Duality of holomorphic function spaces and smoothing properties of the Bergman projection

Let Ω ⋐ C n \Omega \Subset \mathbb {C}^{n} be a domain with smooth boundary, whose Bergman projection B B maps the Sobolev space H k 1 ( Ω ) H^{k_{1}}(\Omega ) (continuously) into H k 2 ( Ω ) H^{k_{2}}(\Omega ) . We establish two smoothing results: (i) the full Sobolev norm ‖ B f ‖ k 2 \|Bf\|_{k_{2}} is controlled by L 2 L^2 derivatives of f f taken along a single, distinguished direction (of order ≤ k 1 \leq k_{1} ), and (ii) the projection of a conjugate holomorphic function in L 2 ( Ω ) L^{2}(\Omega ) is automatically in H k 2 ( Ω ) H^{k_{2}}(\Omega ) . There are obvious corollaries for when B B is globally regular.