Pseudolocal Estimates for ∂̄ on General Pseudoconvex Domains

Extending the well-known results for smoothly bounded case, we show that subelliptic and pseudolocal estimates hold in the neighbourhood of a smooth strictly pseudoconvex (or finite type) boundary point of any pseudoconvex domain (i.e. possibly unbounded or with nonsmooth boundary). As an application, we also prove the corresponding generalization of Kerzman’s and Fefferman-Boutet de Monvel-Sjöstrand’s results on the boundary behaviour of the Bergman kernel. Let Ω be a pseudoconvex domain in C and ∂̄ the weak maximal operator of the antiholomorphic exterior differentiation from the space L(p,q)(Ω) of squareintegrable (p, q)-forms (0 ≤ p ≤ n, 0 ≤ q ≤ n) into L(p,q+1)(Ω). Denote by ∂̄∗ the Hilbert space adjoint of ∂̄, and by θ its formal adjoint, both acting from L(p,q) into L(p,q−1) for each p, q. (Thus if T stands for the operator θ defined on C ∞ 0 (Ω), we have ∂̄ = T ∗ and ∂̄∗ = T ∗∗ = the closure of T ; and ∂̄∗ is the restriction of θ to dom ∂̄∗.) Let ¤ = ∂̄∗∂̄ + ∂̄∂̄∗ be the operator ∂̄θ+θ∂̄ on dom¤ := {u ∈ dom ∂̄∩dom ∂̄∗ : ∂̄u ∈ dom ∂̄∗ and ∂̄∗u ∈ dom ∂̄}. Then ¤ is an (unbounded) self-adjoint operator, with ker¤ = (Ran¤)⊥ = {u ∈ dom ∂̄ ∩ dom ∂̄∗ : ∂̄u = θu = 0}. One defines the Neumann operator N as the inverse of the restriction of ¤ to (ker¤)⊥, i.e. Nα = 0 if α ∈ ker¤, and Nα = φ if α = ¤φ ∈ Ran¤ where φ⊥ ker¤. The operator N is self-adjoint, its domain is ker¤⊕Ran¤, and N is bounded (and defined everywhere) if and only if Ran¤ is closed. Let z be a smooth boundary point (i.e. a point of ∂Ω in some neighbourhood of which ∂Ω is a C∞-submanifold of C). Then there exists a special boundary chart around z, i.e. a system of local coordinates (x1, . . . , x2n) ∈ R ' C on a neighbourhood U of z in which z corresponds to the origin, Ω to {x2n > 0}, and ∂Ω to {x2n = 0}. The advantage of boundary charts is that if a function f supported in U belongs to dom ∂̄∗, then so do any of its translates in the tangential variables x1, . . . , x2n−1 (and similarly for dom ∂̄). For s ∈ R, the tangential Sobolev norm ||| · |||s of a function f supported in U is defined as the usual Sobolev s-norm in which derivatives are taken only with respect to these tangential variables (see [FK], 2000 Mathematics Subject Classification. Primary 32W05; Secondary 32A25.