Geometrical Toeplitz operators and Carleson embeddings over smoothly bounded convex domains of finite type in ℂn
摘要
For a smoothly bounded convex domain Ω ⊂ ℂn of finite type, let Ap(Ω) be the Bergman space on Ω with its reproducing kernel K(·, ·). We geometrically characterize such a nonnegative Borel measure μ that the Toeplitz operator Tμf(z) = ∫Ω f(w)K(z, w)dμ(w) is: (i) bounded from Ap(Ω) to Aq(Ω); (ii) compact from Ap(Ω) to Aq(Ω); (iii) in the Schatten class on A2(Ω). Meanwhile, we can geometrically characterize the boundedness-compactness-Schatten class of the Carleson embedding Iμ: Ap(Ω) → Lq(Ω, dμ).