Compactness of Toeplitz operators with continuous symbols on pseudoconvex domains in ℂⁿ

Let Ω Ω be a bounded pseudoconvex domain in C n C^n with Lipschitz boundary and ϕ φ be a continuous function on Ω ¯ Ω ̄ . We show that the Toeplitz operator T ϕ Tφ with symbol ϕ φ is compact on the weighted Bergman space if and only if ϕ φ vanishes on the boundary of Ω Ω . We also show that compactness of the Toeplitz operator T ϕ p , q Tp,qφ on ∂ ¯ ∂ ̄ -closed ( p , q ) (p,q) -forms for 0 ≤ p ≤ n 0≤ p≤ n and q ≥ 1 q≥ 1 is equivalent to ϕ = 0 φ =0 on Ω Ω .