Local Sobolev—Bergman Kernels of Strictly Pseudoconvex Domains

Abstract

This article grew out of an attempt to understand analytic aspects of Fefferman’s invariant theory [F3] of the Bergman kernel on the diagonal of Ω × Ω for strictly pseudoconvex domains Ω in ℂ n with smooth (C ∞ or real analytic) boundary. The framework of his invariant theory applies equally to the Szego kernel if the surface element on ∂Ω is appropriately chosen, while the Szego kernel is regarded as the reproducing kernel of a Hilbert space of holomorphic functions in Ω which belong to the L 2 Sobolev space of order 1/2. This fact is our starting point. For each s ∈ ℝ, we first globally define the Sobolev-Bergman kernel K s of order s/2 to be the reproducing kernel of the Hilbert space H S /2(Ω) of holomorphic functions which belong to the L 2 Sobolev space of order s/2, where the inner product is specified arbitrarily.