Sobolev regularity of the canonical solutions to ∂ ¯ on product domains

Let Ω be a product domain in ℂ n ,n≥2, where each slice has smooth boundary. We observe that the canonical solution operator for the ∂ ¯ equation on Ω is bounded in W k,p (Ω), k∈ℤ + ,1<p<∞. This Sobolev regularity is sharp in view of Kerzman-type examples.