Global C∞ Irregularity of the ∂̄–neumann Problem for Worm Domains
نویسنده
چکیده
where ρ is a defining function for Ω, = ∂̄∂̄∗ + ∂̄∗∂̄, u, f are (0, 1) forms, and denotes the interior product of forms. Under the stated hypotheses on Ω, this problem is uniquely solvable for every f ∈ L(Ω). The Neumann operator N , mapping f to the solution u, is continuous on L(Ω). The Bergman projection B is the orthogonal projection of L(Ω) onto the closed subspace of L holomorphic functions on Ω, and is related to N by B = I − ∂̄∗N∂̄. N and B are C∞ pseudolocal if Ω is strictly pseudoconvex, or more generally, is of finite type [Ca1]. Both preserve C∞(Ω) under certain weaker hypotheses [BS2], [Ca2]. For any pseudoconvex, smoothly bounded Ω and any finite exponent s, there exists a strictly positive weight w ∈ C∞(Ω) such that the Neumann operator and Bergman projection with respect to the Hilbert space L(Ω, w(x)dx) map the Sobolev space H(Ω) boundedly to H(Ω), for all 0 ≤ t ≤ s [K1]. It has remained an open question whether N and B, defined with respect to the standard metric, preserve C∞(Ω) without further hypotheses on Ω. An affirmative answer would have significant consequences [BL].
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