- The Solid Cauchy transform
Steven R. Bell · 2015
We now wish to express the Bergman projection in terms of the Szego˝ projection. Suppose Ω is a bounded domain with C∞ smooth boundary and suppose v ∈ C∞(Ω). We want to compute Bv, the Bergman projection of v. Define (Λv)(z) = 1 2πi ∫∫ w∈Ω v(w) w¯ − z¯ dw ∧ dw¯ for z ∈ Ω. The operator Λ maps C∞(Ω) into itself and (∂Λv/∂z) = v on Ω (see the remark after Theorem 2.2). Let E denote the Poisson extension operator mapping a function u ∈ C∞(Ω) to the harmonic function on Ω that has the same boundary values as u. In this book, we have expressed E in terms of the Szego˝ projection (see Theorem 10.1 and Chapter 14).