Toeplitz Operators and Carleson Mea- sures on Generalized Bargmann-Fock Spaces

1.1. Definitions Throughout this paper, λ denotes the Lebesgue measure on C and ωo = dd|z| the Euclidean Kähler form in C, where d = √ −1 4 (∂̄ − ∂). Let φ ∈ C (C) be a function, μ a measure in C, and p ∈ [1,∞). One can define the spaces Lp(e−pφdμ) and F (μ, φ) := Lp(e−pφdμ) ∩ O(C). If the measure μ is Lebesgue measure, we simply write F (λ, φ) =: F (φ). Similarly one can define L∞(e−φ, μ) = {f ; μ-Ess. Sup.|f |e−φ < +∞} and F∞(φ) := L∞(e−φ, μ) ∩ O(C). When the measure μ has reasonable properties F (μ, φ) ⊂ Lp(e−pφdμ) (resp. F∞(φ) ⊂ L∞(e−φ)) is a closed subspace. In particular, we have for such μ an orthogonal projection L2(e−2φdμ) → F (μ, φ), which is called the Bergman projection. This projection is an integral operator given by an integral kernel called the Bergman kernel, here denoted K(z, w̄). Later we recall basic and well-known properties of the Bergman kernel.