Remark on Integral Means of Derivatives of Blaschke Products

If B is the Blachke product with zeros {zn}, then , whereMoreover, it is a well-known fact that, for 0 < p < ∞,is bounded if and only if Mp(r, ΨB) is bounded. We find a Blaschke product B0 such that Mp(r, ) and Mp(r, ) are not comparable for any < p < ∞. In addition, it is shown that, if 0 < p < ∞, B is a Carleson–Newman Blaschke product and a weight ω satisfies a certain regularity condition, thenwhere d A(z) is the Lebesgue area measure on the unit disc.

Toeplitz Operators on Bergman Spaces

Let G be a bounded, open, connected, non-empty subset of the complex plane C. We put the usual two dimensional (Lebesgue) area measure on G and consider the Hilbert space L2(G) that consists of the complex-valued, measurable functions defined on G that are square integrable. The inner product on L2(G) is given by the norm ‖h‖2 of a function h in L2(G) is given by ‖h‖2 = (∫G|h|2)1/2.The Bergman space of G, denoted La2(G), is the set of functions in L2(G) that are analytic on G. The Bergman space La2(G) is actually a closed subspace of L2(G) (see [12 , Section 1.4]) and thus it is a Hilbert space.Let G denote the closure of G and let C(G) denote the set of continuous, complex-valued functions defined on G.