundedness-compactness-Schatten class of the Carleson embedding $I_\mu:~A^p(\Omega)\to~L^q(\Omega,{d}\mu)$.