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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 $$I^{(n)}_{h,\varphi}f(z)=\int^z_0 f^{(n)}(\varphi(\xi))h(\zeta)\,d\zeta$$ induced by holomorphic maps $h$ and $\varphi$ of the open unit disk $\Bbb D$, where $\varphi({\Bbb D})\subset\Bbb D$ and $n$ is a non-negative integer. A complete characterization of when $ I^{(n)}_{h,\varphi}$ is bounded on the Bergman space ${\cal A}^2$ 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.

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