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Carleson measures and a class of generalized integration operators on the Bergman space. (English) Zbl 1225.47032

Summary: We consider a linear operator

I h,ϕ (n) f(z)= 0 z f (n) (ϕ(ξ))h(ζ)dζ

induced by holomorphic maps h and ϕ of the open unit disk 𝔻, where ϕ(𝔻)𝔻 and n is a non-negative integer. A complete characterization of when I h,ϕ (n) is bounded on the Bergman space 𝒜 2 is established by using Luecking’s result for Carleson measures. We also compute upper and lower bounds for the essential norm of this operator on the Bergman space.

47B35Toeplitz operators, Hankel operators, Wiener-Hopf operators
46E20Hilbert spaces of continuous, differentiable or analytic functions
30H20Bergman spaces, Fock spaces